On Duality Principles and Related Convex Dual Formulations Suitable for Local and Global Non-Convex Variational Optimization

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On Duality Principles and Related Convex Dual Formulations Suitable for Local and Global Non-Convex Variational Optimization

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A peer-reviewed article of this preprint also exists.

Fabio Botelho^*

This version is not peer-reviewed

Submitted:

02 June 2023

Posted:

06 June 2023

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Abstract

This article develops duality principles and related convex dual formulations suitable for the local and global optimization of non-convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg-Landau type equation.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

1. Introduction

In this article we establish a duality principle and a related convex dual formulation suitable for the local optimization of the primal formulation for a large class of models in non-convex optimization.

The main duality principle is applied to the Ginzburg-Landau system in superconductivity in the absence of a magnetic field.

Such results are based on the works of J.J. Telega and W.R. Bielski [2,3,13,14] and on a D.C. optimization approach developed in Toland [15].

About the other references, details on the Sobolev spaces involved are found in [1]. Related results on convex analysis and duality theory are addressed in [5,6,7,9,12]. Finally, similar models on the superconductivity physics may be found in [4,11].

Remark 1.1.

It is worth highlighting, we may generically denote

\int_{Ω} [{(- γ \nabla^{2} + K I_{d})}^{- 1} v^{*}] v^{*} d x

simply by

\int_{Ω} \frac{{(v^{*})}^{2}}{- γ \nabla^{2} + K} d x,

where

I_{d}

denotes a concerning identity operator.

Other similar notations may be used along this text as their indicated meaning are sufficiently clear.

Finally,

\nabla^{2}

denotes the Laplace operator and for real constants

K_{2} > 0

and

K_{1} > 0

, the notation

K_{2} ≫ K_{1}

means that

K_{2} > 0

is much larger than

K_{1} > 0 .

At this point we start to describe the primal and dual variational formulations.

Let

Ω \subset R^{3}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

For the primal formulation, consider a functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}} . \end{matrix}

(1)

Here

γ > 0,

α > 0

β > 0

and

f \in L^{2} (Ω) \cap L^{\infty} (Ω) .

Moreover,

V = W_{0}^{1, 2} (Ω)

and we denote

Y = Y^{*} = L^{2} (Ω) .

At this point we define

F_{1} : V \times Y \to R

F_{2} : V \to R

and

G : V \times Y \to R

\begin{matrix} F_{1} (u, v_{0}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - \frac{K}{2} \int_{Ω} u^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x + \frac{K_{2}}{2} \int_{Ω} u^{2} d x, \end{matrix}

(2)

F_{2} (u) = \frac{K_{2}}{2} \int_{Ω} u^{2} d x + {〈 u, f 〉}_{L^{2}},

and

G (u, v) = \frac{α}{2} \int_{Ω} {(u^{2} - β + v)}^{2} d x + \frac{K}{2} \int_{Ω} u^{2} d x .

We define also

J_{1} (u, v_{0}^{*}) = F_{1} (u, v_{0}^{*}) - F_{2} (u) + G (u, 0),

J (u) = \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}},

and

F_{1}^{*} : {[Y^{*}]}^{3} \to R,

F_{2}^{*} : Y^{*} \to R,

and

G^{*} : {[Y^{*}]}^{2} \to R,

\begin{matrix} F_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*}) \\ = & sup_{u \in V} {{〈 u, v_{1}^{*} + v_{2}^{*} 〉}_{L^{2}} - F_{1} (u, v_{0}^{*})} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} + v_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f)}^{2}}{(- γ \nabla^{2} - K + K_{2} + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2})} d x \\ - \frac{K_{1}}{2} \int_{Ω} f^{2} d x, \end{matrix}

(3)

\begin{matrix} F_{2}^{*} (v_{2}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u)} \\ = & \frac{1}{2 K_{2}} \int_{Ω} {(v_{2}^{*} - f)}^{2} d x, \end{matrix}

(4)

and

\begin{matrix} G^{*} (v_{1}^{*}, v_{0}^{*}) & = & sup_{(u, v) \in V \times Y} {{〈 u, v_{1}^{*} 〉}_{L^{2}} - {〈 v, v_{0}^{*} 〉}_{L^{2}} - G (u, v)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{2 v_{0}^{*} + K} d x + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x \\ + β \int_{Ω} v_{0}^{*} d x \end{matrix}

(5)

v_{0}^{*} \in B^{*}

where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2} .

Define also

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

A^{+} = {u \in V : u f \geq 0 a . e . in Ω},

V_{1} = V_{2} \cap A^{+},

B_{2}^{*} = {v_{0}^{*} \in Y^{*} : - γ \nabla^{2} - K + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2} > 0},

D_{3}^{*} = {(v_{1}^{*}, v_{2}^{*}) \in Y^{*} \times Y^{*} : - 1 / α + 4 K_{1} [u {(v_{1}^{*}, v_{2}^{*}, v_{0}^{*})}^{2}] + 100 / K_{2} \leq 0, \forall v_{0}^{*} \in B^{*}},

where

u (v_{2}^{*}, v_{0}^{*}) = \frac{φ_{1}}{φ},

φ_{1} = (v_{1}^{*} + v_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f)

and

φ = (- γ \nabla^{2} - K + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2} + K_{2}),

D^{*} = {v_{2}^{*} \in Y^{*}; ∥ v_{2}^{*} ∥_{\infty} < K_{4}}

E^{*} = {v_{1}^{*} \in Y^{*} : ∥ v_{1}^{*} ∥_{\infty} \leq K_{5}},

for some

K_{3}, K_{4}, K_{5} > 0

to be specified,

Finally, we also define

J_{1}^{*} : {[Y^{*}]}^{2} \times B^{*} \to R,

J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*}) = - F_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*}) + F_{2}^{*} (v_{2}^{*}) - G^{*} (v_{1}^{*}, v_{0}^{*}) .

Assume now

K_{1} = 1 / [4 (α + ε) K_{3}^{2}]

K_{2} ≫ K_{1} ≫ max {K_{3}, K_{4}, K_{5}, 1, γ, α, β} .

Observe that, by direct computation, we may obtain

\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} = - \frac{1}{α} + 4 K_{1} u {(v^{*})}^{2} + O (1 / K_{2}) < 0,

for

v_{0}^{*} \in B_{3}^{*}

Considering such statements and definitions, we may prove the following theorem.

Theorem 1.2.

Let

({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \in ((D^{*} \times E^{*}) \cap D_{3}^{*}) \times (B_{2}^{*} \cap B^{*})

be such that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0

and

u_{0} \in V_{1},

where

u_{0} = \frac{{\hat{v}}_{1}^{*} + {\hat{v}}_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f}{K_{2} - K - γ \nabla^{2} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}} .

Under such hypotheses, we have

δ J (u_{0}) = 0,

so that

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} \\ = & inf_{v_{2}^{*} \in D^{*}} \{sup_{(v_{1}^{*}, v_{0}^{*}) \in E^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(6)

Proof.

Observe that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0

so that, since

({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}) \in D_{3}^{*}

{\hat{v}}_{0}^{*} \in B_{2}^{*}

and

J_{1}^{*}

is quadratic in

v_{2}^{*}

, we may infer that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) & = & inf_{v_{2}^{*} \in Y^{*}} J_{1}^{*} (v_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & sup_{(v_{1}^{*}, v_{0}^{*}) \in E^{*} \times B^{*}} J_{1}^{*} ({\hat{v}}_{2}^{*}, v_{1}^{*}, v_{0}^{*}) . \end{matrix}

(7)

Therefore, from a standard saddle point theorem, we have that

J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = inf_{v_{2}^{*} \in Y^{*}} \{sup_{(v_{1}^{*}, v_{0}^{*}) \in E^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})\} .

Now we are going to show that

δ J (u_{0}) = 0 .

From

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = 0,

we have

- u_{0} + \frac{{\hat{v}}_{2}^{*}}{K_{2}} = 0,

and thus

{\hat{v}}_{2}^{*} = K_{2} u_{0} .

From

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{1}^{*}} = 0,

we obtain

- u_{0} - \frac{{\hat{v}}_{1}^{*} - f}{2 {\hat{v}}_{0}^{*} + K} = 0,

and thus

{\hat{v}}_{1}^{*} = - 2 {\hat{v}}_{0}^{*} u_{0} - K u_{0} + f .

Finally, denoting

D = - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f,

from

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{0}^{*}} = 0,

we have

- 2 D u_{0} + u_{0}^{2} - \frac{{\hat{v}}_{0}^{*}}{α} - β = 0,

so that

{\hat{v}}_{0}^{*} = α (u_{0}^{2} - β - 2 D u_{0}) .

(8)

Observe now that

{\hat{v}}_{1}^{*} + {\hat{v}}_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) f = (K_{2} - K - γ \nabla^{2} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}) u_{0}

so that

\begin{matrix} K_{2} u_{0} - 2 {\hat{v}}_{0} u_{0} - K u_{0} + f \\ = & K_{2} u_{0} - K u_{0} - γ \nabla^{2} u_{0} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) (- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f) . \end{matrix}

(9)

The solution for this last system of equations (8) and (9) is obtained through the relations

{\hat{v}}_{0}^{*} = α (u_{0}^{2} - β)

and

- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f = D = 0,

so that

δ J (u_{0}) = - γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f = 0

and

δ \{J (u_{0}) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f)}^{2} d x\} = 0 .

Moreover, from the Legendre transform properties

F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} + {\hat{v}}_{1}^{*} 〉}_{L^{2}} - F_{1} (u_{0}, {\hat{v}}_{0}^{*}),

F_{2}^{*} ({\hat{v}}_{2}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} 〉}_{L^{2}} - F_{2} (u_{0}),

G^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = - {〈 u_{0}, {\hat{v}}_{1}^{*} 〉}_{L^{2}} - {〈 0, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - G (u_{0}, 0),

so that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) & = & - F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) + F_{2}^{*} ({\hat{v}}_{2}^{*}) - G^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & F_{1} (u_{0}, {\hat{v}}_{0}^{*}) - F_{2} (u_{0}) + G (u_{0}, 0) \\ = & J (u_{0}) . \end{matrix}

(10)

Observe now that

\begin{matrix} J (u_{0}) & = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ \leq & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - \frac{K}{2} \int_{Ω} u^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + {\hat{v}}_{0}^{*} u - f)}^{2} d x + {〈 u, {\hat{v}}_{1}^{*} 〈}_{L^{2}} - {〈 u, f 〉}_{L^{2}} \\ - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{2 v_{0}^{*} + K} d x - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x \\ \leq & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - {〈 u, f 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + {\hat{v}}_{0}^{*} u - f)}^{2} d x \\ + sup_{(v_{1}^{*}, v_{0}^{*}) \in D^{*} \times B^{*}} \{+ {〈 u, {\hat{v}}_{1}^{*} 〉}_{L^{2}} - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{2 v_{0}^{*} + K} d x \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x\} \\ = & J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x, \end{matrix}

(11)

\forall u \in V_{1} .

Hence, we have got

J (u_{0}) = inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} .

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} \\ = & inf_{v_{2}^{*} \in Y^{*}} \{sup_{(v_{1}^{*}, v_{0}^{*}) \in E^{*} \times (B^{*} \cap B_{r} ({\hat{v}}_{0}^{*}))} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(12)

The proof is complete.

□

2. Another duality principle suitable for a local optimization of the primal formulation

In this section we develop a second duality principle which the dual formulation is concave.

We start by describing the primal formulation.

Let

Ω \subset R^{3}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

For the primal formulation, consider a functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}} . \end{matrix}

(13)

Here

γ > 0,

α > 0

β > 0

and

f \in L^{2} (Ω) \cap L^{\infty} (Ω) .

Moreover,

V = W_{0}^{1, 2} (Ω)

and we denote

Y = Y^{*} = L^{2} (Ω) .

Define the functionals

F_{1} : V \times Y \to R

F_{2} : V \to R

\begin{matrix} F_{1} (u, v_{0}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ - \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x + \frac{K_{2}}{2} \int_{Ω} {(\nabla^{2} u)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}} - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x \\ - β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(14)

and

F_{2} (u) = \frac{K_{2}}{2} \int_{Ω} {(\nabla^{2} u)}^{2} d x .

We define also

F_{1}^{*} : {[Y^{*}]}^{2} \to R

and

F_{2}^{*} : Y^{*} \to R,

\begin{matrix} F_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{1} (u, v_{0}^{*})} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{2}^{*} + f - K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f)}^{2}}{K_{2} \nabla^{4} - γ \nabla^{2} + 2 v_{0}^{*} - K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}} d x \\ + \frac{K_{1}}{2} \int_{Ω} f^{2} d x \end{matrix}

(15)

and,

\begin{matrix} F_{2}^{*} (v_{2}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u)} \\ = & \frac{1}{2 K_{2}} \int_{Ω} \frac{{(v_{2}^{*})}^{2}}{\nabla^{4}} d x . \end{matrix}

(16)

Here we denote

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ 2 v_{0}^{*} ∥_{\infty} < K / 2},

for an appropriate real constant

K > 0

Furthermore, we define

D^{*} = {v_{2}^{*} \in Y^{*} : ∥ v_{2}^{*} ∥_{\infty} \leq 5 K_{2} / 4}

and

J_{1}^{*} : D^{*} \times B^{*} \to R,

J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) = - F_{1}^{*} (v_{2}^{*}, v_{0}^{*}) + F_{2}^{*} (v_{2}^{*}) .

Assuming

0 < α ≪ 1

(through a re-scaling, if necessary) and

K_{2} ≫ K_{1} ≫ K ≫ {max {∥ f ∥}_{\infty}, α, β, γ, 1}

by directly computing

δ^{2} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})

we may easily obtain that for such specified real constants,

J_{1}^{*}

in concave in

(v_{2}^{*}, v_{0}^{*})

D^{*} \times B^{*} .

2.1. The main duality principle and a concerning convex dual formulation

Considering the statements and definitions presented in the previous section, we may prove the following theorem.

Theorem 2.1.

Let

({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \in D^{*} \times B^{*}

be such that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = 0

and

u_{0} \in V

be such that

u_{0} = \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*})}{\partial v_{2}^{*}} .

Under such hypotheses, we have

δ J (u_{0}) = 0,

and

\begin{matrix} J (u_{0}) & = & inf_{u \in V} \{J (u) + \frac{1}{2 K_{2}} \int_{Ω} {(\frac{{\hat{v}}_{2}^{*}}{- \nabla^{2}} - K_{2} (- \nabla^{2} u))}^{2} d x\} \\ = & sup_{(v_{2}^{*}, v_{0}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(17)

Proof.

Observe that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = 0

so that, since

J_{1}^{*}

is concave in

(v_{2}^{*}, v_{0}^{*})

D^{*} \times B^{*}

, we obtain

J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = sup_{(v_{2}^{*}, v_{0}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) .

Now we are going to show that

δ J (u_{0}) = 0 .

From

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = 0,

and

\frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*})}{\partial v_{2}^{*}} = u_{0}

we have

- \frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*}, v_{0}^{*})}{\partial v_{2}^{*}} + u_{0} = 0

and

{\hat{v}}_{2}^{*} - K_{2} \nabla^{4} u_{0} = 0 .

Observe now that denoting

H (v_{2}^{*}, v_{0}^{*}, u) = {〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{1} (u, v_{0}^{*}),

there exists

\hat{u} \in V

such that

\frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} = 0,

and

F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u}),

so that

\begin{matrix} \frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} & = & \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial v_{2}^{*}} \\ + \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{2}^{*}} \\ = & \hat{u} . \end{matrix}

(18)

Summarizing, we have got

u_{0} = \frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = \hat{u} .

Also, denoting

A (u_{0}, {\hat{v}}_{0}^{*}) = - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f,

from

\frac{\partial H ({\hat{v_{2}}}^{*}, {\hat{v}}_{0}^{*}, u_{0})}{\partial u} = 0,

we have

- (- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f - K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) A (u_{0}, {\hat{v}}_{0}^{*}) - {\hat{v}}_{2}^{*} + K_{2} \nabla^{4} u_{0}) = 0,

so that

A (u_{0}, {\hat{v}}_{0}^{*}) - K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) A (u_{0}, {\hat{v}}_{0}^{*}) = 0 .

(19)

From such results, we may infer that

A (u_{0}, {\hat{v}}_{0}^{*}) = - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} - f = 0, in Ω .

Moreover, from

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{0}^{*}} = 0,

we have

- K_{1} A (u_{0}, {\hat{v}}_{0}^{*}) 2 u_{0} - \frac{{\hat{v}}_{0}^{*}}{α} + u_{0}^{2} - β = 0,

so that

v_{0}^{*} = α (u_{0}^{2} - β) .

From such last results we get

- γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f = 0,

and thus

δ J (u_{0}) = 0 .

Furthermore, also from such last results and the Legendre transform properties, we have

F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} 〉}_{L^{2}} - F_{1} (u_{0}, {\hat{v}}_{0}^{*}),

F_{2}^{*} ({\hat{v}}_{2}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} 〉}_{L^{2}} - F_{2} (u_{0}),

so that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \\ = & - F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) + F_{2}^{*} ({\hat{v}}_{2}^{*}) \\ = & F_{1} (u_{0}, {\hat{v}}_{0}^{*}) - F_{2} (u_{0}) \\ = & J (u_{0}) . \end{matrix}

(20)

Finally, observe that

J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \leq - {〈 u, v_{2}^{*} 〉}_{L^{2}} + F_{1} (u, v_{0}^{*}) + F_{2}^{*} (v_{2}^{*}),

\forall u \in V, v_{2}^{*} \in D^{*}, v_{0}^{*} \in B^{*} .

Thus, we may obtain

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \\ \leq & - {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} \\ - \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x + F_{2} (u) + F_{2}^{*} ({\hat{v}}_{2}^{*}) - {〈 u, f 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {({\hat{v}}_{0}^{*})}^{2} d x - β \int_{Ω} {\hat{v}}_{0}^{*} d x \\ \leq & - {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} \\ + F_{2} (u) + F_{2}^{*} ({\hat{v}}_{2}^{*}) - {〈 u, f 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {({\hat{v}}_{0}^{*})}^{2} d x - β \int_{Ω} {\hat{v}}_{0}^{*} d x \\ \leq & sup_{v_{0}^{*} \in Y^{*}} \{- {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} \\ + F_{2} (u) + F_{2}^{*} ({\hat{v}}_{2}^{*}) - {〈 u, f 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x\} \\ = & J (u) + F_{2} (u) - {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + F_{2}^{*} ({\hat{v}}_{2}^{*}), \forall u \in V . \end{matrix}

(21)

Summarizing, we have got

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \leq J (u) + F_{2} (u) - {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + F_{2}^{*} ({\hat{v}}_{2}^{*}), \forall u \in V . \end{matrix}

(22)

Joining the pieces, from a concerning convexity in u, we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V} \{J (u) + \frac{1}{2 K_{2}} \int_{Ω} {(\frac{{\hat{v}}_{2}^{*}}{- \nabla^{2}} - K_{2} (- \nabla^{2} u))}^{2} d x\} \\ = & sup_{(v_{2}^{*}, v_{0}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(23)

The proof is complete.

□

3. A convex primal dual for a local optimization of the primal formulation

In this section we develop a convex primal dual formulation corresponding to a non-convex primal formulation.

We start by describing the primal formulation.

Let

Ω \subset R^{3}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

For the primal formulation, consider a functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}} . \end{matrix}

(24)

Here

γ > 0,

α > 0

β > 0

and

f \in L^{2} (Ω) \cap L^{\infty} (Ω) .

Moreover,

V = W_{0}^{1, 2} (Ω)

and we denote

Y = Y^{*} = L^{2} (Ω) .

Define the functional

J_{1}^{*} : V \times {[Y^{*}]}^{2} \to R

, by

\begin{matrix} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{3}^{*} u - f)}^{2} d x + \frac{K_{1}}{2} \int_{Ω} {(v_{3}^{*} - α (u^{2} - β))}^{2} d x \\ - {〈 u, f 〉}_{L^{2}} - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x \\ - β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(25)

We define also

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ 2 v_{0}^{*} ∥_{\infty} < K / 2},

for an appropriate real constant

K > 0

Furthermore, we define

D^{*} = {v_{3}^{*} \in Y^{*} : ∥ v_{3}^{*} ∥_{\infty} \leq K_{2}}

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}}

for an appropriate real constant

K_{3} > 0

and

V_{1} = A^{+} \cap V_{2} .

Now observe that denoting

φ_{1} = v_{3}^{*} - α (u^{2} - β)

, we have

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u^{2}} & = & K_{1} {(- γ \nabla^{2} + 2 v_{3}^{*})}^{2} + 4 K_{1} α^{2} u^{2} \\ - 2 K_{1} α φ_{1} - γ \nabla^{2} + 2 v_{0}^{*} \end{matrix}

(26)

and

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{3}^{*})}^{2}} & = & K_{1} + 4 K_{1} u^{2} . \end{matrix}

(27)

Denoting

φ = - γ \nabla^{2} u + 2 v_{0}^{*} u - f

we have also that

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u \partial v_{3}^{*}} & = & K_{1} (- 2 γ \nabla^{2} u + 2 v_{3}^{*} u) + 2 K_{1} φ - 2 K_{1} α u . \end{matrix}

(28)

In such a case, we obtain

\begin{matrix} det \{\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u \partial v_{3}^{*}}\} & = & \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{3}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}{\partial v_{3}^{*} \partial u})}^{2} \\ = & K_{1}^{2} {(- γ \nabla^{2} + 2 v_{3}^{*} + 4 α u^{2})}^{2} \\ + (- γ \nabla^{2} + 2 v_{0}^{*}) O (K_{1}) \\ - 4 K_{1}^{2} φ^{2} - 4 K_{1}^{2} φ [(- γ \nabla^{2} u + 2 v_{0}^{*} u) - 2 α u] \\ - 2 K_{1}^{2} α φ_{1} (1 + 4 u^{2}) . \end{matrix}

(29)

Observe that at a critical point

φ = 0

so that we may set the non-active restriction

C_{1}^{*} = {(u, v_{3}^{*}) \in V_{1} \times D^{*} : {(φ)}^{2} = {(γ \nabla^{2} u + 2 v_{3}^{*} u - f)}^{2} \leq ε u^{2}, in Ω}

for a small parameter

0 < ε ≪ 1

Now we are going to prove that

C_{1}^{*}

is a convex subset of

V_{1} \times D^{*}

For a

K_{7} > 0

observe that

φ^{2} \leq ε u^{2}

is equivalent to

φ^{2} + K_{7} u^{2} \leq (K_{7} + ε) u^{2},

which is equivalent to

\sqrt{φ^{2} + K_{7} u^{2}} - \sqrt{K_{7} + ε} | u | \leq 0, in Ω .

Define

H (u, v_{3}^{*}) = \sqrt{φ^{2} + K_{7} u^{2}} - \sqrt{K_{7} + ε} | u | .

Observe that since for

(u, v_{3}^{*}) \in V_{1} \times D^{*}

we have

u f \geq 0 in Ω

, we have also that

- \sqrt{K_{7} + ε} | u |

is convex on

V_{1} \times D^{*}

Moreover, for

K_{7} > 0

sufficiently large, the function

\sqrt{φ^{2} + K_{7} u^{2}}

is also convex on

V_{1} \times D^{*}

Summarizing,

H (u, v_{3}^{*})

is convex on

V_{1} \times D^{*}

so that from such results, we may infer that

C_{1}^{*}

is a convex set.

On the other hand, at a critical point we have also

φ_{1} = 0 .

Now define the non-active constraint

C_{2}^{*} = {(u, v_{3}^{*}) \in V_{1} \times D^{*} : φ_{1}^{2} = {(v_{3}^{*} - α (u^{2} - β))}^{2} \leq ε, in Ω} .

Similarly as it was made for

C_{1}^{*}

we may prove that

C_{2}^{*}

is convex in

V_{1} \times D^{*}

For a function (or indeed an operator or matrix of functions in a more general context)

M_{1}

to be specified define

C_{3}^{*} = {(u, v_{3}^{*}) \in V_{1} \times D^{*} : \sqrt{4 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} |} and 2 v_{3}^{*} + M_{1} \geq ε_{1}},

for some appropriate small constant

ε_{1} > 0 .

Since for

(u, v_{3}^{*}) \in V_{1} \times D^{*}

we have

u f \geq 0 in Ω

, it is clear that

C_{3}^{*}

is convex on

V_{1} \times D^{*} .

Observe that if

(u, v_{3}^{*}) \in C_{3}^{*}

, then

4 α u^{2} \geq M_{1} + γ \nabla^{2}

and

2 v_{3}^{*} + M_{1} \geq ε_{1}

so that

- γ \nabla^{2} + 2 v_{3}^{*} + 4 α u^{2} \geq ε_{1} .

At this point, we define the convex set

C^{*} = C_{1}^{*} \cap C_{2} \cap C_{3}^{*}

Finally, observe that for

0 < ε ≪ ε_{1} ≪ 1

, we have that

det {δ_{u v_{3}^{*}}^{2} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})}

is positive definite on

C^{*} \times B^{*}

From such results we may infer that

J_{1}^{*}

is convex in

(u, v_{3}^{*})

and concave in

v_{0}^{*}

C^{*} \times B^{*} .

With such results in mind, we may prove the following theorem.

Theorem 3.1.

Let

(u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) \in C^{*} \times B^{*}

be such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, we have

δ J (u_{0}) = - γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f = 0

and

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 α (u^{2} - β) u - f)}^{2} d x\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{3}^{*}) \in V_{1} \times D^{*}} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(30)

Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})

may be done similarly as in the previous sections.

Observe that

J_{1}^{*}

is convex in

(u, v_{3}^{*})

and concave in

v_{0}^{*}

C^{*} \times B^{*}

, where

C^{*}

and

B^{*}

are convex sets.

From such results and Min-Max Theorem, we may infer that

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{3}^{*}) \in V_{1} \times D^{*}} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})\} .

Finally, observe that

\begin{matrix} J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) & \leq & J_{1}^{*} (u, v_{3}^{*}, {\hat{v}}_{0}^{*}), \forall u \in V_{1}, v_{3}^{*} \in D^{*} . \end{matrix}

(31)

In particular for

v_{3}^{*} = α (u^{2} - β)

we obtain

\begin{matrix} J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) & \leq & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {({\hat{v}}_{0}^{*})}^{2} d x - β \int_{Ω} {\hat{v}}_{0}^{*} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 α (u^{2} - β) u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}} \\ \leq & sup_{v_{0}^{*} \in Y^{*}} \{\frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 α (u^{2} - β) u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ = & J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 α (u^{2} - β) u - f)}^{2} d x, \forall u \in V_{1} . \end{matrix}

(32)

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 α (u^{2} - β) u - f)}^{2} d x\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{3}^{*}) \in V_{1} \times D^{*}} J_{1}^{*} (u, v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(33)

The proof is complete. □

4. Conclusions

In this article we have developed convex dual variational formulations suitable for the local optimization of non-convex primal formulations.

It is worth highlighting, the results may be applied to a large class of models in physics and engineering.

We also emphasize the duality principles here presented are applied to a Ginzburg-Landau type equation. In a future research, we intend to extend such results for some models of plates and shells and other models in the elasticity theory.

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On Duality Principles and Related Convex Dual Formulations Suitable for Local and Global Non-Convex Variational Optimization

Abstract

1. Introduction

2. Another duality principle suitable for a local optimization of the primal formulation

2.1. The main duality principle and a concerning convex dual formulation

3. A convex primal dual for a local optimization of the primal formulation

4. Conclusions

References

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