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This version is not peer-reviewed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:

18 May 2023

Posted:

19 May 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

MSC: 49N15

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 the primal formulation global optimization

In this section we establish one more duality principle and related convex dual formulation suitable for a local optimization of the primal variational 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, we define

V = W_{0}^{1, 2} (Ω)

and 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 we assume

f \in L^{2} (Ω)

, and define

Y = Y^{*} = L^{2} (Ω),

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{4}},

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

and

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

for an appropriate constant

K_{4} > 0

to be specified.

Define also the functionals

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

and

G : Y \to R

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

(14)

and

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

for appropriate positive constants

K_{1}, K_{3}, K_{4}

to be specified.

Moreover, define

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

and

G^{*} : Y^{*} \to R,

\begin{matrix} F_{1}^{*} (v_{3}^{*}, v_{0}^{*}) & = & sup_{u \in V} {- F_{2} (u, v_{3}^{*}, v_{0}^{*})} \\ = & \frac{1}{2} \int_{Ω} \frac{{(f + K_{1} K_{3} v_{3}^{*})}^{2}}{- γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(v_{3}^{*})}^{2}} d x \\ - \frac{K_{1}}{2} \int_{Ω} K_{3}^{2} d x \end{matrix}

and

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

(15)

Furthermore, we define

B^{*} = {v_{3}^{*} \in Y^{*} : u_{1} (v_{3}^{*}) \in V_{1}},

where

u_{1} (v_{3}^{*}) = \frac{K_{3}}{v_{3}^{*}} .

Define also

C_{1}^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K_{2}}

for an appropriate real constant

K_{2} > 0

to be specified, and

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

J_{1}^{*} (v_{3}^{*}, v_{0}^{*}) = - F_{2}^{*} (v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*}) .

Moreover, we assume

K_{1} ≫ K_{2} ≫ max {1, K_{3}, K_{4} {, α, β, γ, ∥ f ∥}_{\infty}}

By directly computing

δ^{2} J_{1}^{*} (v_{3}^{*}, v_{0}^{*})

denoting

A = - K_{1} K_{3},

B = 2 K_{1} v_{3}^{*},

φ = - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(v_{3}^{*})}^{2},

φ_{1} = f + K_{1} K_{3} v_{3}^{*},

u = \frac{φ_{1}}{φ},

we may also obtain,

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

(16)

B^{*} \times C_{1}^{*} .

where

u_{1} = u_{1} (v_{3}^{*}) = \frac{K_{3}}{v_{3}^{*}},

H_{1} = K_{3}^{2} (3 u^{2} - 4 u u_{1} + u_{1}^{2}),

and

H_{2} = u^{2} [(- γ \nabla^{2} + 2 v_{0}^{*}) u_{1}] u_{1} .

At a critical point we have

H_{1} = 0

and

H_{2} = u_{0}^{2} f u_{0} > 0, a . e in Ω .

With such results, for a sufficiently small

ε_{1} > 0

, we may define the restrictions

C_{v_{0}^{*}} = {v_{3}^{*} \in B^{*} : | H_{1} (v_{3}^{*}, v_{0}^{*}) | \leq ε_{1} {(v_{3}^{*})}^{2} in Ω}

and

{(C_{3})}_{v_{0}^{*}} = {v_{3}^{*} \in B^{*} : H_{2} (v_{3}^{*}, v_{0}^{*}) \geq 0 in Ω} .

At this point, we prove that

{(C)}_{v_{0}^{*}}

is a convex set.

Firstly, fixing

v_{0}^{*} \in C_{1}^{*}

observe that for

K_{7}

sufficiently large

| H_{1} (v_{3}^{*}, v_{0}^{*}) | - ε_{1} {(v_{3}^{*})}^{2} + K_{7} {(v_{3}^{*})}^{2}

is convex in

v_{3}^{*}

B^{*}

Observe also that

| H_{1} (v_{3}^{*}, v_{0}^{*}) | \leq ε_{1} {(v_{3}^{*})}^{2}

is equivalent to

| H_{1} (v_{3}^{*}, v_{0}^{*}) | - ε_{1} {(v_{3}^{*})}^{2} + K_{7} {(v_{3}^{*})}^{2} \leq K_{7} {(v_{3}^{*})}^{2},

which is equivalent to

\sqrt{| H_{1} (v_{3}^{*}, v_{0}^{*}) | - ε_{1} {(v_{3}^{*})}^{2} + K_{7} {(v_{3}^{*})}^{2}} \leq \sqrt{K_{7}} | v_{3}^{*} | .

Moreover, this last inequality is equivalent to

H_{5} (v_{3}^{*}, v_{0}^{*}) = \sqrt{| H_{1} (v_{3}^{*}, v_{0}^{*}) | - ε_{1} {(v_{3}^{*})}^{2} + K_{7} {(v_{3}^{*})}^{2}} - \sqrt{K_{7}} | v_{3}^{*} | \leq 0 .

Since for

v_{3}^{*} \in B^{*}

we have

v_{3}^{*} f \geq 0, a . e . in Ω

, we may obtain that

- \sqrt{K_{7}} | v_{3}^{*} |

is a convex function on

B^{*}

, so that

H_{5}

is convex in

v_{3}^{*}

B^{*}

From such results, we may easily infer that

C_{v_{0}^{*}}

is a convex set,

\forall v_{0}^{*} \in C_{1}^{*}

Similarly, we may prove that

{(C_{3})}_{v_{0}^{*}}

is a convex set,

\forall v_{0}^{*} \in C_{1}^{*}

On the other hand, clearly we have

\frac{\partial^{2} J_{1}^{*} (v_{3}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0 in B^{*} \times C_{1}^{*} .

2.1. A concerning duality principle and a related 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}}_{3}^{*}, {\hat{v}}_{0}^{*}) \in {(C_{3})}_{{\hat{v}}_{0}^{*}} \times C_{1}^{*}

be such that

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

and

u_{0} \in V_{1}

be such that

u_{0} = \frac{φ_{1}}{φ} = \frac{f + K_{1} K_{3} {\hat{v}}_{3}^{*}}{- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*} + K_{1} {({\hat{v}}_{3}^{*})}^{2}} .

Assume also

u_{0} \neq 0, a . e . in Ω .

Under such hypotheses, denoting

B_{1}^{*} = C_{v_{0}^{*}} \cap {(C_{3})}_{v_{0}^{*}}

, we have

δ J (u_{0}) = 0,

{\hat{v}}_{3}^{*} u_{0} - K_{3} = 0, a . e . in Ω,

and

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {({\hat{v}}_{3}^{*} u - K_{3})}^{2} d x\} \\ = & inf_{v_{3}^{*} \in B_{1}^{*}} \{sup_{v_{0}^{*} \in C_{1}^{*}} J_{1}^{*} (v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(17)

Proof.

Observe that

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

so that, since

{\hat{v}}_{0}^{*} \in C^{*}

and

{\hat{v}}_{3}^{*} \in C_{{\hat{v}}_{0}^{*}}

, from the results in the previous lines, for a sufficiently small

ε_{1} > 0

, we have that

J_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = inf_{v_{3}^{*} \in B_{1}^{*}} J_{1}^{*} (v_{3}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in C_{1}^{*}} J_{1}^{*} ({\hat{v}}_{3}^{*}, v_{0}^{*}) .

Consequently, from this and the Saddle Point Theorem, we have

J_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = inf_{v_{3}^{*} \in B_{1}^{*}} \{sup_{v_{0}^{*} \in C_{1}^{*}} J_{1}^{*} (v_{3}^{*}, v_{0}^{*})\} .

Now we are going to show that

δ J (u_{0}) = 0 .

Firstly, observe that

F_{2}^{*} (v_{3}^{*}, v_{0}^{*}) = sup_{u \in V} {- F_{2} (u, v_{3}^{*}, v_{0}^{*})} .

Denoting

H (v_{3}^{*}, v_{0}^{*}, u) = - F_{2} (u, v_{3}^{*}, v_{0}^{*}),

there exists

\hat{u} \in V

such that

\frac{\partial H ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} = 0,

and

F_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = H ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u}),

so that

\begin{matrix} u_{0}^{2} = \frac{\partial F_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{0}^{*}} & = & \frac{\partial H ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial v_{0}^{*}} \\ + \frac{\partial H ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{0}^{*}} \\ = & {\hat{u}}^{2} . \end{matrix}

(18)

Summarizing, we this last equation is satisfied through the relation

u_{0} = \hat{u} .

Hence from the variation of

J_{1}^{*}

v_{0}^{*}

, we obtain

u_{0}^{2} - \frac{v_{0}^{*}}{α} - β = 0,

so that

v_{0}^{*} = α (u_{0}^{2} - β) .

On the other hand, from the variation of

J_{1}^{*}

v_{3}^{*}

, we have

\begin{matrix} \frac{\partial F_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{3}^{*}} \\ = & - K_{1} ({\hat{v}}_{3}^{*} u_{0} - K_{3}) u_{0} + \frac{\partial H ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{3}^{*}} \\ = & - K_{1} ({\hat{v}}_{3}^{*} u_{0} - K_{3}) u_{0} \\ = & 0 . \end{matrix}

(19)

From such results, since

u_{0} \neq 0, a . e . in Ω,

we get

{\hat{v}}_{3}^{*} u_{0} - K_{3} = 0, a . e . in Ω .

Consequently, from such last results and from

u_{0} = \frac{f + K_{1} K_{3} {\hat{v}}_{3}^{*}}{- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*} + K_{1} {({\hat{v}}_{3}^{*})}^{2}},

we obtain

\begin{matrix} - γ \nabla^{2} u_{0} + 2 v_{0}^{*} u_{0} + K_{1} {({\hat{v}}_{3}^{*})}^{2} u_{0} - f - K_{1} K_{3} {\hat{v}}_{3}^{*} \\ = & - γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f \\ = & δ J (u_{0}) \\ = & 0 . \end{matrix}

(20)

Summarizing,

δ J (u_{0}) = 0 .

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

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

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

so that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{0}^{*}) \\ = & - F_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) - G^{*} ({\hat{v}}_{0}^{*}) \\ = & J (u_{0}) . \end{matrix}

(21)

Finally, observe that

J_{1}^{*} (v_{3}^{*}, v_{0}^{*}) \leq F_{1} (u, v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*}),

\forall u \in V_{1}, v_{3}^{*} \in B^{*}, v_{0}^{*} \in C_{1}^{*} .

Therefore,

\begin{matrix} J_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) & \leq & sup_{v_{0}^{*} \in C_{1}^{*}} {F_{2} (u, {\hat{v}}_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*})} \\ = & J (u) + \frac{K_{1}}{2} \int_{Ω} {({\hat{v}}_{3}^{*} u - K_{3})}^{2} d x, \end{matrix}

(22)

\forall u \in V_{1} .

Summarizing, we have obtained

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {({\hat{v}}_{3}^{*} u - K_{3})}^{2} d x\} \\ = & inf_{v_{3}^{*} \in B_{1}^{*}} \{sup_{v_{0}^{*} \in C_{1}^{*}} J_{1}^{*} (v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(23)

The proof is complete. □

3. Conclusion

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 the primal formulation global optimization

2.1. A concerning duality principle and a related convex dual formulation

3. Conclusion

References

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