Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints under Data Uncertainty

This article investigates robust optimality and duality for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (in short, UNMPVC) via convexificators. Using the properties of convexificators, we introduce generalized standard Abadie constraint qualification (in short, GS-ACQ) for the considered problem UNMPVC. Moreover, we introduce generalized robust version of nonsmooth stationary conditions, namely, weakly stationary point (in short, RW-Stationary), T-stationary point (in short, RT-Stationary), M-stationary point (in short, RM-Stationary) and S-stationary point (in short, RS-Stationary) for UNMPVC. By employing GS-ACQ, we establish that the RS-Stationary is the necessary first-order optimality condition for a local Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we establish sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type (in short, WRD) and Mond-Weir-type (in short, MWRD) robust dual models corresponding to the primal problem UNMPVC.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

MSC: 49J52; 90C17; 90C29; 90C46

1. Introduction

Multiobjective programming problems (in short, MOP) hold significant importance in practical optimization scenarios, such as business, economics and various scientific and engineering fields (see, for instance, [1,2], and the references mentioned therein). It plays a crucial role in making optimal decisions when multiple conflicting objectives must be simultaneously optimized. Several authors have established various results for MOP in various settings (see, for instance, [3,4,5,6,7,8], and the references mentioned therein).

It is well-known that convexity plays an essential role in optimization theory and algorithms. It ensures that a stationary point is also the global minimum, and satisfaction of the first-order necessary optimality conditions guarantees global optimality. To deal with this nonconvex nature of many real-world optimization problems, the convex optimization theory has been significantly enriched by introducing numerous generalizations of convex functions. Mangasarian [9] introduced the concept of pseudoconvex and pseudoconcave functions as generalizations of convex and concave functions, respectively. De Finetti [10] was one of the first persons to recognize some of the characteristics of functions having convex level sets. He did not name this class, but he did note that it includes all convex functions and some nonconvex functions. Fenchel [11] was one of the early pioneers in formalizing, naming and developing the class of quasiconvex functions. For a more comprehensive study, see [12,13,14,15], and the references mentioned therein.

In many real-world optimization problems in the fields of science, engineering and other fields of modern research, nonsmooth phenomena occur inevitably. To address the nonsmooth characteristics of mathematical programming problems, the concepts of generalized derivatives and subdifferentials have been developed and thoroughly examined (see, for instance, [16,17,18,19,20,21]). Convexificators can be seen as a weaker version of the notion of subdifferential, as convexificators, in general, are closed-set and are not necessarily bounded or convex, unlike most known subdifferentials. For a locally Lipschitz function, generalized subdifferentials, such as Clarke [16], Michel-Penot [19], Ioffe-Mordukhovich [18,20] and Treiman [21] are convexificators (see, for instance, [22,23]). Convexificators have emerged as a powerful tool in nonsmooth analysis, enabling the derivation and extension of numerous key results (see, for instance, [24,25,26,27,28], and the references mentioned therein).

Mathematical programming problems with vanishing constraints (in short, MPVC), introduced by Achtziger and Kanzow [29], constitute a distinctive class of constrained optimization problems that frequently arise in many fields of modern research (see, for instance, [30,31,32], and the references mentioned therein). The nomenclature "vanishing constraints" aptly captures a critical feature of these problems. In many real-world applications, certain constraints become inactive for specific feasible solutions within the decision space. For a more comprehensive study of MPVC and its applications, we refer to [33,34,35,36,37,38,39,40,41,42,43] and the references mentioned therein.

Robust optimization deals with uncertain data in an optimization problem. Mathematical programming problems often incorporate parameters with inherent uncertainty due to estimation errors, rounding discrepancies, or implementation issues. Robust optimization aims to find solutions that remain optimal even when faced with uncertainties in the problem’s parameters. Generally, robust optimization reformulates the original problem into a deterministic equivalent, called the robust counterpart. This allows us to use the standard optimization algorithms to find solutions that are resilient to these uncertainties. Such problems arise frequently in various areas of modern research (see, for instance, [44,45,46,47], and the references mentioned therein). Several authors have studied optimization problems with data uncertainty comprehensively under various assumptions (see, for instance, [48,49,50,51,52,53,54,55], and the references mentioned therein). To deal with the nonsmooth nature of data uncertainty problems, many authors have studied robust nonsmooth mathematical programming problems with uncertain data comprehensively under various assumptions (see, for instance, [56,57,58], and the references mentioned therein). Recently, Gadhi and Ohda [59] established necessary optimality conditions for robust nonsmooth multiobjective optimization problems. Using convexificators, Chen et al. [60] established optimality criteria and formulated dual models for nonconvex multiobjective optimization problems with uncertain data.

It is worthwhile to mention that the optimality conditions and duality theorems have so far been studied by several researchers for nonsmooth robust optimization problems defined in Euclidean spaces (see, for instance, [44,56,57,59], and the references mentioned therein). However, to the best of our knowledge, prior work has not addressed problems involving vanishing constraints and data uncertainty. This article aims to bridge this research gap. In this article, motivated by the works [37,41,44,56,59], we examine a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty via convexificators. We introduce GS-ACQ for UNMPVC and introduce generalized robust versions of various stationary points, namely, RW-stationary, RT-stationary, RM-stationary and RS-stationary points for UNMPVC. Subsequently, under generalized convexity assumptions, we establish sufficient optimality conditions for UNMPVC. We further enrich the analysis of the primal problem UNMPVC by formulating WRD and MWRD-type dual models. This enables us to establish weak, strong, and strict converse duality relationships between the original problem and its dual model. The significance of these findings is demonstrated through the inclusion of non-trivial illustrative examples that demonstrate the practical applicability of the theoretical developments presented in this article.

The primary contributions and novel aspects of the present article are twofold. Firstly, we introduce GS-ACQ constraint qualification and robust stationary conditions for UNMPVC, which is employed to extend the optimality conditions established in [37] for nonsmooth MPVC to a more general mathematical programming problem dealing with data uncertainty. Moreover, we extend the duality theorems established in [41] for MPVC to a wider class of mathematical programming problems dealing with data uncertainty. Secondly, we generalize the robust optimality and duality results established in [59,60] to a more general robust mathematical programming problem incorporating vanishing constraints. To the best of our knowledge, optimality and duality results in terms of convexificators presented in this paper have not been given before.

The present article is structured subsequently: In Section 2, we revisit some basic definitions and mathematical preliminaries. In Section 3, we introduce GS-ACQ constraint qualification and various stationary conditions for UNMPVC. Moreover, we establish sufficient optimality conditions for UNMPVC. In Sections 4 and 5, we formulate WRD and MWRD dual models and derive weak, strong and strict converse duality theorems relating to UNMPVC. Section 6 concludes the paper by summarizing the key findings and outlining potential avenues for future research.

2. Preliminaries and Notations

Throughout the article, the symbols

R^{n}

and

N

denote the n-dimensional Euclidean space and the set of all natural numbers, respectively. Let

\bar{R} : = R \cup {\infty}

denote the extended real line. Let

p, q \in R^{n}

. Then the following notations are employed in the article:

\begin{matrix} p ≺ q ⟺ & p_{j} < q_{j} \forall j \in {1, \dots, n}, \\ p ⪯ q ⟺ & \{\begin{matrix} p_{j} \leq q_{j} \forall j \in {1, \dots, n}, \\ p_{r} < q_{r} for at least one r \in {1, \dots, n} . \end{matrix} \end{matrix}

Let

B

be a nonempty subset of

R^{n}

. We use the symbols

cl B

co B

and

cone B

to denote the closure of

B

, convex hull of

B

and the convex cone (including the origin) generated by

B

, respectively.

Now, we define the following sets that will be utilized in the subsequent sections:

B^{-} : = \{z \in R^{n} : 〈 z, x 〉 \leq 0, \forall x \in B\},

B^{s} : = \{z \in R^{n} : 〈 z, x 〉 < 0, \forall x \in B\} .

We recall the subsequent definitions from [61].

Definition 1.

Let

B

be a nonempty subset of

R^{n}

and

z \in cl B

. The contingent cone

T (B, z)

of the set

B

at z is defined as

T (B, z) : = \{v \in R^{n} : \exists α_{n} ↓ 0 & v_{n} \to v such that z + α_{n} v_{n} \in B, \forall n \in N\} .

Definition 2.

Let

Φ : R^{n} \to \bar{R}

be an extended real-valued function and

z \in dom (Φ)

, where

dom (Φ) : = {z \in R^{n} : Φ (z) \neq \infty}

. Then, the lower and upper Dini derivatives of Φ at z in the direction

v \in R^{n}

are defined, respectively, by

Φ^{-} (z; v) : = \underset{λ ↓ 0}{lim inf} \frac{Φ (z + λ v) - Φ (z)}{λ},

Φ^{+} (z; v) : = \underset{λ ↓ 0}{lim sup} \frac{Φ (z + λ v) - Φ (z)}{λ} .

Definition 3.

Let

Φ : R^{n} \to \bar{R}

be an extended real-valued function. We say that Φ has an upper semi-regular convexificator (in short, USRC),

\partial^{*} Φ (z) \subset R^{n}

z \in dom (Φ)

if the set

\partial^{*} Φ (z)

is closed and for every

v \in R^{n}

we have

Φ^{+} (z; v) \leq sup_{ζ \in \partial^{*} Φ (z)} 〈 ζ, v 〉 .

Definition 4.

Let

Φ : R^{n} \to \bar{R}

be an extended real-valued function. We say that Φ has a lower semi-regular convexificator (in short, LSRC),

\partial_{*} Φ (z) \subset R^{n}

z \in dom (Φ)

if the set

\partial_{*} Φ (z)

is closed and for every

v \in R^{n}

we have

Φ^{-} (z; v) \geq inf_{ζ \in \partial_{*} Φ (z)} 〈 ζ, v 〉 .

We recall the following definition from [62].

Definition 5.

Let

Φ : R^{n} \to \bar{R}

be an extended real-valued function. Further, assume that

z \in R^{n}

is a point such that

Φ (z)

is finite and Φ admits an upper semi-regular convexificator

\partial^{*} Φ (z)

at z. Then,

Φ is $\partial^{*}$ -convex at z if, for each $γ \in R^{n}$ ,

$Φ (γ) - Φ (z) \geq 〈 ξ, γ - z 〉, \forall ξ \in \partial^{*} Φ (z) .$
Φ is strictly $\partial^{*}$ -convex at z if, for each $γ (\neq z) \in R^{n}$ ,

$Φ (γ) - Φ (z) > 〈 ξ, γ - z 〉, \forall ξ \in \partial^{*} Φ (z) .$
Φ is $\partial^{*}$ -pseudoconvex at z if, for each $γ \in R^{n}$ ,

$Φ (γ) < Φ (z) \Rightarrow 〈 ξ, γ - z 〉 < 0, \forall ξ \in \partial^{*} Φ (z) .$
Φ is strictly $\partial^{*}$ -pseudoconvex at z if, for each $γ (\neq z) \in R^{n}$ ,

$Φ (γ) \leq Φ (z) \Rightarrow 〈 ξ, γ - z 〉 < 0, \forall ξ \in \partial^{*} Φ (z) .$
Φ is $\partial^{*}$ -quasiconvex at z if, for each $γ \in R^{n}$ ,

$Φ (γ) \leq Φ (z) \Rightarrow 〈 ξ, γ - z 〉 \leq 0, \forall ξ \in \partial^{*} Φ (z) .$

We recall the following definition of locally Lipschitz function from [63].

Definition 6.

A function

φ : R^{n} \to R

is called locally Lipschitz if every point

d \in R^{n}

possesses a neighbourhood

N_{d} \subset R^{n}

such that

| φ (d_{1}) - φ (d_{2}) | \leq M ‖ d_{1} - d_{2} ‖, \forall d_{1}, d_{2} \in N_{d},

for a constant

M > 0

depending on

N_{d}

Next, we recall the following theorems derived by Chen et al. [60], which are utilized in establishing various results in this article. We use the notation

T

to denote the set defined as

T : = {1, \dots, r}

Theorem 1.

Let

f_{t} : R^{n} \to R

t \in T

be a locally Lipschitz function. Moreover, we assume that

f_{t}

admit USRC

\partial^{*} f_{t} (z)

z \in R^{n}

for all

t \in T

. Then

F (z) : = max {f_{1} (z), \dots, f_{r} (z)}

admits an USRC which is convex and is given as

\partial^{*} F (z) : = co \{⋃_{t \in T (z)} \partial^{*} f_{t} (z)\},

where

T (z) : = {t \in T : f_{t} (z) = F (z)} .

Theorem 2.

Let

F_{j} (z) : = {max}_{w_{j} \in Ω_{j}} f_{j} (z, w_{j}), j \in J : = {1, \dots, p}

and

Ω_{j} (\bar{z}) : = {{\bar{w}}_{j} \in Ω_{j} : F_{j} (\bar{z}) = f_{j} (\bar{z}, {\bar{w}}_{j})}

. Then

\partial^{*} F_{j} (\bar{z}) = \{\partial^{*} f_{j} (\bar{z}, {\bar{w}}_{j}) : {\bar{w}}_{j} \in Ω_{j} (\bar{z})\} = ⋃_{{\bar{w}}_{j} \in Ω_{j} (\bar{z})} \partial^{*} f_{j} (\bar{z}, {\bar{w}}_{j}) .

(1)

3. Optimality Criteria

In this section, we consider a nonsmooth multiobjective programming problem with vanishing constraints under data uncertainty. We introduce GS-ACQ at a feasible point of UNMPVC. Moreover, we introduce RW-stationary, RT-stationary, RM-stationary and RS-stationary points for UNMPVC. By employing GS-ACQ, we establish that the RS-Stationary is the necessary first-order optimality condition for UNMPVC. Subsequently, we establish sufficient optimality conditions for UNMPVC.

We consider the following nonsmooth multiobjective programming problem with vanishing constraints under data uncertainty:

\begin{matrix} UNMPVC Minimize & Φ (z) : = (Φ_{1} (z), \dots, Φ_{r} (z)), \\ subject to & Ψ_{j} (z, w_{j}) \leq 0, j = 1, \dots, p, \\ Θ_{s} (z, w_{s}) = 0, s = 1, \dots, l, \\ K_{i} (z, w_{i}) \geq 0, i = 1, \dots, m, \\ L_{i} (z, w_{i}) K_{i} (z, w_{i}) \leq 0 i = 1, \dots, m, \end{matrix}

where

Φ_{t} : R^{n} \to R

t \in T : = {1, \dots, r}

Ψ_{j} : R^{n} \times R^{n_{j}} \to R

j \in J : = {1, \dots, p}

Θ_{s} : R^{n} \times R^{n_{s}} \to R

s \in S : = {1, \dots, l}

K_{i} : R^{n} \times R^{n_{i}} \to R

i \in I : = {1, \dots, m}

and

L_{i} : R^{n} \times R^{n_{i}} \to R

i \in I : = {1, \dots, m}

are real valued functions,

z \in R^{n}

is the decision variable and

w_{j} \in Ω_{j}, j \in J

w_{s} \in Ω_{s}, s \in S

and

w_{i} \in Ω_{i}, i \in I

are uncertain parameters,

Ω_{j}

Ω_{s}

Ω_{i}

are nonempty convex subsets of

R^{n_{j}}

R^{n_{s}}

and

R^{n_{i}}

, respectively.

The robust multiobjective optimization problem (in short, RNMPVC) associated with UNMPVC is defined as follows:

\begin{matrix} RNMPVC Minimize & Φ (z) : = (Φ_{1} (z), \dots, Φ_{r} (z)), \\ subject to & Ψ_{j} (z, w_{j}) \leq 0, \forall w_{j} \in Ω_{j}, j \in J, \\ Θ_{s} (z, w_{s}) = 0, \forall w_{s} \in Ω_{s}, s \in S, \\ K_{i} (z, w_{i}) \geq 0, \forall w_{i} \in Ω_{i}, i \in I, \\ L_{i} (z, w_{i}) K_{i} (z, w_{i}) \leq 0, \forall w_{i} \in Ω_{i}, i \in I . \end{matrix}

The set of all feasible solutions

F

of RNMPVC is:

\begin{matrix} F = {z \in R^{n} : & Ψ_{j} (z, w_{j}) \leq 0, \forall w_{j} \in Ω_{j}, j \in J, Θ_{s} (z, w_{s}) = 0, \\ \forall w_{s} \in Ω_{s}, s \in S, K_{i} (z, w_{i}) \geq 0, \forall w_{i} \in Ω_{i}, i \in I, \\ L_{i} (z, w_{i}) K_{i} (z, w_{i}) \leq 0, \forall w_{i} \in Ω_{i}, i \in I} . \end{matrix}

Let us define

Ψ_{j}^{*} (z) : = max_{w_{j} \in Ω_{j}} Ψ_{j} (z, w_{j}), j \in J,

Θ_{s}^{*} (z) : = max_{w_{s} \in Ω_{s}} Θ_{s} (z, w_{s}), s \in S,

K_{i}^{*} (z) : = max_{w_{i} \in Ω_{i}} K_{i} (z, w_{i}), i \in I,

L_{i}^{*} (z) : = max_{w_{i} \in Ω_{i}} L_{i} (z, w_{i}), i \in I,

L_{i}^{*} (z) K_{i}^{*} (z) : = max_{w_{i} \in Ω_{i}} L_{i} (z, w_{i}) K_{i} (z, w_{i}), i \in I,

and

\begin{matrix} Ω_{j} (z) : = {w_{j} \in Ω_{j} : & Ψ_{j}^{*} (z) = Ψ_{j} (z, w_{j}), j \in J}, \\ Ω_{s} (z) : = {w_{s} \in Ω_{s} : & Θ_{s}^{*} (z) = Θ_{s} (z, w_{s}), s \in S}, \\ Ω_{i} (z) : = {w_{i} \in Ω_{i} : & K_{i}^{*} (z) = K_{i} (z, w_{i}), L_{i}^{*} (z) = L_{i} (z, w_{i}) \\ L_{i}^{*} (z) K_{i}^{*} (z) = L_{i} (z, w_{i}) K_{i} (z, w_{i}), i \in I} . \end{matrix}

Therefore, the problem RNMPVC can be transformed into the following nonsmooth multiobjective programming problems with vanishing constraints (in short, NMPVC*):

\begin{matrix} NMPVC * Minimize & Φ (z) : = (Φ_{1} (z), \dots, Φ_{r} (z)), \\ subject to & Ψ_{j}^{*} (z) \leq 0, j \in J, \\ Θ_{s}^{*} (z) = 0, s \in S, \\ K_{i}^{*} (z) \geq 0, i \in I, \\ L_{i}^{*} (z) K_{i}^{*} (z) \leq 0, i \in I . \end{matrix}

The set of all feasible solutions

F

of NMPVC* is:

\begin{matrix} F^{*} = {z \in R^{n} : & Ψ_{j}^{*} (z) \leq 0, j \in J, Θ_{s}^{*} (z) = 0, s \in S, \\ K_{i}^{*} (z) \geq 0, i \in I, L_{i}^{*} (z) K_{i}^{*} (z) \leq 0, i \in I} . \end{matrix}

The subsequent definitions of robust Pareto solutions, robust local Pareto solutions, robust weak Pareto solutions and robust local weak Pareto solutions for UNMPVC will be utilized in the sequel.

Definition 7.

Let

\bar{z} \in F

\bar{z}

is said to be a robust Pareto solution of UNMPVC if there does not exist

z \in F

such that

Φ (z) ⪯ Φ (\bar{z}) .

Definition 8.

Let

\bar{z} \in F

\bar{z}

is said to be a robust local Pareto solution of UNMPVC if for any neighbourhood

N

\bar{z}

there does not exist

z \in N \cap F

such that

Φ (z) ⪯ Φ (\bar{z}) .

Definition 9.

Let

\bar{z} \in F

\bar{z}

is said to be a robust weak Pareto solution of UNMPVC if there does not exist

z \in F

such that

Φ (z) ≺ Φ (\bar{z}) .

Definition 10.

Let

\bar{z} \in F

\bar{z}

is said to be a robust local weak Pareto solution of UNMPVC if for any neighbourhood

N

\bar{z}

there does not exist

z \in N \cap F

such that

Φ (z) ≺ Φ (\bar{z}) .

For

\bar{z} \in F^{*}

, we define the following index sets which will be utilized in the upcoming sections of the article.

\begin{matrix} I_{Ψ} = {j : Ψ_{j}^{*} (\bar{z}) = 0}, & I_{+ 0} = {i : K_{i}^{*} (\bar{z}) > 0, L_{i}^{*} (\bar{z}) = 0}, \\ I_{+ -} = {i : K_{i}^{*} (\bar{z}) > 0, L_{i}^{*} (\bar{z}) < 0}, & I_{0 +} = {i : K_{i}^{*} (\bar{z}) = 0, L_{i}^{*} (\bar{z}) > 0}, \\ I_{0 -} = {i : K_{i}^{*} (\bar{z}) = 0, L_{i}^{*} (\bar{z}) < 0}, & I_{00} = {i : K_{i}^{*} (\bar{z}) = 0, L_{i}^{*} (\bar{z}) = 0} . \end{matrix}

Assuming that all functions admit USRC, we introduce the following notation which will be utilized in the subsequent sections of the article:

\begin{matrix} F : = ⋃_{t \in T} co \partial^{*} Φ_{t} (\bar{z}), F^{t} : = ⋃_{q \in T ∖ {t}} co \partial^{*} Φ_{q} (\bar{z}), \\ G : = ⋃_{j \in I_{Ψ}} co \partial^{*} Ψ_{j}^{*} (\bar{z}), H : = ⋃_{s \in S} co \partial^{*} Θ_{s}^{*} (\bar{z}) \cup co \partial^{*} (- Θ_{s}^{*}) (\bar{z}), \\ L_{+ 0} : = ⋃_{i \in I_{+ 0}} co \partial^{*} L_{i}^{*} (\bar{z}), \\ K_{0 +} : = ⋃_{i \in I_{0 +}} co \partial^{*} K_{i}^{*} (\bar{z}) \cup co \partial^{*} (- K_{i}^{*}) (\bar{z}), \end{matrix}

\begin{matrix} K_{0 -} : = ⋃_{i \in I_{0 -}} co \partial^{*} (- K_{i}^{*}) (\bar{z}), \\ K_{00} : = ⋃_{i \in I_{00}} co \partial^{*} (- K_{i}^{*}) (\bar{z}), \\ Γ (\bar{z}) : = {(F^{t_{0}})}^{-} \cap G^{-} \cap H^{-} \cap L_{+ 0}^{-} \cap K_{0 +}^{-} \cap K_{0 -}^{-} \cap K_{00}^{-}, \end{matrix}

\begin{matrix} S : = {z \in R^{n} : & Ψ_{j}^{*} (z) \leq 0, j \in J, Θ_{s}^{*} (z) = 0, s \in S, \\ K_{i}^{*} (z) \geq 0, i \in I, L_{i}^{*} (z) K_{i}^{*} (z) \leq 0, i \in I}, \end{matrix}

\begin{matrix} S^{t} : = {z \in R_{+}^{n} : & Φ_{q} (z) \leq Φ_{q} (\bar{z}), \forall q \in T ∖ {t}, Ψ_{j}^{*} (z) \leq 0, j \in J, \\ Θ_{s}^{*} (z) = 0, s \in S, K_{i}^{*} (z) \geq 0, i \in I, L_{i}^{*} (z) K_{i}^{*} (z) \leq 0, i \in I} . \end{matrix}

In the following definition, we introduce generalized standard Abadie constraint qualification (in short, GS-ACQ) for UNMPVC, which will be useful in establishing first-order necessary optimality conditions for the local weak Pareto solution of UNMPVC.

Definition 11.

Let

\bar{z} \in S

. We say that the generalized standard Abadie constraint qualification (GS-ACQ) holds at

\bar{z}

if at least one of the dual sets used in the definition of

Γ (\bar{z})

is non-zero and

Γ (\bar{z}) \subset T (S^{t_{0}}, \bar{z}) .

In the subsequent definitions, we introduce various stationary concepts for UNMPVC in terms of convexificators.

Definition 12.

Let

\bar{z} \in F

\bar{z}

is a generalized robust weakly stationary (RW-Stationary) point if there exist

ν = (ν^{Φ}, ν^{Ψ}, ν^{Θ}, ν^{K}) \in R^{r + p + l + m}, τ = (τ^{Θ}, τ^{K}, τ^{L}) \in R^{l + 2 m}

and

{\bar{w}}_{j} \in J, {\bar{w}}_{s} \in S, {\bar{w}}_{i} \in I

such that

\begin{matrix} 0 \in \sum_{t \in T} ν_{t}^{Φ} co \partial^{*} Φ_{t} (\bar{z}) & + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} co \partial^{*} Ψ_{j} (\bar{z}, {\bar{w}}_{j}) \\ + \sum_{s \in S} [ν_{s}^{Θ} co \partial^{*} Θ_{s} (\bar{z}, {\bar{w}}_{s}) + τ_{s}^{Θ} co \partial^{*} (- Θ_{s}) (\bar{z}, {\bar{w}}_{s})] \\ + \sum_{i \in I} ν_{i}^{K} co \partial^{*} (- K_{i}) (\bar{z}, {\bar{w}}_{i}) \\ + \sum_{i \in I} [τ_{i}^{K} co \partial^{*} K_{i} (\bar{z}, {\bar{w}}_{i}) + τ_{i}^{L} co \partial^{*} L_{i} (\bar{z}, {\bar{w}}_{i})], \end{matrix}

(2)

ν_{t}^{Φ} > 0, t \in T, ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}, ν_{s}^{Θ}, τ_{s}^{Θ} \geq 0, s \in S, ν_{i}^{K}, τ_{i}^{K}, τ_{i}^{L}, \geq 0, i \in I,

(3)

ν_{I_{+ 0} \cup I_{+ -}}^{K} = τ_{I_{+ 0} \cup I_{+ -}}^{K} = τ_{I_{0 +} \cup I_{+ -} \cup I_{0 -}}^{L} = 0, ν_{i}^{K} - τ_{i}^{K} \geq 0, i \in I_{0 -} .

(4)

Definition 13.

Let

\bar{z} \in F

\bar{z}

is a generalized robust T stationary (RT-Stationary) point if there exist

ν = (ν^{Φ}, ν^{Ψ}, ν^{Θ}, ν^{K}) \in R^{r + p + l + m}, τ = (τ^{Θ}, τ^{K}, τ^{L}) \in R^{l + 2 m}

and

{\bar{w}}_{j} \in J, {\bar{w}}_{s} \in S, {\bar{w}}_{i} \in I

such that they satisfy conditions (2)-(4) along with the subsequent condition:

\forall i \in I_{00}, τ_{i}^{L} (ν_{i}^{K} - τ_{i}^{K}) \leq 0 .

Definition 14.

Let

\bar{z} \in F

\bar{z}

is a generalized robust M stationary (RM-Stationary) point if there exist

ν = (ν^{Φ}, ν^{Ψ}, ν^{Θ}, ν^{K}) \in R^{r + p + l + m}, τ = (τ^{Θ}, τ^{K}, τ^{L}) \in R^{l + 2 m}

and

{\bar{w}}_{j} \in J, {\bar{w}}_{s} \in S, {\bar{w}}_{i} \in I

such that they satisfy conditions (2)-(4) along with the subsequent condition:

\forall i \in I_{00}, τ_{i}^{L} (ν_{i}^{K} - τ_{i}^{K}) = 0 .

Definition 15.

Let

\bar{z} \in F

\bar{z}

is a generalized robust S stationary (RS-Stationary) point if there exist

ν = (ν^{Φ}, ν^{Ψ}, ν^{Θ}, ν^{K}) \in R^{r + p + l + m}, τ = (τ^{Θ}, τ^{K}, τ^{L}) \in R^{l + 2 m}

and

{\bar{w}}_{j} \in J, {\bar{w}}_{s} \in S, {\bar{w}}_{i} \in I

such that they satisfy conditions (2)-(4) along with the subsequent condition:

\forall i \in I_{00}, τ_{i}^{L} = 0, (ν_{i}^{K} - τ_{i}^{K}) \geq 0 .

From the above definitions, we conclude that

RS - Stationary \Rightarrow RM - Stationary \Rightarrow RT - Stationary \Rightarrow RW - Stationary .

In the following theorem, by employing GS-ACQ, we establish that the RS-Stationary is the necessary first-order optimality criteria for a robust local weak Pareto solution of UNMPVC.

Theorem 3.

Let

\bar{z} \in F

be a robust local weak Pareto solution of UNMPVC. Assume that

Φ_{t}, t \in T,

Ψ_{j}, j \in J

Θ_{s}, s \in S

K_{i}, L_{i}, i \in I

admit bounded USRC and are locally Lipschitz functions. If GS-ACQ holds at

\bar{z}

and the cone

\begin{matrix} D^{t_{0}} = cone co F^{t_{0}} & + cone co G + cone co H + cone co K_{0 +} \\ + cone co K_{0 -} + cone co L_{0 +} + cone co K_{00} \end{matrix}

is closed, then there exist

{\tilde{w}}_{j} \in Ω_{j} (\bar{z})

{\tilde{w}}_{s} \in Ω_{s} (\bar{z})

{\tilde{w}}_{i} \in Ω_{i} (\bar{z})

such that

\bar{z}

is a RS-Stationary point of UNMPVC.

Proof.

Let

\bar{z} \in F

be a robust local weak Pareto solution of UNMPVC. Then

\bar{z}

is a robust local weak Pareto solution of NMPVC*. To derive the above result, it is enough to demonstrate that for every

t \in T

0 \in co \partial^{*} Φ_{t} (\bar{z}) + D^{t} .

(5)

Alternatively, let us suppose that there exists

t_{0} \in T

such that

0 \notin co \partial^{*} Φ_{t_{0}} (\bar{z}) + D^{t_{0}} .

(6)

Since

Φ_{t_{0}}

admits bounded USRC, thus

co \partial^{*} Φ_{t_{0}} (\bar{z})

is compact and convex set in

R^{n}

. Since

D^{t_{0}}

is a closed convex set. Hence,

co \partial^{*} Φ_{t_{0}} (\bar{z}) + D^{t_{0}}

is closed and convex set in

R^{n}

. By employing the convex separation theorem there exists

υ \in R^{n}

such that

〈 ξ + ζ, υ 〉 < 0, \forall ξ \in co \partial^{*} Φ_{t_{0}} (\bar{z}), \forall ζ \in D^{t_{0}} .

Since every cone contains zero we get

〈 ξ, υ 〉 < 0, \forall ξ \in co \partial^{*} Φ_{t_{0}} (\bar{z}) .

(7)

Thus,

Φ_{t_{0}}^{+} (\bar{z}, υ) < 0 .

(8)

Hence

\exists δ > 0

such that

Φ_{t_{0}} (\bar{z} + λ υ) < Φ_{t_{0}} (\bar{z}), \forall λ \in (0, δ) .

(9)

Also, we deduce that

〈 ζ, υ 〉 \leq 0, \forall ζ \in D^{t_{0}} .

Consequently,

〈 ζ_{t}^{1}, υ 〉 \leq 0, \forall ζ_{t}^{1} \in co \partial^{*} Φ_{t} (\bar{z}), \forall t \in T ∖ {t_{0}},

(10)

〈 ζ_{j}^{2}, υ 〉 \leq 0, \forall ζ_{j}^{2} \in co \partial^{*} Ψ_{j}^{*} (\bar{z}), \forall j \in I_{Ψ},

(11)

〈 ζ_{s}^{3}, υ 〉 \leq 0, \forall ζ_{s}^{3} \in co \partial^{*} Θ_{s}^{*} (\bar{z}) \cup co \partial^{*} (- Θ_{s}^{*}) (\bar{z}), \forall s \in S,

(12)

〈 ζ_{i}^{4}, υ 〉 \leq 0, \forall ζ_{i}^{4} \in co \partial^{*} K_{i}^{*} (\bar{z}) \cup co \partial^{*} (- K_{i}^{*}) (\bar{z}), \forall i \in I_{0 +},

(13)

〈 ζ_{i}^{5}, υ 〉 \leq 0, \forall ζ_{i}^{5} \in co \partial^{*} (- K_{i}^{*}) (\bar{z}), \forall i \in I_{0 -},

(14)

〈 ζ_{i}^{6}, υ 〉 \leq 0, \forall ζ_{i}^{6} \in co \partial^{*} L_{i}^{*} (\bar{z}), \forall i \in I_{+ 0},

(15)

〈 ζ_{i}^{7}, υ 〉 \leq 0, \forall ζ_{i}^{7} \in co \partial^{*} (- K_{i}^{*}) (\bar{z}), \forall i \in I_{00} .

(16)

From (10)-(16) we get

υ \in Γ (\bar{z}) .

By GS-ACQ we get

υ \in T (S^{t_{0}}, \bar{z}) .

(17)

Therefore, there exist

λ_{n} ↓ 0

and

υ_{n} \to υ

such that

\bar{z} + λ_{n} υ_{n} \in S^{t_{0}}

(18)

Let

C_{t_{0}} > 0

be the Lipschitzian constant for

Φ_{t_{0}}

near

\bar{z},

then for n large enough we have

Φ_{t_{0}} (z_{n} + λ_{n} v_{n}) \leq Φ_{t_{0}} (\bar{z} + λ_{n} v) + C_{t_{0}} λ_{n} ∥ v_{n} - v ∥ .

Hence, for sufficiently large n we have

Φ_{t_{0}} (\bar{z} + λ_{n} v_{n}) < Φ_{t_{0}} (\bar{z}) .

(19)

From (18) and (19) we arrive at a contradiction with the local weak Pareto efficiency at

\bar{z} .

Hence (5) holds. Therefore, there exist non-negative multipliers

ν_{t}^{Φ} > 0, t \in T, ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}, ν_{s}^{Θ}, τ_{s}^{Θ} \geq 0, s \in S, τ_{i}^{K}, i \in I_{0 +} ν_{i}^{K}, i \in I_{0 +} \cup I_{0 -} \cup I_{00}, τ_{i}^{L}, i \in I_{+ 0}

such that

\begin{matrix} 0 \in \sum_{t \in T} ν_{t}^{Φ} co \partial^{*} Φ_{t} (\bar{z}) & + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} co \partial^{*} Ψ_{j}^{*} (\bar{z}) + \sum_{s \in S} ν_{s}^{Θ} co \partial^{*} Θ_{s}^{*} (\bar{z}) \\ + \sum_{s \in S} τ_{s}^{Θ} co \partial^{*} (- Θ_{s}^{*}) (\bar{z}) + \sum_{i \in I_{0 +} \cup I_{0 -} \cup I_{00}} ν_{i}^{K} co \partial^{*} (- K_{i}^{*}) (\bar{z}) \\ + \sum_{i \in I_{0 +}} τ_{i}^{K} co \partial^{*} K_{i}^{*} (\bar{z}) + \sum_{i \in I_{+ 0}} τ_{i}^{L} co \partial^{*} L_{i}^{*} (\bar{z}) . \end{matrix}

(20)

Taking

ν_{I_{+ 0} \cup I_{+ -}}^{K} = 0, τ_{I_{00} \cup I_{0 -} \cup I_{+ -} \cup I_{+ 0}}^{K} = 0, τ_{I_{+ -} \cup I_{0 +} \cup I_{0 -} \cup I_{00}}^{L} = 0,

from (20) we get

\begin{matrix} 0 \in \sum_{t \in T} ν_{t}^{Φ} co \partial^{*} Φ_{t} (\bar{z}) & + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} co \partial^{*} Ψ_{j}^{*} (\bar{z}) \\ + \sum_{s \in S} [ν_{s}^{Θ} co \partial^{*} Θ_{s}^{*} (\bar{z}) + τ_{s}^{Θ} co \partial^{*} (- Θ_{s}^{*}) (\bar{z})] \\ + \sum_{i \in I} ν_{i}^{K} co \partial^{*} (- K_{i}^{*}) (\bar{z}) \\ + \sum_{i \in I} [τ_{i}^{K} co \partial^{*} K_{i}^{*} (\bar{z}) + τ_{i}^{L} co \partial^{*} L_{i}^{*} (\bar{z})], \end{matrix}

ν_{t}^{Φ} > 0, t \in T, ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}, ν_{s}^{Θ}, τ_{s}^{Θ} \geq 0, s \in S, τ_{i}^{K}, ν_{i}^{K}, τ_{i}^{L}, i \in I

ν_{I_{+ 0} \cup I_{+ -}}^{K} = 0, τ_{I_{00} \cup I_{0 -} \cup I_{+ -} \cup I_{+ 0}}^{K} = 0, τ_{I_{+ -} \cup I_{0 +} \cup I_{0 -}}^{L} = 0,

\forall i \in I_{00}, τ_{i}^{L} = 0, (ν_{i}^{K} - τ_{i}^{K}) = ν_{i}^{K} \geq 0 .

Hence,

\bar{z}

is an RS-Stationary point of NMPVC*. From Theorem 2 there exist

{\tilde{w}}_{j} \in Ω_{j} (\bar{z})

{\tilde{w}}_{s} \in Ω_{s} (\bar{z})

{\tilde{w}}_{i} \in Ω_{i} (\bar{z})

such that

\bar{z}

is a RS-Stationary point of UNMPVC. This completes the proof of the theorem. □

Remark 1.

It is worthwhile to note that Theorem 3 generalizes Theorem 3.1 established in Hu et al. [37] for a more general programming problem involving data uncertainty. If $Ω_{j}, j \in J,$ $Ω_{s}, s \in S$ and $Ω_{i}, i \in I$ are singleton sets, then Theorem 3 reduces to Theorem 3.1 established in [37].
In the domain of robust optimization, Theorem 3 extends the robust first-order optimality conditions established by Chen et al. [60] for a more general programming problems UNMPVC. For $S = \emptyset$ and $I = \emptyset$ Theorem 3 reduces to Theorem 3.3 established by Chen et al. [60].

We consider the following non-trivial example in the Euclidean space setting to highlight the importance of the necessary optimality condition derived in Theorem 3.

Example 1.

Let us examine the following nonsmooth multiobjective programming problem with vanishing constraints under data uncertainty given by

\begin{matrix} UNMPVC 1 Minimize & Φ (z) = (Φ_{1} (z), Φ_{2} (z)) : = (| z_{1} |, | z_{2} |), \\ subject to & Ψ (z, w_{1}) : = z_{1} sin w_{1} \leq 0, \forall w_{1} \in Ω_{1} : = [0, \frac{π}{2}], \\ K (z, w_{2}) : = z_{2} w_{2} \leq 0, \forall w_{2} \in Ω_{2} : = [0, 1], \\ L (z, w_{2}) K (z, w_{2}) : = z_{1} z_{2} w_{2} \leq 0, \forall w_{2} \in Ω_{2} : = [0, 1], \end{matrix}

where

Φ_{t} : R^{2} \to \bar{R}, t \in {1, 2}

Ψ : R^{2} \times Ω_{1} \to \bar{R}

K : R^{2} \times Ω_{2} \to \bar{R}

L : R^{2} \times Ω_{2} \to \bar{R}

and

z = (z_{1}, z_{2}) \in R^{2} .

It is evident that

\begin{matrix} Ψ^{*} (z) = max_{w_{1} \in Ω_{1}} z_{1} sin w_{1} = z_{1}, & K^{*} (z) = max_{w_{2} \in Ω_{2}} z_{2} w_{2} = z_{2}, \\ L^{*} (z) K^{*} (z) = max_{w_{2} \in Ω_{2}} z_{1} z_{2} w_{2} = z_{1} z_{2}, \end{matrix}

and

\bar{z} = (0, 0)

is a robust local weak Pareto solution in UNMPVC1. Moreover, in view of Definition 3, the upper semi-regular convexificators at

\bar{z}

of each function constituting UNMPVC1 are given by

\begin{matrix} \partial^{*} Φ_{1} (\bar{z}) = {(- 1, 0), (1, 0)}, & \partial^{*} Φ_{2} (\bar{z}) = {(0, - 1), (0, 1)}, \\ \partial^{*} Ψ^{*} (\bar{z}) = {(1, 0)}, & \partial^{*} K^{*} (\bar{z}) = {(0, 1)}, \\ \partial^{*} (- K^{*}) (\bar{z}) = {(0, - 1)}, & \partial^{*} L^{*} (\bar{z}) = {(1, 0)} . \end{matrix}

We have

G^{-} = {z : z_{1} \leq 0},

K_{00}^{-} = K_{0 -}^{-} = {z : z_{2} \geq 0} .

Hence GS-ACQ is satisfied at

\bar{z} = (0, 0)

and

D^{t}

is a closed set. Moreover, there exist

ν_{1}^{Φ} = 1, ν_{2}^{Φ} = 1, ν^{Ψ} = 1, ν^{K} = 1, τ^{K} = 0, τ^{L} = 0

and

ξ_{1}^{Φ} = (- 1, 0) \in \partial^{*} Φ_{1}^{*} (\bar{z}), ξ_{2}^{Φ} = (0, 1) \in \partial^{*} Φ_{2}^{*} (\bar{z}), η^{Ψ} = (1, 0) \in \partial^{*} Ψ^{*} (\bar{z}), η^{K} = (0, 1) \in \partial^{*} K^{*} (\bar{z}), η^{- K} = (0, - 1) \in \partial^{*} (- K^{*}) (\bar{z}), η^{L} = (1, 0) \in \partial^{*} L^{*} (\bar{z})

such that

\bar{z} = (0, 0)

is a RS-Stationary point of UNMPVC1.

Now, under the assumptions of generalized convexity, we prove the sufficient optimality conditions for robust local weak Pareto solution and robust local Pareto solution of UNMPVC.

Theorem 4.

Let us assume that

\bar{z}

be a feasible RS-Stationary point of UNMPVC and consider the index sets:

I_{0 -}^{K} : = {i \in I_{0 -} : τ_{i}^{K} > 0},

I_{0 +}^{K} : = {i \in I_{0 +} : τ_{i}^{K} > 0},

I_{+ 0}^{L} : = {i \in I_{+ 0} : τ_{i}^{L} > 0},

Assume that

Ψ_{j}, j \in I_{Ψ}

\pm Θ_{s}, s \in S

- K_{i}, i \in I_{0 +} \cup I_{0 -} \cup I_{00}

K_{i}, i \in I_{0 +}

and

L_{i}, i \in I_{+ 0}

, are

\partial^{*}

-quasiconvex at

\bar{z}

. Then,

if $Φ_{t}, t \in T$ are $\partial^{*}$ -pseudoconvex at $\bar{z}$ and $I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset$ , then $\bar{z}$ is a robust local weak Pareto solution of UNMPVC;
if $Φ_{t}, t \in T$ are strictly $\partial^{*}$ -pseudoconvex at $\bar{z}$ and $I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset$ , then $\bar{z}$ is a robust local Pareto solution of UNMPVC.

Proof.

Assume, on the contrary, that $\bar{z}$ is not a local robust weak Pareto solution of UNMPVC. Thus there exists $y \in F$ such that $Φ (y) ≺ Φ (\bar{z})$ .

From the $\partial^{*}$ -pseudoconvexity of $Φ_{t}, t \in T$ at $\bar{z}$ we have

$〈 ξ_{t}^{1}, y - \bar{z} 〉 < 0, \forall ξ_{t}^{1} \in \partial^{*} Φ_{t} (\bar{z}), \forall t \in T .$

(21)

Since at any feasible point y of UNMPVC we have

$Ψ_{j} (y, w_{j}) \leq Ψ_{j} (\bar{z}, {\bar{w}}_{j}), \forall j \in I_{Ψ}, {\bar{w}}_{j} \in Ω_{j} (\bar{z}),$

therefore, from the $\partial^{*}$ -quasiconvexity of $Ψ_{j}, j \in J$ we have

$〈 ξ_{j}^{2}, y - \bar{z} 〉 \leq 0, \forall ξ_{j}^{2} \in \partial^{*} Ψ_{j} (\bar{z}, {\bar{w}}_{j}), \forall j \in I_{Ψ} .$

(22)

Similarly we get

$〈 ξ_{s}^{3}, y - \bar{z} 〉 \leq 0, \forall ξ_{s}^{3} \in \partial^{*} Θ_{s} (\bar{z}, {\bar{w}}_{s}), \forall s \in S, {\bar{w}}_{s} \in Ω_{s} (\bar{z}),$

(23)

$〈 ξ_{s}^{4}, y - \bar{z} 〉 \leq 0, \forall ξ_{s}^{4} \in \partial^{*} (- Θ_{s}) (\bar{z}, {\bar{w}}_{s}), \forall s \in S, {\bar{w}}_{s} \in Ω_{s} (\bar{z}),$

(24)

$\begin{matrix} 〈 ξ_{i}^{5}, y - \bar{z} 〉 \leq 0, \forall ξ_{i}^{5} \in \partial^{*} (- K_{i}) (\bar{z}, {\bar{w}}_{i}), \forall i \in I_{0 +} \cup I_{0 -} \cup I_{00}, {\bar{w}}_{i} \in Ω_{i} (\bar{z}), \end{matrix}$

(25)

$〈 ξ_{i}^{6}, y - \bar{z} 〉 \leq 0, \forall ξ_{i}^{6} \in \partial^{*} L_{i} (\bar{z}, {\bar{w}}_{i}), \forall i \in I_{+ 0}, {\bar{w}}_{i} \in Ω_{i} (\bar{z}),$

(26)

$〈 ξ_{i}^{7}, y - \bar{z} 〉 \leq 0, \forall ξ_{i}^{7} \in \partial^{*} K_{i} (\bar{z}, {\bar{w}}_{i}), \forall i \in I_{0 +}, {\bar{w}}_{i} \in Ω_{i} (\bar{z}) .$

(27)

From (21)-(27) and from assumption $I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset$ there exist multipliers $ν_{t}^{Φ} > 0, t \in T, ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}, ν_{s}^{Θ}, τ_{s}^{Θ} \geq 0, s \in S, ν_{i}^{K} \geq 0, i \in I_{0 +} \cup I_{0 -} \cup I_{00}, τ_{i}^{L} \geq 0, i \in I_{+ 0}, τ_{i}^{K} \geq 0, i \in I_{0 +}$ such that

$\begin{matrix} 〈 \sum_{t \in T} ν_{t}^{Φ} ξ_{t}^{1} + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} ξ_{j}^{2} & + \sum_{s \in S} [ν_{i}^{Θ} ξ_{i}^{3} + τ_{i}^{Θ} ξ_{i}^{4}] \\ + \sum_{i \in I} ν_{i}^{K} ξ_{i}^{5} + \sum_{i \in I} [τ_{i}^{K} ξ_{i}^{6} + τ_{i}^{L} ξ_{i}^{7}], y - \bar{z} 〉 < 0, \end{matrix}$

(28)

which contradicts the fact that $\bar{z}$ is a RS-stationary point of UNMPVC.
Using the definition of strictly $\partial^{*}$ -pseudoconvex function, the proof is similar to the first part. Hence, we omit it for brevity.

□

Remark 2.

Theorem 4 generalizes Theorem 3.4 established in [60] for a more general programming problem UNMPVC. For $I = \emptyset$ and $S = \emptyset$ Theorem 4 reduces to Theorem 3.4 derived in [60].
Theorem 4 generalizes and extends Theorem 3.3 derived in [37] for a wider class of mathematical programming problems UNMPVC. If $Ω_{j}, j \in J,$ $Ω_{s}, s \in S$ and $Ω_{i}, i \in I$ are singleton sets, then Theorem 4 is similar to the sufficient optimality criteria derived in [37].

To illustrate the practical relevance of the results established in Theorem 4, we investigate the following example.

Example 2.

Let us consider the following nonsmooth multiobjective programming problems incorporating vanishing constraints under data uncertainty given by

\begin{matrix} UNMPVC 2 Minimize & Φ (z) = (Φ_{1} (z), Φ_{2} (z)) : = (z_{1}^{2}, z_{2}), \\ subject to & Ψ (z, w_{1}) : = - z_{2} e^{- w_{1}} \leq 0, \forall w_{1} \in Ω_{1} : = [0, \infty), \\ K (z, w_{2}) : = z_{1} w_{2} \geq 0, \forall w_{2} \in Ω_{2} : = [0, 1], \\ L (z, w_{2}) K (z, w_{2}) : = - | z_{2} | z_{1} w_{2} \leq 0, \forall w_{2} \in Ω_{2} : = [0, 1], \end{matrix}

where

Φ_{t} : R^{2} \to \bar{R}, t \in {1, 2}

Ψ : R^{2} \times Ω_{1} \to \bar{R}

K (z, w_{2}) : R^{2} \times Ω_{2} \to \bar{R}

L (z, w_{2}) : R^{2} \times Ω_{2} \to \bar{R}

and

z = (z_{1}, z_{2}) \in R^{2} .

It is evident that

\begin{matrix} Ψ^{*} (z) = max_{w_{1} \in Ω_{1}} - z_{2} e^{- w_{1}} = - z_{2}, \\ K^{*} (z) = max_{w_{2} \in Ω_{2}} z_{1} w_{2} = z_{1}, \\ L^{*} (z) K^{*} (z) = max_{w_{2} \in Ω_{2}} z_{1} z_{2} w_{2} = z_{1} z_{2}, \end{matrix}

and

\bar{z} = (0, 0)

is a local weak pareto solution of UNMPVC2.

Furthermore, by Definition 3, the upper semi-regular convexificators at

\bar{z}

of each function constituting UNMPVC2 are given by

\begin{matrix} \partial^{*} Φ_{1} (\bar{z}) = {(0, 0)}, & \partial^{*} Φ_{2} (\bar{z}) = {(1, 0)}, \\ \partial^{*} Ψ^{*} (\bar{z}) = {(0, - 1)}, & \partial^{*} K^{*} (\bar{z}) = {(1, 0)}, \\ \partial^{*} (- K^{*}) (\bar{z}) = {(- 1, 0)}, & \partial^{*} L^{*} (\bar{z}) = {(0, - 1), (0, 1)} . \end{matrix}

We have

\bar{z} = (0, 0)

is a RS-Stationary point and

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

. Moreover,

Φ_{t}, t \in {1, 2}

are

\partial^{*}

-pseudoconvex at

\bar{z}

and

Ψ_{j}, j \in I_{Ψ}

\pm Θ_{s}, s \in S

- K_{i}, i \in I_{0 +} \cup I_{0 -} \cup I_{00}

K_{i}, i \in I_{0 +}

and

L_{i}, i \in I_{+ 0}

, are

\partial^{*}

-quasiconvex at

\bar{z}

. Hence, conditions of Theorem 4 are satisfied at

\bar{z}

4. Wolfe-type Robust Dual Model

This section deals with formulating a Wolfe-type robust dual model (WRD) relating to the primal problem UNMPVC. We further extend our analysis by establishing weak, strong, and strict converse duality theorems. These theorems illuminate the profound interrelationships between the primal problem UNMPVC and its corresponding WRD dual model.

We formulate WRD corresponding to the primal problem UNMPVC as follows:

\begin{matrix} \underset{(y, ν)}{Maximize} {Φ (y) & + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) e \\ + \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (y, w_{s}) e + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) e \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) e \\ + \sum_{i \in I} τ_{i}^{K} K_{i} (y, w_{i}) e + \sum_{i \in I} τ_{i}^{L} L_{i} (y, w_{i}) e, e = (1, 1, \dots, 1) \in R^{r}}, \end{matrix}

subject to (y, ν) \in F_{W R D},

where

Φ (y) = (Φ_{1} (y), \dots, Φ_{r} (y))

and

F_{W R D}

is feasible set of WRD given by

\begin{matrix} F_{W R D} : = {ν = & (ν^{Φ}, ν^{Ψ}, ν^{Θ}, τ^{Θ}, ν^{K}, τ^{K}, τ^{L}) \in R^{r} \times R^{p} \times R^{2 l} \times R^{3 m}, {(ν^{Φ})}^{T} e = 1, \\ y \in R_{+}^{n} : 0 \in \sum_{t \in T} ν_{t}^{Φ} c o \partial^{*} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} c o \partial^{*} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} [ν_{s}^{Θ} c o \partial^{*} Θ_{s} (y, w_{s}) + τ_{s}^{Θ} c o \partial^{*} (- Θ_{s}) (y, w_{s})] \\ + \sum_{i \in I} ν_{i}^{K} c o \partial^{*} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} [τ_{i}^{K} c o \partial^{*} K_{i} (y, w_{i}) + τ_{i}^{L} c o \partial^{*} L_{i} (y, w_{i})], \\ ν_{t}^{Φ} > 0, t \in T, ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}, ν_{s}^{Θ}, τ_{s}^{Θ} \geq 0, s \in S, \\ ν_{i}^{K}, τ_{i}^{K}, τ_{i}^{L} \geq 0, i \in I, \\ ν_{I_{+ 0} \cup I_{+ -}}^{K} = τ_{I_{+ 0} \cup I_{+ -}}^{K} = τ_{I_{0 +} \cup I_{+ -} \cup I_{0 -}}^{L} = 0, \\ ν_{i}^{K} - τ_{i}^{K} \geq 0, i \in I_{0 -}, τ_{i}^{L} = 0, (ν_{i}^{K} - τ_{i}^{K}) \geq 0, \forall i \in I_{00}} . \end{matrix}

The subsequent theorem delves into the weak duality theorem between UNMPVC and WRD.

Theorem 5

(Weak Duality). Let z

\in F

and

(y, ν) \in F_{W R D}

. Suppose that

Φ_{t}, t \in T, Ψ_{j}, j \in I_{Ψ}, \pm Θ_{s}, s \in S, - K_{i}, i \in I_{0 +} \cup I_{0 -} \cup I_{00},

K_{i}, i \in I_{0 +}

and

L_{i}, i \in I_{+ 0}

admit bounded USRC and are

\partial^{*}

-convex functions at y. Assume also that

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

. Then we have

\begin{matrix} Φ (z) \neg ≺ Φ (y) & + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) e + \sum_{i \in I} ν_{s}^{Θ} Θ_{s} (y, w_{s}) e + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) e \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) e + \sum_{i \in I} τ_{i}^{K} K_{i} (y, w_{i}) e + \sum_{i \in I} τ_{i}^{L} L_{i} (y, w_{i}) e . \end{matrix}

(29)

Proof.

Assume, on the contrary, that

\begin{matrix} Φ (z) ≺ Φ (y) & + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) e + \sum_{i \in I} ν_{s}^{Θ} Θ_{s} (y, w_{s}) e + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) e \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) e + \sum_{i \in I} τ_{i}^{K} K_{i} (y, w_{i}) e + \sum_{i \in I} τ_{i}^{L} L_{i} (y, w_{i}) e . \end{matrix}

(30)

Multiplying (30) by

ν_{t}^{Φ} > 0, t \in T

and since

{(ν^{Φ})}^{T} e = 1

we get

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & < \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (y, w_{s}) + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} [ν_{i}^{K} K_{i} (y, w_{i}) + ν_{i}^{L} L_{i} (y, w_{i})] . \end{matrix}

(31)

Let z be any feasible solution of UNMPVC. Since

Φ_{t}, t \in T

\partial^{*}

-convex at y, then we have

Φ_{t} (z) - Φ_{t} (y) \geq 〈 ξ_{t}^{1}, z - y 〉, \forall ξ_{t}^{1} \in \partial^{*} Φ_{t} (y), \forall t \in T .

(32)

Similarly, by the

\partial^{*}

-convexity of

Ψ_{j} (\cdot, w_{j}),

j \in I_{Ψ},

\pm Θ_{s} (\cdot, w_{s}),

s \in S,

- K_{i} (\cdot, w_{i}),

i \in I_{0 +} \cup I_{0 -} \cup I_{00}

K_{i} (\cdot, w_{i}), i \in I_{0 +}

and

L_{i} (\cdot, w_{i}), i \in I_{+ 0}

at y we have

\begin{matrix} Ψ_{j} (z, w_{j}) - Ψ_{j} (y, w_{j}) \geq 〈 ξ_{j}^{2}, z - y 〉, \forall ξ_{j}^{2} \in \partial^{*} Ψ_{j} (y, w_{j}), \forall j \in I_{Ψ}, \end{matrix}

(33)

Θ_{s} (z, w_{s}) - Θ_{i} (y, w_{s}) \geq 〈 ξ_{s}^{3}, z - y 〉, \forall ξ_{s}^{3} \in \partial^{*} Θ_{i} (y, w_{s}), \forall s \in S,

(34)

\begin{matrix} (- Θ_{s}) (z, w_{s}) - (- Θ_{s}) (y, w_{s}) \geq 〈 ξ_{s}^{4}, z - y 〉, \forall ξ_{s}^{4} \in \partial^{*} (- Θ_{s}) & (y, w_{s}), \\ \forall s \in S, \end{matrix}

(35)

\begin{matrix} (- K_{i}) (z, w_{i}) - (- K_{i}) (y, w_{i}) \geq 〈 ξ_{i}^{5}, z - y 〉, & \forall ξ_{i}^{5} \in \partial^{*} (- K_{i}) (y, w_{i}), \\ \forall i \in I_{0 +} \cup I_{0 -} \cup I_{00}, \end{matrix}

(36)

K_{i} (z, w_{i}) - K_{i} (y, w_{i}) \geq 〈 ξ_{i}^{6}, z - y 〉, \forall ξ_{i}^{6} \in \partial^{*} K_{i} (y, w_{i}), \forall i \in I_{0 +},

(37)

L_{i} (z, w_{i}) - L_{i} (y, w_{i}) \geq 〈 ξ_{i}^{7}, z - y 〉, \forall ξ_{i}^{7} \in \partial^{*} K_{i} (y, w_{i}), \forall i \in I_{+ 0},

(38)

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

, then multiplying (32)-(38) by

ν_{t}^{Φ} > 0, t \in T

ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}

ν_{s}^{Θ} \geq 0, s \in S

τ_{s}^{Θ} \geq 0, s \in S

ν_{i}^{K} \geq 0, i \in I_{0 +} \cup I_{0 -} \cup I_{00}

τ_{i}^{K} \geq 0, i \in I_{0 +}, τ_{i}^{L} \geq 0, i \in I_{0 +}

, respectively and then adding them subsequently, we get

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & - \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (z, w_{j}) - \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (z, w_{s}) - \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (y, w_{s}) + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (z, w_{s}) \\ - \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (z, w_{i}) - \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} τ_{i}^{K} K_{i} (z, w_{i}) - \sum_{i \in I} τ_{i}^{K} K_{i} (y, w_{i}) + \sum_{i \in I} τ_{i}^{L} L_{i} (z, w_{i}) \\ - \sum_{i \in I} τ_{i}^{L} L_{i} (y, w_{i}) \geq 〈 \sum_{t \in T} ν_{t}^{Φ} ξ_{t}^{1} + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} ξ_{j}^{2} + \sum_{s \in S} [ν_{s}^{Θ} ξ_{s}^{3} + τ_{s}^{Θ} ξ_{s}^{4}] \\ + \sum_{i \in I} ν_{i}^{K} ξ_{i}^{5} + \sum_{i \in I} [τ_{i}^{K} ξ_{i}^{6} + τ_{i}^{L} ξ_{i}^{7}], z - y 〉 . \end{matrix}

From the feasibility conditions of

y \in F_{W R D}

, there exist

{\bar{ξ}}_{t}^{1} \in c o \partial^{*} Φ_{t} (y),

t \in T

{\bar{ξ}}_{j}^{2} \in c o \partial^{*} Ψ_{j} (y, w_{j}),

j \in I_{Ψ}

{\bar{ξ}}_{s}^{3} \in c o \partial^{*} Θ_{s} (y, w_{s}),

s \in S

{\bar{ξ}}_{s}^{4} \in c o \partial^{*} (- Θ_{s}) (y, w_{s}),

s \in S

{\bar{ξ}}_{i}^{5} \in c o \partial^{*} (- K_{i}) (y, w_{i}),

i \in I

{\bar{ξ}}_{i}^{6} \in c o \partial^{*} K_{i} (y, w_{i}),

i \in I

and

{\bar{ξ}}_{i}^{7} \in c o \partial^{*} L_{i} (y, w_{i}),

i \in I

such that

\sum_{t \in T} ν_{t}^{Φ} {\bar{ξ}}_{t}^{1} + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} {\bar{ξ}}_{j}^{2} + \sum_{s \in S} [ν_{s}^{Θ} {\bar{ξ}}_{s}^{3} + τ_{s}^{Θ} {\bar{ξ}}_{s}^{4}] + \sum_{i \in I} ν_{i}^{K} {\bar{ξ}}_{i}^{5} + \sum_{i \in I} [τ_{i}^{K} {\bar{ξ}}_{i}^{5} + τ_{i}^{L} {\bar{ξ}}_{i}^{6}] = 0 .

Therefore,

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & - \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (z, w_{j}) - \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (z, w_{s}) - \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (y, w_{s}) + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (z, w_{s}) \\ - \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (z, w_{i}) - \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} τ_{i}^{K} K_{i} (z, w_{i}) - \sum_{i \in I} τ_{i}^{K} K_{i} (y, w_{i}) + \sum_{i \in I} τ_{i}^{L} L_{i} (z, w_{i}) \\ - \sum_{i \in I} τ_{i}^{L} L_{i} (y, w_{i}) \geq 0 . \end{matrix}

Since

z \in F

, we have

Ψ_{j} (z, w_{j}) \leq 0, j \in I_{Ψ}, Θ_{s} (z, w_{s}) = 0, s \in S, - K_{i} (z, w_{i}) < 0, i \in I_{+}, K_{i} (z, w_{i}) = 0, i \in I_{0}, L_{i} (z, w_{i}) > 0, i \in I_{0 +}, L_{i} (z, w_{i}) = 0, i \in I_{00} \cup I_{+ 0}, L_{i} (z, w_{i}) = 0, i \in I_{0 -} \cup I_{+ -} .

Therefore,

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & \geq \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} [ν_{s}^{Θ} Θ_{s} (y, w_{s}) + τ_{s}^{Θ} (- Θ_{s}) (y, w_{s})] + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} [ν_{i}^{K} K_{i} (y, w_{i}) + ν_{i}^{L} L_{i} (y, w_{i})] . \end{matrix}

(39)

Hence, from (31), we arrive at a contradiction. This completes the proof of the weak duality theorem. □

In the subsequent theorem, we establish the strong duality theorem for WRD relating to the primal problem UNMPVC.

Theorem 6

(Strong Duality). Let

\bar{z} \in F

be a robust local weak Pareto solution of UNMPVC. Suppose that

Φ_{t}, t \in T, Ψ_{j}, j \in I_{Ψ}, \pm Θ_{s}, s \in S, - K_{i}, i \in I_{0 +} \cup I_{0 -} \cup I_{00},

K_{i}, i \in I_{0 +}

and

L_{i}, i \in I_{+ 0}

admit bounded USRC and are

\partial^{*}

-convex functions at

\bar{z}

. Suppose also that at

\bar{z}

, GS-ACQ holds. Then there exists

\bar{ν} = ({\bar{ν}}^{Φ}, {\bar{ν}}^{Ψ}, {\bar{ν}}^{Θ}, {\bar{τ}}^{Θ}, {\bar{ν}}^{K}, {\bar{τ}}^{K}, {\bar{τ}}^{L}) \in R^{r} \times R^{p} \times R^{2 l} \times R^{3 m}

such that

(\bar{z}, \bar{ν})

becomes a robust local weak solution of the WRD. Moreover, the corresponding objective function values are equal.

Proof.

Since

\bar{z}

is a robust local weak solution of UNMPVC and GS-ACQ holds at

\bar{z}

, then from Theorem 3 there exists

\bar{ν} = ({\bar{ν}}^{Φ}, {\bar{ν}}^{Ψ}, {\bar{ν}}^{Θ}, {\bar{τ}}^{Θ}, {\bar{ν}}^{K}, {\bar{τ}}^{K}, {\bar{τ}}^{L}) \in R^{r} \times R^{p} \times R^{2 l} \times R^{3 m},

such that

\bar{z}

is a RS-Stationary point of UNMPVC. Therefore, there exist

{\bar{ξ}}_{t}^{1} \in c o \partial^{*} Φ_{t} (\bar{z}),

t \in T

{\bar{ξ}}_{j}^{2} \in c o \partial^{*} Ψ_{j} (\bar{z},

{\bar{w}}_{j}),

j \in I_{Ψ},

{\bar{w}}_{j} \in Ω_{j} (\bar{z})

{\bar{ξ}}_{s}^{3} \in c o \partial^{*} Θ_{s} (\bar{z},

{\bar{w}}_{s}),

{\bar{ξ}}_{s}^{4} \in c o \partial^{*} (- Θ_{s}) (\bar{z}, {\bar{w}}_{s}),

s \in S, {\bar{w}}_{s} \in Ω_{s} (\bar{z})

{\bar{ξ}}_{i}^{5} \in c o \partial^{*} (- K_{i}) (\bar{z}, {\bar{w}}_{i}),

i \in I

{\bar{ξ}}_{i}^{6} \in c o \partial^{*} K_{i} (\bar{z}, {\bar{w}}_{i}), i \in I,

and

{\bar{ξ}}_{i}^{7} \in c o \partial^{*} L_{i} (\bar{z}, {\bar{w}}_{i}), i \in I, {\bar{w}}_{i} \in Ω_{i} (\bar{z})

such that

\sum_{t \in T} {\bar{ν}}_{t}^{Φ} {\bar{ξ}}_{t}^{1} + \sum_{j \in I_{Ψ}} {\bar{ν}}_{j}^{Ψ} {\bar{ξ}}_{j}^{2} + \sum_{s \in S} [{\bar{ν}}_{s}^{Θ} {\bar{ξ}}_{s}^{3} + {\bar{τ}}_{s}^{Θ} {\bar{ξ}}_{s}^{4}] + \sum_{i \in I} {\bar{ν}}_{i}^{K} {\bar{ξ}}_{i}^{5} + \sum_{i \in I} [{\bar{τ}}_{i}^{K} {\bar{ξ}}_{i}^{6} + {\bar{τ}}_{i}^{L} {\bar{ξ}}_{i}^{7}] = 0,

\begin{matrix} {\bar{ν}}_{t}^{Φ} > 0, t \in T, {\bar{ν}}_{j}^{Ψ} \geq 0, j \in I_{Ψ}, {\bar{ν}}_{s}^{Θ}, {\bar{τ}}_{s}^{Θ} \geq 0, s \in S, {\bar{ν}}_{i}^{K}, {\bar{τ}}_{i}^{K}, {\bar{τ}}_{i}^{L} \geq 0, i \in I . \end{matrix}

Therefore,

(\bar{z}, \bar{ν}) \in F_{W}

. By Theorem 5 and for any

(y, ν) \in F_{W}

we have

\begin{matrix} Φ (\bar{z}) \neg ≺ Φ (y) & + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} [ν_{s}^{Θ} Θ_{s} (y, w_{s}) + τ_{s}^{Θ} (- Θ_{i}) (y, w_{s})] \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) + \sum_{i \in I} [τ_{i}^{K} K_{i} (y, w_{i}) + τ_{i}^{L} L_{i} (y, w_{i})] . \end{matrix}

(40)

From the feasibility conditions of UNMPVC and WRD we have

Ψ_{j} (\bar{z}, {\bar{w}}_{j}) = 0, j \in I_{Ψ}, Θ_{s} (\bar{z}, {\bar{w}}_{s}) = 0, s \in S, K_{i} (\bar{z}, {\bar{w}}_{i}) = 0, \forall i \in I_{0}, L_{i} (\bar{z}, {\bar{w}}_{i}) = 0, \forall i \in I_{+ 0} .

Therefore, we have

\begin{matrix} Φ (\bar{z}) = Φ (\bar{z}) & + \sum_{j \in I_{Ψ}} {\bar{ν}}_{j}^{Ψ} Ψ_{j} (\bar{z}, {\bar{w}}_{j}) + \sum_{s \in S} [{\bar{ν}}_{s}^{Θ} Θ_{s} (\bar{z}, {\bar{w}}_{s}) + {\bar{τ}}_{s}^{Θ} (- Θ_{s}) (\bar{z}, {\bar{w}}_{s})] \\ + \sum_{i \in I} {\bar{ν}}_{i}^{K} (- K_{i}) (\bar{z}, {\bar{w}}_{i}) + \sum_{i \in I} [{\bar{τ}}_{i}^{K} K_{i} (\bar{z}, {\bar{w}}_{i}) + {\bar{τ}}_{i}^{L} L_{i} (\bar{z}, {\bar{w}}_{i})] . \end{matrix}

(41)

From Equations (40) and (41) we have

\begin{matrix} Φ (\bar{z}) & + \sum_{j \in I_{Ψ}} {\bar{ν}}_{j}^{Ψ} Ψ_{j} (\bar{z}, {\bar{w}}_{j}) + \sum_{s \in S} [{\bar{ν}}_{s}^{Θ} Θ_{s} (\bar{z}, {\bar{w}}_{s}) + {\bar{τ}}_{s}^{Θ} (- Θ_{s}) (\bar{z}, {\bar{w}}_{s})] \\ + \sum_{i \in I} {\bar{ν}}_{i}^{K} (- K_{i}) (\bar{z}, {\bar{w}}_{i}) + \sum_{i \in I} [{\bar{τ}}_{i}^{K} K_{i} (\bar{z}, {\bar{w}}_{i}) + {\bar{τ}}_{i}^{L} L_{i} (\bar{z}, {\bar{w}}_{i})] \\ \neg ≺ Φ (y) + \sum_{i \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{i}) + \sum_{s \in S} [ν_{s}^{Θ} Θ_{s} (y, w_{s}) + τ_{s}^{Θ} (- Θ_{s}) (y, w_{s})] \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{s}) + \sum_{i \in I} [τ_{i}^{K} K_{i} (y, w_{i}) + τ_{i}^{L} L_{i} (y, w_{i})] . \end{matrix}

Hence,

(\bar{z}, \bar{ν})

is a robust local weak solution of WRD. Moreover, the corresponding objective values are equal. □

Now, we establish the strict converse duality theorem relating WRD and UNMPVC.

Theorem 7

(Strict Converse Duality). Suppose that

\bar{z}

is a robust local weak Pareto solution of UNMPVC. Let

(\hat{y}, \hat{ν})

be the global weak Pareto solution of WRD. Suppose that the assumptions of strong duality theorem hold and Φ is strictly

\partial^{*}

-convex at

\hat{y}

. Assume that

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

. Then

\bar{z} = \hat{y}

Proof.

Let us assume that

\bar{z} \neq \hat{y}

. By the Theorem 6 we have

\begin{matrix} Φ (\bar{z}) = Φ (\bar{z}) & + \sum_{j \in I_{Ψ}} {\bar{ν}}_{j}^{Ψ} Ψ_{j} (\bar{z}, {\bar{w}}_{j}) e + \sum_{s \in S} [{\bar{ν}}_{s}^{Θ} Θ_{s} (\bar{z}, {\bar{w}}_{s}) + {\bar{τ}}_{s}^{Θ} (- Θ_{s}) (\bar{z}, {\bar{w}}_{s})] e \\ + \sum_{i \in I} {\bar{ν}}_{i}^{K} (- K_{i}) (\bar{z}, {\bar{w}}_{i}) e + \sum_{i \in I} [{\bar{τ}}_{i}^{K} K_{i} (\bar{z}, {\bar{w}}_{i}) + {\bar{τ}}_{i}^{L} L_{i} (\bar{z}, {\bar{w}}_{i})] e . \\ = Φ (\hat{y}) + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\hat{y}, w_{j}) e + \sum_{s \in S} [{\hat{ν}}_{s}^{Θ} Θ_{s} (\hat{y}, w_{s}) + {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\hat{y}, w_{s})] e \\ + \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\hat{y}, w_{i}) e + \sum_{i \in I} [{\hat{τ}}_{i}^{K} K_{i} (\hat{y}, w_{i}) + {\hat{τ}}_{i}^{L} L_{i} (\hat{y}, w_{i})] e . \end{matrix}

(42)

Multiplying (42) by

ν_{t}^{Φ} > 0, t \in T

and since

{(ν^{Φ})}^{T} e = 1

we get

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & = \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (y, w_{s}) + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} [ν_{i}^{K} K_{i} (y, w_{i}) + ν_{i}^{L} L_{i} (y, w_{i})] . \end{matrix}

(43)

Since

Φ

is strictly

\partial^{*}

-convex at

\hat{y}

, therefore

Φ_{t} (\bar{z}) - Φ_{t} (\hat{y}) > 〈 ξ_{t}^{1}, \bar{z} - \hat{y} 〉, \forall ξ_{t}^{1} \in \partial^{*} Φ (\hat{y}), \forall t \in T

(44)

Similarly, by the

\partial^{*}

-convexity of

Ψ_{j} (\cdot, w_{j}),

j \in I_{Ψ},

\pm Θ_{s} (\cdot, w_{s}),

s \in S,

- K_{i} (\cdot, w_{i}),

i \in I_{0 +} \cup I_{0 -} \cup I_{00},

K_{i} (\cdot, w_{i}), i \in I_{0 +},

L_{i} (\cdot, w_{i}), i \in I_{+ 0}

\hat{y}

we have

Ψ_{j} (\bar{z}, w_{j}) - Ψ_{j} (\hat{y}, w_{j}) \geq 〈 ξ_{j}^{2}, \bar{z} - \hat{y} 〉, \forall ξ_{j}^{2} \in \partial^{*} Ψ_{j} (\hat{y},, w_{j}), \forall j \in I_{Ψ},

(45)

Θ_{s} (\bar{z}, w_{s}) - Θ_{s} (\hat{y}, w_{s}) \geq 〈 ξ_{s}^{3}, \bar{z} - \hat{y} 〉, \forall ξ_{s}^{3} \in \partial^{*} Θ_{s} (\hat{y}, w_{s}), \forall s \in S,

(46)

\begin{matrix} (- Θ_{s}) (\bar{z}, w_{s}) - (- Θ_{s}) (\hat{y}, w_{s}) \geq 〈 ξ_{s}^{4}, \bar{z} - \hat{y} 〉, \forall ξ_{s}^{4} \in & \partial^{*} (- Θ_{s}) (\hat{y}, w_{s}), \\ \forall s \in S, \end{matrix}

(47)

\begin{matrix} (- K_{i}) (\bar{z}, w_{i}) - (- K_{i}) (\hat{y}, w_{i}) \geq 〈 ξ_{i}^{5}, \bar{z} - \hat{y} 〉, & \forall ξ_{i}^{5} \in \partial^{*} (- K_{i}) (\hat{y}, w_{i}), \\ \forall i \in I_{0 +} \cup I_{0 -} \cup I_{00}, \end{matrix}

(48)

K_{i} (\bar{z}, w_{i}) - K_{i} (\hat{y}, w_{i}) \geq 〈 ξ_{i}^{6}, \bar{z} - \hat{y} 〉, \forall ξ_{i}^{6} \in \partial^{*} L_{i} (\hat{y}, w_{i}), \forall i \in I_{0 +} .

(49)

L_{i} (\bar{z}, w_{i}) - L_{i} (\hat{y}, w_{i}) \geq 〈 ξ_{i}^{7}, \bar{z} - \hat{y} 〉, \forall ξ_{i}^{7} \in \partial^{*} L_{i} (\hat{y}, w_{i}), \forall i \in I_{+ 0} .

(50)

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

, then multiplying (44)-(50) by

{\hat{ν}}_{t}^{Φ} > 0, t \in T

{\hat{ν}}_{j}^{Ψ} \geq 0, j \in I_{Ψ}

{\hat{ν}}_{s}^{Θ} \geq 0, s \in S

{\hat{τ}}_{s}^{Θ} \geq 0, s \in S

{\hat{ν}}_{i}^{K} > 0, i \in I_{0 +} \cup I_{0 -} \cup I_{00}

{\hat{τ}}_{i}^{K} > 0, i \in I_{0 +}

{\hat{τ}}_{i}^{L} > 0, i \in I_{+ 0}

, respectively and then adding them subsequently we get

\begin{matrix} \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\bar{z}) & - \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\hat{y}) + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\bar{z}, w_{j}) - \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\hat{y}, w_{j}) \\ + \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{s} (\bar{z}, w_{j}) - \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{s} (\hat{y}, w_{s}) + \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\bar{z}, w_{s}) \\ - \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\hat{y}, {\hat{w}}_{s}) + \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\hat{y}, w_{i}) \\ + \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\hat{y}, w_{i}) + \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\bar{z}, w_{i}) \\ - \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\hat{y}, w_{i}) > 〈 \sum_{t \in T} {\hat{ν}}_{t}^{Φ} ξ_{t}^{1} + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} ξ_{j}^{2} + \sum_{s \in S} [{\hat{ν}}_{s}^{Θ} ξ_{s}^{3} + {\hat{τ}}_{s}^{Θ} ξ_{s}^{4}] \\ + \sum_{i \in I} {\hat{ν}}_{i}^{K} ξ_{i}^{5} + \sum_{i \in I} [{\hat{τ}}_{i}^{K} ξ_{i}^{6} + {\hat{ν}}_{i}^{L} ξ_{i}^{7}], \bar{z} - \hat{y} 〉 . \end{matrix}

From the feasibility conditions of

\hat{y} \in F_{W R D}

, there exist

{\hat{ξ}}_{t}^{1} \in c o \partial^{*} Φ_{t} (\hat{y}),

t \in T

{\hat{ξ}}_{j}^{2} \in c o \partial^{*} Ψ_{j} (\hat{y}, w_{j}),

j \in I_{Ψ}

{\hat{ξ}}_{s}^{3} \in c o \partial^{*} Θ_{s} (\hat{y}, w_{s}),

s \in S

{\hat{ξ}}_{s}^{4} \in c o \partial^{*} (- Θ_{s}) (\hat{y}, w_{s}),

s \in S

{\hat{ξ}}_{i}^{5} \in c o \partial^{*} (- K_{i}) (\hat{y}, w_{i}),

i \in I

{\hat{ξ}}_{i}^{6} \in c o \partial^{*} K_{i} (\hat{y}, w_{i}),

i \in I

and

{\hat{ξ}}_{i}^{7} \in c o \partial^{*} L_{i} (\hat{y}, w_{i}),

i \in I

such that

\sum_{t \in T} {\hat{ν}}_{t}^{Φ} {\hat{ξ}}_{t}^{1} + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} {\hat{ξ}}_{j}^{2} + \sum_{s \in S} [{\hat{ν}}_{s}^{Θ} {\hat{ξ}}_{s}^{3} + {\hat{τ}}_{s}^{Θ} {\hat{ξ}}_{s}^{4}] + \sum_{i \in I} {\hat{ν}}_{i}^{K} {\hat{ξ}}_{i}^{5} + \sum_{i \in I} [{\hat{τ}}_{i}^{K} {\hat{ξ}}_{i}^{6} + {\hat{τ}}_{i}^{L} {\hat{ξ}}_{i}^{6}] = 0 .

Therefore,

\begin{matrix} \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\bar{z}) & - \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\hat{y}) + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\bar{z}, w_{j}) - \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\hat{y}, w_{j}) \\ + \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{s} (\bar{z}, w_{s}) - \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{i} (\hat{y}, w_{s}) + \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\bar{z}, w_{s}) \\ - \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\hat{y}, w_{s}) + \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\hat{y}, w_{i}) \\ + \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\hat{y}, w_{i}) + \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\bar{z}, w_{i}) \\ - \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\hat{y}, w_{i}) \geq 0 . \end{matrix}

Since

z \in F

, we have

Ψ_{j} (z, w_{j}) \leq 0, j \in I_{Ψ}, Θ_{s} (z, w_{s}) = 0, s \in S, - K_{i} (z, w_{i}) < 0, i \in I_{+}, K_{i} (z, w_{i}) = 0, i \in I_{0}, L_{i} (z, w_{i}) > 0, i \in I_{0 +}, L_{i} (z, w_{i}) = 0, i \in I_{00} \cup I_{+ 0}, L_{i} (z, w_{i}) = 0, i \in I_{0 -} \cup I_{+ -} .

Therefore,

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & > \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} [ν_{s}^{Θ} Θ_{s} (y, w_{s}) + τ_{s}^{Θ} (- Θ_{s}) (y, w_{s})] \\ + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} [ν_{i}^{K} K_{i} (y, w_{i}) + ν_{i}^{L} L_{i} (y, w_{i})] . \end{matrix}

which contradicts (43). Therefore,

\bar{z} = \hat{y}

. This completes the proof. □

Remark 3.

It is worthwhile to note that the weak, strong and strict converse duality theorems presented in this article for Wolfe-type robust dual model, generalize Theorem 3, Theorem 4 and Theorem 7, respectively derived by Mishra et al. [41] for a more general programming problem UNMPVC, involving data uncertainty.
If all the functions are continuously differentiable and if $Ω_{j}, j \in J,$ $Ω_{s}, s \in S$ and $Ω_{i}, i \in I$ are singleton sets, then the Theorem 5, Theorem 6 and Theorem 7 reduces to Theorem 3, Theorem 4 and Theorem 7, respectively derived by Mishra et al. [41].

5. Mond-Weir-Type Robust Dual Model

This section deals with formulating a Mond-Weir-type robust dual model (MWRD) corresponding to the considered primal problem UNMPVC. Subsequently, we derive the duality theorems, namely, weak, strong and strict converse duality theorems. These theorems illuminate the profound interrelationships between the UNMPVC and MWRD.

Now, corresponding to the primal problem UNMPVC, we formulate the Mond-Weir-type robust (MWRD) dual model as follows:

\begin{matrix} \underset{(y, ν)}{Maximize} Φ (y) = (Φ_{1} (y), \dots, Φ_{r} (y)), \\ subject to : (y, ν) \in F_{M W}, \end{matrix}

where

F_{M W}

denotes the set of all feasible solutions of MWRD and is defined as

\begin{matrix} F_{M W} = & {ν = (ν^{Φ}, ν^{Ψ}, ν^{Θ}, τ^{Θ}, ν^{K}, τ^{K}, τ^{L}) \in R^{r} \times R^{p} \times R^{2 l} \times R^{3 m}, \\ y \in R_{+}^{n} : 0 \in \sum_{t \in T} ν_{t}^{Φ} c o \partial^{*} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} c o \partial^{*} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} [ν_{s}^{Θ} c o \partial^{*} Θ_{s} (y, w_{s}) + τ_{s}^{Θ} c o \partial^{*} (- Θ_{s}) (y, w_{s})] \\ + \sum_{i \in I} ν_{i}^{K} c o \partial^{*} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} [τ_{i}^{K} c o \partial^{*} K_{i} (y, w_{i}) + τ_{i}^{L} c o \partial^{*} L_{i} (y, w_{i})], \\ Ψ_{j} (y, w_{j}) \geq 0, j \in I_{Ψ}, Θ_{s} (y, w_{s}) = 0, s \in S, K_{i} (y, w_{i}) \leq 0, i \in I_{+ 0}, \\ - K_{i} (y, w_{i}) \geq 0, i \in I_{0 +} \cup I_{0 -} \cup I_{00}, L_{i} (y, w_{i}) \geq 0, i \in I_{+ 0}, \\ ν_{t}^{Φ} > 0, t \in T, ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}, ν_{s}^{Θ}, τ_{s}^{Θ} \geq 0, s \in S, \\ ν_{i}^{K}, τ_{i}^{K}, τ_{i}^{L} \geq 0, i \in I, \\ ν_{I_{+ 0} \cup I_{+ -}}^{K} = τ_{I_{+ 0} \cup I_{+ -}}^{K} = τ_{I_{0 +} \cup I_{+ -} \cup I_{0 -}}^{L} = 0, \\ ν_{i}^{K} - τ_{i}^{K} \geq 0, i \in I_{0 -}, τ_{i}^{L} = 0, (ν_{i}^{K} - τ_{i}^{K}) \geq 0, \forall i \in I_{00}} . \end{matrix}

In the subsequent theorem, we establish the weak duality theorem that relates UNMPVC and WRD.

Theorem 8

(Weak Duality). Let

z \in F

and

(y, ν) \in F_{M W}

. Suppose that

Φ_{t}, t \in T, Ψ_{j}, j \in I_{Ψ}, \pm Θ_{s} s \in S, - K_{i}, i \in I_{0 +} \cup I_{0 -} \cup I_{00}, K_{i}, i \in I_{0 +}, L_{i}, i \in I_{+ 0}

admit bounded USRC and are

\partial^{*}

-convex at y. Assume also that

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

. Then

Φ (z) \neg ≺ Φ (y) .

(51)

Proof.

Assume, on the contrary, that

\begin{matrix} Φ (z) ≺ Φ (y) . \end{matrix}

(52)

Multiplying (52) by

ν_{t}^{Φ} > 0, t \in T

we get

\sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) < \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) .

(53)

Since

Φ_{t}, t \in T

\partial^{*}

-convex at y we have

Φ_{t} (z) - Φ_{t} (y) \geq 〈 ξ_{t}^{1}, z - y 〉, \forall ξ_{t}^{1} \in \partial^{*} Φ_{t} (y), \forall t \in T .

(54)

Similarly, by the

\partial^{*}

-convexity of

Ψ_{j} (\cdot, w_{j})

j \in I_{Ψ},

\pm Θ_{s} (\cdot, w_{s}),

s \in S,

- K_{i} (\cdot, w_{i}),

i \in I_{0 +} \cup I_{0 -} \cup I_{00}

K_{i} (\cdot, w_{i}),

i \in I_{0 +}

and

L_{i} (\cdot, w_{i}),

i \in I_{+ 0}

at y we have

\begin{matrix} Ψ_{j} (z, w_{j}) - Ψ_{j} (y, w_{j}) \geq 〈 ξ_{j}^{2}, z - y 〉, \forall ξ_{j}^{2} \in \partial^{*} Ψ_{j} (y, w_{j}), \forall j \in I_{Ψ}, \end{matrix}

(55)

Θ_{s} (z, w_{s}) - Θ_{i} (y, w_{s}) \geq 〈 ξ_{s}^{3}, z - y 〉, \forall ξ_{s}^{3} \in \partial^{*} Θ_{i} (y, w_{s}), \forall s \in S,

(56)

\begin{matrix} (- Θ_{s}) (z, w_{s}) - (- Θ_{s}) (y, w_{s}) \geq 〈 ξ_{s}^{4}, z - y 〉, \forall ξ_{s}^{4} \in \partial^{*} (- Θ_{s}) & (y, w_{s}), \\ \forall s \in S, \end{matrix}

(57)

\begin{matrix} (- K_{i}) (z, w_{i}) - (- K_{i}) (y, w_{i}) \geq 〈 ξ_{i}^{5}, z - y 〉, & \forall ξ_{i}^{5} \in \partial^{*} (- K_{i}) (y, w_{i}), \\ \forall i \in I_{0 +} \cup I_{0 -} \cup I_{00}, \end{matrix}

(58)

K_{i} (z, w_{i}) - K_{i} (y, w_{i}) \geq 〈 ξ_{i}^{6}, z - y 〉, \forall ξ_{i}^{6} \in \partial^{*} K_{i} (y, w_{i}), \forall i \in I_{0 +},

(59)

L_{i} (z, w_{i}) - L_{i} (y, w_{i}) \geq 〈 ξ_{i}^{7}, z - y 〉, \forall ξ_{i}^{7} \in \partial^{*} K_{i} (y, w_{i}), \forall i \in I_{+ 0},

(60)

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

, then multiplying (54)-(60) by

ν_{t}^{Φ} > 0, t \in T

ν_{j}^{Ψ} \geq 0, j \in I_{Ψ}

ν_{s}^{Θ} \geq 0, s \in S

τ_{s}^{Θ} \geq 0, s \in S

ν_{i}^{K} \geq 0, i \in I_{0 +} \cup I_{0 -} \cup I_{00}

τ_{i}^{K} \geq 0, i \in I_{0 +}, τ_{i}^{L} \geq 0, i \in I_{+ 0}

, respectively and then adding them subsequently we get

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & - \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (z, w_{j}) - \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (z, w_{s}) - \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (y, w_{s}) + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (z, w_{s}) \\ - \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (z, w_{i}) - \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} τ_{i}^{K} K_{i} (z, w_{i}) - \sum_{i \in I} τ_{i}^{K} K_{i} (y, w_{i}) + \sum_{i \in I} τ_{i}^{L} L_{i} (z, w_{i}) \\ - \sum_{i \in I} τ_{i}^{L} L_{i} (y, w_{i}) \geq 〈 \sum_{t \in T} ν_{t}^{Φ} ξ_{t}^{1} + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} ξ_{j}^{2} + \sum_{s \in S} [ν_{s}^{Θ} ξ_{s}^{3} + τ_{s}^{Θ} ξ_{s}^{4}] \\ + \sum_{i \in I} ν_{i}^{K} ξ_{i}^{5} + \sum_{i \in I} [τ_{i}^{K} ξ_{i}^{6} + τ_{i}^{L} ξ_{i}^{7}], z - y 〉 . \end{matrix}

From the feasibility conditions of

y \in F_{M W}

, there exist

{\bar{ξ}}_{t}^{1} \in c o \partial^{*} Φ_{t} (y),

t \in T

{\bar{ξ}}_{j}^{2} \in c o \partial^{*} Ψ_{j} (y, w_{j}),

j \in I_{Ψ}

{\bar{ξ}}_{s}^{3} \in c o \partial^{*} Θ_{s} (y, w_{s}),

s \in S

{\bar{ξ}}_{s}^{4} \in c o \partial^{*} (- Θ_{s}) (y, w_{s}),

s \in S

{\bar{ξ}}_{i}^{5} \in c o \partial^{*} (- K_{i}) (y, w_{i}),

i \in I

{\bar{ξ}}_{i}^{6} \in c o \partial^{*} K_{i} (y, w_{i}),

i \in I

and

{\bar{ξ}}_{i}^{7} \in c o \partial^{*} L_{i} (y, w_{i}),

i \in I

such that

\sum_{t \in T} ν_{t}^{Φ} {\bar{ξ}}_{t}^{1} + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} {\bar{ξ}}_{j}^{2} + \sum_{s \in S} [ν_{s}^{Θ} {\bar{ξ}}_{s}^{3} + τ_{s}^{Θ} {\bar{ξ}}_{s}^{4}] + \sum_{i \in I} ν_{i}^{K} {\bar{ξ}}_{i}^{5} + \sum_{i \in I} [τ_{i}^{K} {\bar{ξ}}_{i}^{5} + τ_{i}^{L} {\bar{ξ}}_{i}^{6}] = 0 .

Therefore,

\begin{matrix} \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) & - \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) + \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (z, w_{j}) - \sum_{j \in I_{Ψ}} ν_{j}^{Ψ} Ψ_{j} (y, w_{j}) \\ + \sum_{s \in S} ν_{s}^{Θ} Θ_{s} (z, w_{s}) - \sum_{s \in S} ν_{s}^{Θ} Θ_{i} (y, w_{s}) + \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (z, w_{s}) \\ - \sum_{s \in S} τ_{s}^{Θ} (- Θ_{s}) (y, w_{s}) + \sum_{i \in I} ν_{i}^{K} (- K_{i}) (z, w_{i}) - \sum_{i \in I} ν_{i}^{K} (- K_{i}) (y, w_{i}) \\ + \sum_{i \in I} τ_{i}^{K} K_{i} (z, w_{i}) - \sum_{i \in I} τ_{i}^{K} K_{i} (y, w_{i}) + \sum_{i \in I} τ_{i}^{L} L_{i} (z, w_{i}) \\ - \sum_{i \in I} τ_{i}^{L} L_{i} (y, w_{i}) \geq 0 . \end{matrix}

Since

z \in F

, we have

Ψ_{j} (z, w_{j}) \leq 0,

j \in I_{Ψ},

Θ_{s} (z, w_{s}) = 0,

s \in S,

- K_{i} (z, w_{i}) < 0,

i \in I_{+},

K_{i} (z, w_{i}) = 0,

i \in I_{0},

L_{i} (z, w_{i}) > 0,

i \in I_{0 +},

L_{i} (z, w_{i}) = 0,

i \in I_{00} \cup I_{+ 0},

L_{i} (z, w_{i}) = 0,

i \in I_{0 -} \cup I_{+ -} .

Since

y \in F_{M W},

we have

Ψ_{j} (y, w_{j}) \geq 0,

j \in I_{Ψ},

Θ_{s} (y, w_{s}) = 0,

s \in S,

- K_{i} (y, w_{i}) > 0,

i \in I_{+},

K_{i} (y, w_{i}) = 0,

i \in I_{0},

L_{i} (y, w_{i}) < 0,

i \in I_{0 +},

L_{i} (y, w_{i}) = 0,

i \in I_{00} \cup I_{+ 0},

L_{i} (y, w_{i}) = 0,

i \in I_{0 -} \cup I_{+ -} .

Hence,

\sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) \geq \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) .

Hence, from (53), we arrive at a contradiction. This completes the proof of the weak duality theorem. □

In the subsequent theorem, we establish the strong duality theorem for WRD relating to the primal problem UNMPVC.

Theorem 9

(Strong Duality). Let

\bar{z}

be a local robust Pareto solution UNMPVC. Let at

\bar{z}

Φ_{t}, t \in T, Ψ_{j}, j \in I_{Ψ}, \pm Θ_{s}, s \in S, - K_{i}, i \in I_{0 +} \cup I_{0 -} \cup I_{00},

K_{i}, i \in I_{0 +}

and

L_{i}, i \in I_{+ 0}

admit bounded USRC and are

\partial^{*}

-convex functions at

\bar{z}

. Suppose that at

\bar{z}

, GS-ACQ holds. Then there exists

\bar{ν} = ({\bar{ν}}^{Φ}, {\bar{ν}}^{Ψ}, {\bar{ν}}^{Θ}, {\bar{τ}}^{Θ}, {\bar{ν}}^{K}, {\bar{τ}}^{K}, {\bar{τ}}^{L}) \in R^{r} \times R^{p} \times R^{2 l} \times R^{3 m}

such that

(\bar{z}, \bar{ν})

is a robust local weak solution of MWRD. Moreover, the corresponding objective values are equal.

Proof.

Since

\bar{z}

is a locally Pareto solution of UNMPVC and GS-ACQ holds at

\bar{z}

. Then from Theorem 3 there exists

\bar{ν} = ({\bar{ν}}^{Φ}, {\bar{ν}}^{Ψ}, {\bar{ν}}^{Θ}, {\bar{τ}}^{Θ}, {\bar{ν}}^{K}, {\bar{τ}}^{K}, {\bar{τ}}^{L}) \in R^{r} \times R^{p} \times R^{2 l} \times R^{3 m}

such that

\bar{z}

is a RS-Stationary point of UNMPVC. Thus, there exist

{\bar{ξ}}_{t}^{1} \in c o \partial^{*} Φ_{t} (\bar{z}),

t \in T,

{\bar{ξ}}_{j}^{2} \in c o \partial^{*} Ψ_{i} (\bar{z},

{\bar{w}}_{j}),

j \in I_{Ψ},

{\bar{w}}_{j} \in Ω_{j} (\bar{z})

{\bar{ξ}}_{s}^{3} \in c o \partial^{*} Θ_{s} (\bar{z}, {\bar{w}}_{s}),

{\bar{ξ}}_{s}^{4} \in c o \partial^{*} (- Θ_{s}) (\bar{z}, {\bar{w}}_{s}),

s \in S, {\bar{w}}_{s} \in Ω_{s} (\bar{z})

{\bar{ξ}}_{i}^{5} \in c o \partial^{*} (- K_{i}) (\bar{z}, {\bar{w}}_{i}),

i \in I

{\bar{ξ}}_{i}^{6} \in c o \partial^{*} K_{i} (\bar{z}, {\bar{w}}_{i}),

i \in I,

and

{\bar{ξ}}_{i}^{7} \in c o \partial^{*} L_{i} (\bar{z}, {\bar{w}}_{i}),

i \in I, {\bar{w}}_{i} \in Ω_{i} (\bar{z})

such that

\sum_{t \in T} {\bar{ν}}_{t}^{Φ} {\bar{ξ}}_{t}^{1} + \sum_{j \in I_{Ψ}} {\bar{ν}}_{j}^{Ψ} {\bar{ξ}}_{j}^{2} + \sum_{s \in S} [{\bar{ν}}_{s}^{Θ} {\bar{ξ}}_{s}^{3} + {\bar{τ}}_{s}^{Θ} {\bar{ξ}}_{s}^{4}] + \sum_{i \in I} {\bar{ν}}_{i}^{K} {\bar{ξ}}_{i}^{5} + \sum_{i \in I} [{\bar{τ}}_{i}^{K} {\bar{ξ}}_{i}^{6} + {\bar{τ}}_{i}^{L} {\bar{ξ}}_{i}^{7}] = 0,

\begin{matrix} {\bar{ν}}_{t}^{Φ} > 0, t \in T, {\bar{ν}}_{j}^{Ψ} \geq 0, j \in I_{Ψ}, {\bar{ν}}_{s}^{Θ}, {\bar{τ}}_{s}^{Θ} \geq 0, s \in S, {\bar{ν}}_{i}^{K}, {\bar{τ}}_{i}^{K}, {\bar{τ}}_{i}^{L} \geq 0, i \in I . \end{matrix}

Therefore,

(\bar{z}, \bar{ν}) \in F_{M W}

. From Theorem 8, for any feasible solution

(y, ν)

of MWRD we have

\begin{matrix} Φ (\bar{z}) \neg ≺ Φ (y) . \end{matrix}

(61)

Hence,

(\bar{z}, \bar{ν})

is a robust local weak solution of MWRD. Moreover, the respective objective values are equal. This completes the proof. □

Now, we establish the strict converse duality theorem relating WRD and UNMPVC.

Theorem 10

(Strict Converse Duality). Suppose that

\bar{z}

is a robust local weak Pareto solution of UNMPVC and

(\hat{y}, \hat{ν})

is a robust global weak Pareto solution of MWRD. If the assumptions of the strong duality theorem hold and

Φ_{t}, t \in T

is strictly

\partial^{*}

-convex at

\hat{y}

we have

\bar{z} = \hat{y}

Proof.

Let us assume that

\bar{z} \neq \hat{y}

. By Theorem 9 there exists

\bar{ν} = ({\bar{ν}}^{Φ}, {\bar{ν}}^{Ψ}, {\bar{ν}}^{Θ}, {\bar{τ}}^{Θ}, {\bar{ν}}^{K}, {\bar{τ}}^{K}, {\bar{τ}}^{L}) \in R^{r} \times R^{p} \times R^{2 l} \times R^{3 m},

such that

(\bar{z}, \bar{ν}) \in F_{M W}

and

Φ (\bar{z}) = Φ (\hat{y}) .

(62)

Multiplying (62) by

ν_{t}^{Φ}, t \in T

we get

\sum_{t \in T} ν_{t}^{Φ} Φ_{t} (z) = \sum_{t \in T} ν_{t}^{Φ} Φ_{t} (y) .

(63)

Since

Φ

is strictly

\partial^{*}

-convex at

\hat{y}

, therefore

Φ_{t} (\bar{z}) - Φ_{t} (\hat{y}) > 〈 ξ_{t}^{1}, \bar{z} - \hat{y} 〉, \forall ξ_{t}^{1} \in \partial^{*} Φ (\hat{y}), t \in T .

(64)

Similarly, by the

\partial^{*}

-convexity of

Ψ_{j},

j \in I_{Ψ},

\pm Θ_{s},

s \in S,

(- K_{i}),

i \in I_{0 +} \cup I_{0 -} \cup I_{00},

K_{i},

i \in I_{0 +},

L_{i},

i \in I_{+ 0}

\hat{y}

we have

Ψ_{j} (\bar{z}, w_{j}) - Ψ_{j} (\hat{y}, w_{j}) \geq 〈 ξ_{j}^{2}, \bar{z} - \hat{y} 〉, \forall ξ_{j}^{2} \in \partial^{*} Ψ_{j} (\hat{y},, w_{j}), \forall j \in I_{Ψ},

(65)

Θ_{s} (\bar{z}, w_{s}) - Θ_{s} (\hat{y}, w_{s}) \geq 〈 ξ_{s}^{3}, \bar{z} - \hat{y} 〉, \forall ξ_{s}^{3} \in \partial^{*} Θ_{s} (\hat{y}, w_{s}), \forall s \in S,

(66)

\begin{matrix} (- Θ_{s}) (\bar{z}, w_{s}) - (- Θ_{s}) (\hat{y}, w_{s}) \geq 〈 ξ_{s}^{4}, \bar{z} - \hat{y} 〉, \forall ξ_{s}^{4} \in \partial^{*} & (- Θ_{s}) (\hat{y}, w_{s}), \\ \forall s \in S, \end{matrix}

(67)

\begin{matrix} (- K_{i}) (\bar{z}, w_{i}) - (- K_{i}, w_{i}) (\hat{y}, w_{i}) \geq 〈 ξ_{i}^{5}, \bar{z} - \hat{y} 〉, & \forall ξ_{i}^{5} \in \partial^{*} (- K_{i}) (\hat{y}, w_{i}), \\ \forall i \in I_{0 +} \cup I_{0 -} \cup I_{00}, \end{matrix}

(68)

K_{i} (\bar{z}, w_{i}) - K_{i} (\hat{y}, w_{i}) \geq 〈 ξ_{i}^{6}, \bar{z} - \hat{y} 〉, \forall ξ_{i}^{6} \in \partial^{*} L_{i} (\hat{y}, w_{i}), \forall i \in I_{0 +} .

(69)

L_{i} (\bar{z}, w_{i}) - L_{i} (\hat{y}, w_{i}) \geq 〈 ξ_{i}^{7}, \bar{z} - \hat{y} 〉, \forall ξ_{i}^{7} \in \partial^{*} L_{i} (\hat{y}, w_{i}), \forall i \in I_{+ 0} .

(70)

I_{0 -}^{K} \cup I_{0 +}^{K} \cup I_{+ 0}^{L} = \emptyset

, then multiplying (64)-(70) by

{\hat{ν}}_{t}^{Φ} > 0, t \in T

{\hat{ν}}_{j}^{Ψ} \geq 0, j \in I_{Ψ}

{\hat{ν}}_{s}^{Θ} \geq 0, s \in S

{\hat{τ}}_{s}^{Θ} \geq 0, s \in S

{\hat{ν}}_{i}^{K} > 0, i \in I_{0 +} \cup I_{0 -} \cup I_{00}

{\hat{τ}}_{i}^{K} > 0, i \in I_{0 +}

{\hat{τ}}_{i}^{L} > 0, i \in I_{+ 0}

, respectively and then adding them subsequently we get

\begin{matrix} \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\bar{z}) & - \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\hat{y}) + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\bar{z}, w_{j}) - \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\hat{y}, w_{j}) \\ + \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{s} (\bar{z}, w_{j}) - \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{s} (\hat{y}, w_{s}) + \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\bar{z}, w_{s}) \\ - \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\hat{y}, {\hat{w}}_{s}) + \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\hat{y}, w_{i}) \\ + \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\hat{y}, w_{i}) + \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\bar{z}, w_{i}) \\ - \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\hat{y}, w_{i}) > 〈 \sum_{t \in T} {\hat{ν}}_{t}^{Φ} ξ_{t}^{1} + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} ξ_{j}^{2} + \sum_{s \in S} [{\hat{ν}}_{s}^{Θ} ξ_{s}^{3} + {\hat{τ}}_{s}^{Θ} ξ_{s}^{4}] \\ + \sum_{i \in I} {\hat{ν}}_{i}^{K} ξ_{i}^{5} + \sum_{i \in I} [{\hat{τ}}_{i}^{K} ξ_{i}^{6} + {\hat{ν}}_{i}^{L} ξ_{i}^{7}], \bar{z} - \hat{y} 〉 . \end{matrix}

From the feasibility conditions of

\hat{y} \in F_{W R D}

, there exist

{\hat{ξ}}_{t}^{1} \in c o \partial^{*} Φ_{t} (\hat{y}),

t \in T

{\hat{ξ}}_{j}^{2} \in c o \partial^{*} Ψ_{j} (\hat{y}, w_{j}),

j \in I_{Ψ}

{\hat{ξ}}_{s}^{3} \in c o \partial^{*} Θ_{s} (\hat{y}, w_{s}),

s \in S

{\hat{ξ}}_{s}^{4} \in c o \partial^{*} (- Θ_{s}) (\hat{y}, w_{s}),

s \in S

{\hat{ξ}}_{i}^{5} \in c o \partial^{*} (- K_{i}) (\hat{y}, w_{i}),

i \in I

{\hat{ξ}}_{i}^{6} \in c o \partial^{*} K_{i} (\hat{y}, w_{i}), i \in I

and

{\hat{ξ}}_{i}^{7} \in c o \partial^{*} L_{i} (\hat{y}, w_{i}), i \in I

such that

\sum_{t \in T} {\hat{ν}}_{t}^{Φ} {\hat{ξ}}_{t}^{1} + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} {\hat{ξ}}_{j}^{2} + \sum_{s \in S} [{\hat{ν}}_{s}^{Θ} {\hat{ξ}}_{s}^{3} + {\hat{τ}}_{s}^{Θ} {\hat{ξ}}_{s}^{4}] + \sum_{i \in I} {\hat{ν}}_{i}^{K} {\hat{ξ}}_{i}^{5} + \sum_{i \in I} [{\hat{τ}}_{i}^{K} {\hat{ξ}}_{i}^{6} + {\hat{τ}}_{i}^{L} {\hat{ξ}}_{i}^{6}] = 0 .

Therefore,

\begin{matrix} \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\bar{z}) & - \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\hat{y}) + \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\bar{z}, w_{j}) - \sum_{j \in I_{Ψ}} {\hat{ν}}_{j}^{Ψ} Ψ_{j} (\hat{y}, w_{j}) \\ + \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{s} (\bar{z}, w_{s}) - \sum_{s \in S} {\hat{ν}}_{s}^{Θ} Θ_{i} (\hat{y}, w_{s}) + \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\bar{z}, w_{s}) \\ - \sum_{s \in S} {\hat{τ}}_{s}^{Θ} (- Θ_{s}) (\hat{y}, w_{s}) + \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{ν}}_{i}^{K} (- K_{i}) (\hat{y}, w_{i}) \\ + \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\bar{z}, w_{i}) - \sum_{i \in I} {\hat{τ}}_{i}^{K} K_{i} (\hat{y}, w_{i}) + \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\bar{z}, w_{i}) \\ - \sum_{i \in I} {\hat{τ}}_{i}^{L} L_{i} (\hat{y}, w_{i}) > 0 . \end{matrix}

Since

\bar{z} \in F

and

\hat{y} \in F_{W D}

we have

\begin{matrix} \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\bar{z}) > & \sum_{t \in T} {\hat{ν}}_{t}^{Φ} Φ_{t} (\hat{y}) . \end{matrix}

Hence, from (63), we get a contradiction. This completes the proof. □

Remark 4.

Theorem 8, Theorem 9 and Theorem 10 presented in this article for Mond-Weir-type robust dual model, generalize Theorem 8, Theorem 9 and Theorem 12, respectively derived in Mishra et al. [41] for a more general programming problem involving data uncertainty.

To illustrate the duality results established in this article for WRD and MWRD-type dual problems related to the primal problem UNMPVC, we present the following non-trivial example in the Euclidean space setting.

Example 3.

Let us examine the subsequent robust multiobjective programming problem with vanishing constraints under data uncertainty given by

\begin{matrix} P_{1} Minimize & Φ (z) = (Φ_{1} (z), Φ_{2} (z)) : = (e^{- z_{1}}, e^{- 2 z_{2}}), \\ subject to & Ψ_{1} (z, w_{1}) : = - | z_{1} | w_{1}^{2} \leq 0, \forall w_{1} \in Ω_{1} : = [1, \infty), \\ Ψ_{2} (z, w_{2}) : = z_{2} cos w_{2} \leq 0, \forall w_{2} \in Ω_{2} : = [0, \frac{π}{2}], \\ K (z, w_{3}) = - z_{1} sin w_{3} \geq 0, \forall w_{3} \in Ω_{3} : = [0, \frac{π}{2}], \\ L (z, w_{3}) K (z, w_{3}) = - z_{1} z_{2} sin w_{3} \leq 0, \forall w_{3} \in Ω_{3} : = [0, \frac{π}{2}], \end{matrix}

where

Φ_{t} : R^{2} \to \bar{R}, t \in {1, 2}

Ψ : R^{2} \times Ω_{1} \to \bar{R}

K (z, w_{2}) : R^{2} \times Ω_{2} \to \bar{R}

L (z, w_{2}) : R^{2} \times Ω_{2} \to \bar{R}

and

z = (z_{1}, z_{2}) \in R^{2} .

The set of all feasible solutions of

P_{1}

is defined as

\begin{matrix} F_{P_{1}} = {(z_{1}, z_{2}) \in R^{2} : & - | z_{1} | w_{1}^{2} \leq 0, z_{2} cos w_{2} \leq 0, - z_{1} sin w_{3} \geq 0, \\ - z_{1} z_{2} sin w_{3} \geq 0, w_{k} \in Ω_{k}, k \in {1, 2, 3}} . \end{matrix}

It is evident that

\begin{matrix} Ψ_{1}^{*} (z) = max_{w_{1} \in Ω_{1}} - | z_{1} | w_{1}^{2} = - | z_{1} |, \\ Ψ_{2}^{*} (z) = max_{w_{2} \in Ω_{2}} z_{2} cos w_{2} = z_{2}, \\ K^{*} (z) = max_{w_{3} \in Ω_{3}} - z_{1} sin w_{3} = - z_{1}, \\ L^{*} (z) = max_{w_{2} \in Ω_{2}} z_{2} = z_{2} . \end{matrix}

It can be verified that given any feasible solution to the primal problem, inequality (29) is satisfied. Hence, Theorem 5 holds for the considered problem

P_{1}

. Furthermore, GS-ACQ is satisfied at

\bar{z} = (0, 0)

and there exists

\bar{ν} = ({\bar{ν}}^{Φ}, {\bar{ν}}^{Ψ}, {\bar{ν}}^{K}, {\bar{τ}}^{K}, {\bar{τ}}^{L}) \in R^{r} \times R^{p} \times R^{3 m}

such that

(\bar{z}, \bar{v})

is a feasible solution of the dual problem. Hence, Theorem 6 holds for the considered problem

P_{1}

. Moreover, objective functions are strictly

\partial^{*}

-convex at

\bar{z}

and from Theorem 4,

\bar{z}

is a robust local weak Pareto solution of the primal problem

P_{1}

. It can be verified from the weak duality theorem that

\bar{z} = (0, 0)

is the global robust weak Pareto solution of the Wolfe-type dual problem of

P_{1}

. Hence, Theorem 7 is verified for the problem

P_{1}

Now corresponding to the primal problem

P_{1}

, the Mond-Weir-type dual problem can be formulated as

\begin{matrix} \underset{(y, ν)}{Maximize} Φ (y) = (e^{- z_{1}}, e^{- 2 z_{2}}), \\ subject to Ψ_{1} (z, w_{1}) : = - | z_{1} | w_{1}^{2} \geq 0, \forall w_{1} \in Ω_{1} : = [1, \infty), \\ Ψ_{2} (z, w_{2}) : = z_{2} cos w_{2} \geq 0, \forall w_{2} \in Ω_{2} : = [0, \frac{π}{2}], \\ K (z, w_{3}) = - z_{1} sin w_{3} \leq 0, \forall w_{3} \in Ω_{3} : = [0, \frac{π}{2}], \\ L (z, w_{3}) K (z, w_{3}) = - z_{1} z_{2} sin w_{3} \geq 0, \forall w_{3} \in Ω_{3} : = [0, \frac{π}{2}], \end{matrix}

Similar to the Wolfe-type robust dual model, it can be verified that given any feasible solution to the primal problem, inequality (51) is satisfied. Hence, Theorem 8 holds for the considered problem

P_{1}

. Moreover, since GS-ACQ is satisfied at

\bar{z} = (0, 0)

and there exists

\bar{ν} = ({\bar{ν}}^{Φ}, {\bar{ν}}^{Ψ}, {\bar{ν}}^{K}, {\bar{τ}}^{K}, {\bar{τ}}^{L}) \in R^{r} \times R^{p} \times R^{3 m}

such that

(\bar{z}, \bar{v})

is a feasible solution of the dual problem. Hence, Theorem 9 holds for the considered problem

P_{1}

. Moreover, objective functions are strictly

\partial^{*}

-convex at

\bar{z}

and from Theorem 4,

\bar{z}

is a robust local weak Pareto solution of the primal problem

P_{1}

. It can be verified from the weak duality theorem that

\bar{z} = (0, 0)

is the global robust weak Pareto solution of the Mond-Weir-type dual problem of

P_{1}

. Hence, Theorem 10 is verified for the problem

P_{1}

6. Conclusion and Future Directions

In this article, a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC) has been investigated. We introduced GS-ACQ for the considered problem UNMPVC. Moreover, we have introduced RW-stationary, RT-stationary, RM-stationary and RS-stationary conditions for the problem UNMPVC. By employing GS-ACQ, we have established that the RS-Stationary is the necessary first-order optimality condition for UNMPVC. Subsequently, under generalized convexity assumptions, we have derived sufficient optimality conditions for a feasible solution to be a robust local weak Pareto solution in UNMPVC. Furthermore, we have formulated WRD and MWRD dual models and derived duality theorems, namely, weak, strong and strict converse duality theorem that relates the primal problem with the corresponding dual models.

The optimality conditions and duality results established in this article extend several well-known results existing in the literature for nonsmooth multiobjective programming problems to a more general programming problem UNMPVC in terms of convexificators. In particular, the optimality conditions and duality results presented in this article extend the corresponding optimality conditions established in [37,60] and duality results established in [41] to a wider class of programming problem UNMPVC.

The results derived in this article leave various avenues for future research. In view of the work presented in this article, it would be interesting to introduce various other constraint qualifications and establish interrelationships among them for nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty.

Author Contributions:

B.B. Upadhyay: Supervision, Conceptualization, Validation; S.K. Singh: Writing, Validation, Methodology; I.M. Stancu-Minasian: Methodology, Formal analysis, Validation; A.M. Rusu-Stancu: Methodology, Formal analysis, Validation.

Conflicts of Interest

The authors declare that there is no actual or potential conflict of interest in relation to this article.

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Consent for Publication

All the authors have read and approved the final manuscript..

References

Branke, J.; Deb, K.; Miettinen, K.; Slowiński, R. Multiobjective Optimization: Interactive and Evolutionary Approaches; Springer: Berlin, Germany, 2008. [Google Scholar]
Miettinen, K. Nonlinear Multiobjective Optimization; Kluwer Academic Publishers: Boston, MA, USA, 1999. [Google Scholar]
Ghosh, A.; Upadhyay, B.B.; Stancu-Minasian, I.M. Constraint qualifications for multiobjective programming problems on Hadamard manifolds. Aust. J. Math. Anal. Appl. 2023, 20, Article 2, 17 pp.
Mishra, S.K.; Upadhyay, B.B. Efficiency and duality in nonsmooth multiobjective fractional programming involving η-pseudolinear functions. Yugosl. J. Oper. Res. 2012, 22, 3–18. [Google Scholar] [CrossRef]
Upadhyay, B.B.; Ghosh, A.; Mishra, P.; Treanţă, S. Optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds using generalized geodesic convexity. RAIRO Oper. Res. 2022, 56, 2037–2065. [Google Scholar] [CrossRef]
Upadhyay, B.B.; Stancu-Minasian, I.M.; Mishra, P. On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities. Optimization 2023, 72, 2635–2659. [Google Scholar] [CrossRef]
Upadhyay, B.B.; Treanţă, S.; Mishra, P. On Minty variational principle for nonsmooth multiobjective optimization problems on Hadamard manifolds. Optimization 2023, 72, 3081–3100. [Google Scholar] [CrossRef]
Upadhyay, B.B.; Mishra, P. On interval-valued multiobjective programming problems and vector variational-like inequalities using limiting subdifferential. In Computational Management, Patnaik, S.; Tajeddini, K., Jain, V., Eds.; Springer: Cham, Switzerland, 2021; pp. 325–343. [Google Scholar]
Mangasarian, O.L. Nonlinear Programming; McGraw-Hill: New York, USA, 1969. [Google Scholar]
De Finetti, B. Sulle stratificazioni convesse. Ann. Mat. Pura Appl. 1949, 30, 173–183. [Google Scholar] [CrossRef]
Fenchel, W. Convex Cones, Sets, and Functions; Lecture notes, Princeton University: NJ, USA 1953. [Google Scholar]
Cambini, A.; Martein, L. Generalized Convexity and Optimization: Theory and Applications; Springer: Berlin, Germany, 2009. [Google Scholar]
Dutta, J.; Chandra, S. Convexifactors, generalized convexity, and optimality conditions. J. Optim. Theory Appl. 2002, 113, 41–64. [Google Scholar] [CrossRef]
Dutta, J.; Chandra, S. Convexifactors, generalized convexity and vector optimization. Optimization 2004, 53, 77–94. [Google Scholar] [CrossRef]
Upadhyay, B.B.; Mishra, S.K. Nonsmooth semi-infinite minmax programming involving generalized (Φ,ρ)-invexity. J. Syst. Sci. Complex. 2015, 28, 857–875. [Google Scholar] [CrossRef]
Clarke, F.H. Optimization and Nonsmooth Analysis, 2nd ed.; SIAM: Philadelphia, PA, USA, 1990. [Google Scholar]
Demyanov, V.F.; Jeyakumar, V. Hunting for a smaller convex subdifferential. J. Global Optim. 1997, 10, 305–326. [Google Scholar] [CrossRef]
Ioffe, A.D. Approximate subdifferentials and applications. II. Mathematika 1986, 33, 111–128. [Google Scholar] [CrossRef]
Michel, P.; Penot, J.-P. A generalized derivative for calm and stable functions. Differ. Integral Equ. 1992, 5, 433–454. [Google Scholar] [CrossRef]
Mordukhovich, B.S.; Shao, Y. On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 1995, 2, 211–227. [Google Scholar]
Treiman, J.S. The linear nonconvex generalized gradient and Lagrange multipliers. SIAM J. Optim. 1995, 5, 670–680. [Google Scholar] [CrossRef]
Demyanov, V.F. Convexification and concavification of positively homogeneous functions by the same family of linear functions. Technical Report; University of Pisa: Italy, 1994; pp. 1–11. [Google Scholar]
Jeyakumar, V.; Luc, D.T. Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 1999, 101, 599–621. [Google Scholar] [CrossRef]
Luu, D.V. Convexificators and necessary conditions for efficiency. Optimization 2014, 63, 321–335. [Google Scholar] [CrossRef]
Luu, D.V. Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 2014, 160, 510–526. [Google Scholar]
Rimpi; Lalitha, C. S. Constraint qualifications in terms of convexificators for nonsmooth programming problems with mixed constraints. Optimization 2023, 72, 2019–2038. [Google Scholar] [CrossRef]
Upadhyay, B.B.; Mishra, P.; Mohapatra, R.N.; Mishra, S.K. On the applications of nonsmooth vector optimization problems to solve generalized vector variational inequalities using convexificators. In Optimization of Complex Systems: Theory, Models, Algorithms and Applications; Le Thi, H., Le, H., Pham Dinh, T., Eds.; Springer: Cham, Switzerland, 2020; pp. 660–671. [Google Scholar]
Upadhyay, B.B.; Singh, S.K. Optimality and duality for nonsmooth semidefinite multiobjective fractional programming problems using convexificators. UPB Sci. Bull. Ser. A 2023, 85, 79–90. [Google Scholar]
Achtziger, W.; Kanzow, C. Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math. Program. 2008, 114, 69–99. [Google Scholar] [CrossRef]
Jabr, R.A. Solution to Economic Dispatching with Disjoint Feasible Regions via Semidefinite Programming. IEEE Trans. Power Syst. 2012, 27, 572–573. [Google Scholar] [CrossRef]
Jung, M.N.; Kirches, C.; Sager, S. On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control. In Facets of Combinatorial Optimization; Jünger, M., Reinelt, G., Eds.; Springer: Heidelberg, Berlin, 2013; pp. 387–417. [Google Scholar]
Palagachev, K.; Gerdts, M. Mathematical Programs with Blocks of Vanishing Constraints Arising in Discretized Mixed-Integer Optimal Control Problems. Set-Valued Var. Anal. 2015, 23, 149–167.
Hoheisel, T.; Kanzow, C. First- and second-order optimality conditions for mathematical programs with vanishing constraints. Appl. Math. 2007, 52, 495–514. [Google Scholar] [CrossRef]
Hoheisel, T.; Kanzow, C. Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications. J. Math. Anal. Appl. 2008, 337, 292–310. [Google Scholar] [CrossRef]
Hoheisel, T.; Kanzow, C. On the Abadie and Guignard constraint qualifications for mathematical programmes with vanishing constraints. Optimization 2009, 58, 431–448. [Google Scholar] [CrossRef]
Hu, Q.; Wang, J.; Chen, Y. New dualities for mathematical programs with vanishing constraints. Ann. Oper. Res. 2020, 287, 233–255. [Google Scholar] [CrossRef]
Hu, Q.J.; Zhou, Z.J.; Chen, Y. Some Convexificators-Based Optimality Conditions for Nonsmooth Mathematical Program with Vanishing Constraints. Am. J. Oper. Res. 2021, 11, 324–337. [Google Scholar] [CrossRef]
Kazemi, S. , Kanzi, N.: Constraint qualifications and stationary conditions for mathematical programming with non-differentiable vanishing constraints. J. Optim. Theory Appl. 179(3), 800-819 (2018).
Kazemi, S.; Kanzi, N.; Ebadian, A. Estimating the Fréchet normal cone in optimization problems with nonsmooth vanishing constraints. Iran. J. Sci. Technol. Trans. A Sci. 2019, 43, 2299–2306. [Google Scholar] [CrossRef]
Lai, K.K.; Hassan, M.; Singh, S.K.; Maurya, J.K.; Mishra, S.K. Semidefinite multiobjective mathematical programming problems with vanishing constraints using convexificators. Fractal Fract. 2022, 6, Paper 3, 19 pages.
Mishra, S.K.; Singh, V.; Laha, V. On duality for mathematical programs with vanishing constraints. Ann. Oper. Res. 2016, 243, 249–272. [Google Scholar] [CrossRef]
Sadeghieh, A.; Kanzi, N.; Caristi, G. On stationarity for nonsmooth multiobjective problems with vanishing constraints. J. Glob. Optim. 2022, 82, 929–949. [Google Scholar] [CrossRef]
Shaker Ardakani, J.; Farahmand Rad, S.H.; Kanzi, N.; Reihani Ardabili, P. Necessary stationary conditions for multiobjective optimization problems with nondifferentiable convex vanishing constraints. Iran. J. Sci. Technol. Trans. A Sci. 2019, 43, 2913–2919. [Google Scholar] [CrossRef]
Ben-Tal, A.; El-Ghaoui, L.; Nemirovski, A. Robust Optimization; Princeton University Press: Princeton, NJ, USA, 2009. [Google Scholar]
Bertsimas, D.; Brown, D.; Caramanis, C. Theory and applications of robust optimization. SIAM Rev. 2011, 53, 464–501. [Google Scholar] [CrossRef]
Goh, J.; Sim, M. Robust optimization made easy with ROME. Oper. Res. 2011, 59, 973–985. [Google Scholar] [CrossRef]
Löfberg, J. Automatic robust convex programming. Optim. Methods Softw. 2012, 27, 115–129. [Google Scholar] [CrossRef]
Ben-Tal, A.; Nemirovski, A. Selected topics in robust convex optimization. Math. Program. 2008, 112, 125–158. [Google Scholar] [CrossRef]
Chuong, T.D. Optimality and duality for robust multiobjective optimization problems. Nonlinear Anal. 2016, 134, 127–143. [Google Scholar] [CrossRef]
Chuong, T.D. Robust optimality and duality in multiobjective optimization problems under data uncertainty. SIAM J. Optim. 2020, 30, 1501–1526. [Google Scholar] [CrossRef]
Goberna, M.A.; Jeyakumar, V.; Li, G.; Lopez, M.A. Robust linear semi-infinite programming duality under uncertainty. Math. Program. 2013, 139, 185–203. [Google Scholar] [CrossRef]
Goberna, M.A.; Jeyakumar, V.; Li, G.; Vicente-Pérez, J. Robust solutions to multi-objective linear programs with uncertain data. European J. Oper. Res. 2015, 242, 730–743. [Google Scholar] [CrossRef]
Jeyakumar, V.; Li, G.Y. Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 2010, 20, 3384–3407. [Google Scholar] [CrossRef]
Jeyakumar, V.; Li, G.; Lee, G.M. Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Anal. 2012, 75, 1362–1373. [Google Scholar] [CrossRef]
Lee, J.H.; Lee, G.M. On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems. Ann. Oper. Res. 2018, 269, 419–438. [Google Scholar] [CrossRef]
Chen, J.W.; Köbis, E.; Yao, J.C. Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 2019, 181, 411–436. [Google Scholar] [CrossRef]
Fakhar, M.; Mahyarinia, M.R.; Zafarani, J. On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization. European J. Oper. Res. 2018, 265, 39–48. [Google Scholar] [CrossRef]
Lee, G.M.; Lee, J.H. On nonsmooth optimality theorems for robust multiobjective optimization problems. J. Nonlinear Convex Anal. 2015, 16, 2039–2052. [Google Scholar]
Gadhi, N.A.; Ohda, M. Necessary optimality conditions for robust nonsmooth multiobjective optimization problems. Control Cybern. 2022, 51, 289–302. [Google Scholar] [CrossRef]
Chen, J.W.; Yang, R.; Köbis, E.; Ou, X. Convexificators for nonconvex multiobjective optimization problems with uncertain data: robust optimality and duality. Optimization, (2023). [CrossRef]
Golestani, M.; Nobakhtian, S. Nonsmooth multiobjective programming and constraint qualifications. Optimization 2013, 62, 783–795. [Google Scholar] [CrossRef]
Pandey, Y.; Mishra, S.K. Duality for Nonsmooth Optimization Problems with Equilibrium Constraints, Using Convexificators. J. Optim. Theory Appl. 2016, 171, 694–707. [Google Scholar] [CrossRef]
Costea, N.; Kristály, A.; Varga, C. Locally Lipschitz functionals. In Variational and Monotonicity Methods in Nonsmooth Analysis; Birkhäuser: Cham, Switzerland, 2021; pp. 21–51. [Google Scholar]

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Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints under Data Uncertainty

Abstract

1. Introduction

2. Preliminaries and Notations

3. Optimality Criteria

4. Wolfe-type Robust Dual Model

5. Mond-Weir-Type Robust Dual Model

6. Conclusion and Future Directions

Author Contributions:

Conflicts of Interest

Data Availability Statement

Consent for Publication

References

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