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This version is not peer-reviewed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:

04 January 2023

Posted:

04 January 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].

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) .

Define the functionals

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

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 - {〈 u, f 〉}_{L^{2}} .

We define also

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}^{*})} \\ = & \int_{Ω} \frac{{(v_{1}^{*} + v_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f)}^{2}}{2 [K_{2} - K - γ \nabla^{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}^{*})}^{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}^{*} - f)}^{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^{*} : {∥ 2 v_{0}^{*} ∥}_{\infty} < K / 8 and - γ \nabla^{2} + 2 v_{0}^{*} > ε I_{d}\},

for a small parameter

0 < ε ≪ 1

Furthermore, we define

D^{*} = {v_{1}^{*} \in Y^{*} : ∥ v_{1}^{*} ∥_{\infty} \leq (3 / 2) K}

and

J_{1}^{*} : Y^{*} \times D^{*} \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}^{*}) .

Assuming

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

by directly computing

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

we may obtain that for such specified real constants,

J_{1}^{*}

in convex in

v_{2}^{*}

and it is concave in

(v_{1}^{*}, v_{0}^{*})

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

2. 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 1.

Let

({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \in Y^{*} \times D^{*} \times B^{*}

be such that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\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{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 D^{*} \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

J_{1}^{*}

is convex in

v_{2}^{*}

and concave in

(v_{1}^{*}, v_{0}^{*})

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

, from the Min-Max theorem, we obtain

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

and

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

we have

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

and

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

Observe now that denoting

H (v_{2}^{*}, v_{1}^{*}, v_{0}^{*}, u) = {〈 u, v_{1}^{*} + 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}}_{1}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} = 0,

and

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

so that

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

(7)

Summarizing, we have got

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

we have

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

so that

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

(8)

From such results, we may infer that

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

(9)

Now observe that from the variation of

J_{1}^{*}

v_{1}^{*}

, we have

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

so that

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

that is

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

From this and (8), we may infer that

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

so that

- 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}^{*}) = 0 .

From this and the concerning boundary conditions, since

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

we may obtain

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

Moreover, from

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

we have

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}}_{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)

Finally, observe that

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

\forall u \in V, v_{2}^{*} \in Y^{*}, v_{1}^{*} \in D^{*}, v_{0}^{*} \in B^{*} .

Thus, we may obtain

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

(11)

From this and (11), we obtain

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

(12)

Joining the pieces, from a concerning convexity in u, 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 D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(13)

The proof is complete.

□

Remark 1.

We could have also defined

B^{*} = \{v_{0}^{*} \in Y^{*} : {∥ 2 v_{0}^{*} ∥}_{\infty} < K / 8 and - γ \nabla^{2} + 2 v_{0}^{*} < - ε I_{d}\},

for a small parameter

0 < ε ≪ 1

. This corresponds to

- γ \nabla^{2} + 2 v_{0}^{*}

be negative definite, whereas the previous case corresponds to

- γ \nabla^{2} + 2 v_{0}^{*}

be positive definite. It is worth recalling the inequality

- γ \nabla^{2} + 2 v_{0}^{*} < - ε I_{d}

necessarily refers to a finite dimensional version for the model in question, in a finite elements or finite differences context.

3. One More 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 global 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}

(14)

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_{1},

for an appropriate constant

K_{4} > 0

to be specified.

Define also the functionals

F_{1} : V \to R

F_{2} : V \times Y \to R

and

G : Y \to R

F_{1} (u) = \frac{K_{2}}{2} \int_{Ω} {(\nabla^{2} u)}^{2} d x - {〈 u, f 〉}_{L^{2}},

\begin{matrix} F_{2} (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_{2}}{2} \int_{Ω} {(\nabla^{2} u)}^{2} d x \\ - \frac{K_{1}}{2} \int_{Ω} {(v_{3}^{*} u - K_{3})}^{2} d x, \end{matrix}

(15)

and

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

for appropriate positive constants

K_{1}, K_{2}, K_{3}, K_{4}

to be specified.

Moreover, define

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

and

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

and

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

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

(16)

and

\begin{matrix} F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u, v_{3}^{*}, v_{0}^{*})} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{2}^{*} - K_{1} K_{3} v_{3}^{*})}^{2}}{K_{2} \nabla^{4} + γ \nabla^{2} - 2 v_{0}^{*} - K_{1} {(v_{3}^{*})}^{2}} \\ - \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}

(17)

Furthermore, we define

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

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_{4}} .

and

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

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

Moreover, assuming

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

By directly computing

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

denoting

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

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

φ = - K_{2} \nabla^{4} - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(v_{3}^{*})}^{2}),

φ_{1} = v_{2}^{*} - K_{1} K_{3} v_{3}^{*},

u = - \frac{φ_{1}}{φ},

we may obtain, considering that

φ < 0

\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{3}^{*})}^{2}} =

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

Moreover,

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

(18)

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

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, we may define the restrictions

C_{2}^{*} = {v_{0}^{*} \in Y^{*} : H_{1} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) \geq 0, in Ω, \forall v_{2}^{*} \in D^{*}, v_{3}^{*} \in B^{*}} .

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

Here, we define

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

On the other hand, clearly we have

\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0

From such results, we may obtain that

J_{1}^{*}

in convex in

(v_{2}^{*}, v_{3}^{*})

and it is concave in

v_{0}^{*}

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

3.1. The main 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.

Let

({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*} {\hat{v}}_{0}^{*}) \in D^{*} \times B^{*} \times C^{*}

be such that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = 0

and

u_{0} \in V_{1}

be such that

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

Assume also

u_{0} \neq 0, a . e . in Ω .

Under such hypotheses, 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) \\ = & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(19)

Proof.

Observe that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = 0

so that, since

J_{1}^{*}

is convex in

(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*} \times C^{*}

and

\frac{\partial^{2} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} > 0, \forall v_{0}^{*} \in C_{1}^{*},

we obtain

J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, {\hat{v}}_{0}^{*}) sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, v_{0}^{*}) .

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

J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} .

Now we are going to show that

δ J (u_{0}) = 0 .

From

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = 0,

and

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

we have

- \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} - u_{0} = 0

and

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

Observe now that

F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, v_{0}^{*}) = sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u, v_{3}^{*}, v_{0}^{*})} .

Denoting

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

there exists

\hat{u} \in V

such that

\frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} = 0,

and

F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u}),

so that

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

(20)

Summarizing, we have got

u_{0} = \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = \hat{u} .

From such results and the Legendre tranform proprieties we get

v_{2}^{*} = \frac{\partial F_{1} (u_{0})}{\partial u}

and

v_{2}^{*} = \frac{\partial F_{2} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u} .

On the other hand, from the variation of

J_{1}^{*}

v_{3}^{*}

, we have

\begin{matrix} \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\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}}_{2}^{*}, {\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}

(21)

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 .

Finally, from the variation of

J_{1}^{*}

v_{0}^{*}

we obtain

\frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{0}^{*}} - \frac{\partial G^{*} (v_{0}^{*})}{\partial v_{0}^{*}} = 0,

so that

u_{0}^{2} + \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{0}^{*}} - \frac{v_{0}^{*}}{α} - β = 0 .

Thus,

v_{0}^{*} = α (u_{0}^{2} - β) .

Consequently, from such last results, we have

\begin{matrix} 0 = {\hat{v}}_{2}^{*} - {\hat{v}}_{2}^{*} \\ = & \frac{\partial F_{1} (u_{0})}{\partial u} - \frac{\partial F_{2} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u} \\ = & K_{2} \nabla^{4} u_{0} - f - K_{2} \nabla^{4} u_{0} - γ \nabla^{2} u_{0} + 2 v_{0}^{*} u_{0} \\ = & - γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f \\ = & δ J (u_{0}) . \end{matrix}

(22)

Summarizing,

δ J (u_{0}) = 0 .

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

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

F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*} {\hat{v}}_{0}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} 〉}_{L^{2}} - F_{2} (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}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) \\ = & - F_{1}^{*} ({\hat{v}}_{2}^{*}) + F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) - G^{*} ({\hat{v}}_{0}^{*}) \\ = & J (u_{0}) . \end{matrix}

(23)

Finally, observe that

J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) \leq F_{1} (u) - {〈 u, v_{2}^{*} 〉}_{L^{2}} + F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*}),

\forall u \in V_{1}, v_{2}^{*} \in D^{*}, v_{3}^{*} \in B^{*}, v_{0}^{*} \in C^{*} .

Therefore,

sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) \leq sup_{v_{0}^{*} \in C_{1}^{*}} {- {〈 u, v_{2}^{*} 〉}_{L^{2}} + F_{1} (u) + F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*})},

so that

\begin{matrix} inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ \leq & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C_{1}^{*}} {- {〈 u, v_{2}^{*} 〉}_{L^{2}} + F_{1} (u) + F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*})}\} \\ = & J (u), \forall u \in V_{1} . \end{matrix}

(24)

Summarizing, we have got

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) & = & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ \leq & inf_{u \in V_{1}} J (u) . \end{matrix}

(25)

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} J (u) \\ = & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(26)

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.

Author Contributions

Funding

Conflicts of Interest

References

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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. The Main Duality Principle and a Concerning Convex Dual Formulation

3. One More Duality Principle Suitable for the Primal Formulation Global Optimization

3.1. The main duality principle and a related convex dual formulation

4. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

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