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This version is not peer-reviewed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:

22 July 2023

Posted:

24 July 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 .

Now we present some basic definitions and statements.

Definition 1.2.

Let V be a Banach space. We define the topological dual space of V, denoted by

V^{'}

, as the set of all continuous and linear functionals defined on V.

We assume

V^{'}

may be represented through another Banach space denoted by

V^{*}

and a bilinear form

{〈 \cdot, \cdot 〉}_{V} : V \times V^{*} \to R .

More specifically, for each

f \in V^{'}

, we suppose there exists a unique

u^{*} \in V^{*}

such that

f (u) = {〈 u, u^{*} 〉}_{V}, \forall u \in V .

Moreover, we define the norm of f, denoted by

{∥ f ∥}_{V^{*}}

{∥ f ∥}_{V^{*}} = sup {| {〈 u, u^{*} 〉}_{V} : u \in {V and ∥ u ∥}_{V} \leq 1} \equiv ∥ u^{*} ∥_{V^{*}} .

For an open, bounded and connected set

Ω \subset R^{N}

and

Y = Y^{*} = L^{2} (Ω)

we recall that

{〈 u, u^{*} 〉}_{L^{2}} = \int_{Ω} u u^{*} d x .

More specifically, for each continuous and linear functional

f : Y \to R

there exists a unique

u^{*} \in Y^{*} = L^{2} (Ω)

such that

f (u) = \int_{Ω} u u^{*} d x, \forall u \in Y = L^{2} (Ω) .

Definition 1.3

(Polar functional). Let V be a Banach space and let

F : V \to R

be a functional.

We define the polar functional of F, denoted by

F^{*} : V^{*} \to R

, by

F^{*} (u^{*}) = sup_{u \in V} {{〈 u, u^{*} 〉}_{L^{2}} - F (u)}, \forall u^{*} \in V^{*} .

Another important definition refers to the Legendre transform one and respective relevant propriety, which are summarized in the next theorem.

Theorem 1.4

(Legendre transform theorem). Let V be a Banach space and let

F : V \to R

be a twice continuously Fréchet differentiable functional.

Let

u^{*} \in V^{*}

. Assume there exists a unique

\hat{u} \in V

such that

u^{*} = \frac{\partial F (\hat{u})}{\partial u} .

Suppose also

det \{\frac{\partial^{2} F (u)}{\partial u^{2}}\} \neq 0,

in a neighborhood of

\hat{u} .

Under such hypotheses, defining the Legendre tranform of F at

u^{*}

F_{L}^{*} (u^{*})

where

F_{L}^{*} (u^{*}) = {〈 \hat{u}, u^{*} 〉}_{V} - F (\hat{u})

we have that

\hat{u} = \frac{\partial F_{L}^{*} (u^{*})}{\partial u^{*}} .

Remark 1.5.

Concerning such a last definition, observe that if F is convex on V, then the extremal condition

u^{*} = \frac{\partial F (\hat{u})}{\partial u},

corresponds to globally maximize

H (u) = {〈 u, u^{*} 〉}_{V} - F (u)

on V, so that, in such a case,

F^{*} (u^{*}) = H (\hat{u}) = {〈 \hat{u}, u^{*} 〉}_{V} - F (\hat{u}) = F_{L}^{*} (u^{*}) .

Summarizing, if F is convex, under the hypotheses of the last theorem, the polar functional

F^{*} (u^{*})

coincides with the Legendre transform of F on

V^{*}

already denoted by

F_{L}^{*}

, that is,

F^{*} (u) = F_{L}^{*} (u^{*}), \forall u^{*} \in V^{*} .

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 .

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

\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}

(2)

and

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

At this point we assume a finite dimensional version for this concerning model. For example, we may define a new domain for the primal functional considering the projection of V on the space spanned by the first N (in general N=10, is enough) eigen-vectors of the Laplace operator, corresponding to the first N eigen-values. On this new not relabeled finite dimensional space V, we recall that the Laplace operator is bounded, so that we assume the following restriction,

- γ \nabla^{2} + 2 v_{0}^{*} \leq 0 .

Hence, we denote

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

for an appropriate real constant

K > 0

We define also

F_{1}^{*} : Y^{*} \times B^{*} \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{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x, + \frac{K_{1}}{2} \int_{Ω} f^{2} d x \end{matrix}

(3)

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}

(4)

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. The main duality principle and a concerning convex (in fact concave) 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}

(5)

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}

(6)

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 .

(7)

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}

(8)

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}, 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}

(9)

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}

(10)

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}

(11)

The proof is complete.

□

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

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}

(12)

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}

(13)

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}

(14)

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}

(15)

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}

(16)

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}

(17)

Observe that at a critical point

φ = 0

and

φ_{1} = 0 .

From such results we may infer that

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

around any critical point.

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

Theorem 3.1.

Let

(u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) \in V_{1} \times D^{*} \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 there exists

r > 0

such that

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

(18)

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, as previously obtained, there exists

r > 0

such that

det \{\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u \partial v_{3}^{*}}\} > 0, \forall (u, v_{3}^{*}) \in B_{r} (u_{0}, {\hat{v}}_{3}^{*})

and

\frac{\partial^{2} J_{1}^{*} (u_{0}, {\hat{v}}_{3}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0, \forall v_{0}^{*} \in B^{*} .

Since for a sufficiently large

K_{1} > 0

we have

\frac{\partial^{2} J_{1}^{*} (u, v_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u^{2}} > 0, in B_{r} (u_{0}, {\hat{v}}_{3}^{*}),

from these last results and the standard Saddle point theorem, we have

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

The proof is complete. □

4. A duality principle for a related relaxed formulation concerning the vectorial approach in the calculus of variations

In this section we develop a duality principle for a related vectorial model in the calculus of variations.

Let

Ω \subset R^{n}

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

\partial Ω = Γ .

For

1 < p < + \infty

, consider a functional

J : V \to R

where

J (u) = G (\nabla u) + F (u) - {〈 u, f 〉}_{L^{2}},

where

V = \{u \in W^{1, p} (Ω; R^{N}) : u = u_{0} in \partial Ω\}

and

f \in L^{2} (Ω; R^{N}) .

We assume

G : Y \to R

and

F : V \to R

are Fréchet differentiable and F is also convex.

Also

G (\nabla u) = \int_{Ω} g (\nabla u) d x,

where

g : R^{N \times n}

it is supposed to be Fréchet differentiable. Here we have denoted

Y = L^{p} (Ω; R^{N \times n})

We define also

J_{1} : V \times Y_{1} \to R

J_{1} (u, ϕ) = G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}},

where

Y_{1} = W^{1, p} (Ω \times Ω; R^{N})

and

G_{1} (\nabla u + \nabla_{y} ϕ) = \frac{1}{| Ω |} \int_{Ω} \int_{Ω} g (\nabla u (x) + \nabla_{y} ϕ (x, y)) d x d y .

Moreover, we define the relaxed functional

J_{2} : V \to R

J_{2} (u) = inf_{ϕ \in V_{0}} J_{1} (u, ϕ),

where

V_{0} = {ϕ \in Y_{1} : ϕ (x, y) = 0, in Ω \times \partial Ω} .

Now observe that

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + G_{1} (\nabla u + \nabla_{y} ϕ) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{2}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x + G_{1} (v)\} \\ + inf_{(v, ϕ) \in V \times V_{0}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}}\} \\ = & - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) \\ + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \cdot u_{0} d Γ, \end{matrix}

(19)

\forall (u, ϕ) \in V \times V_{0}, v^{*} \in A^{*}

, where

A^{*} = {v^{*} \in Y_{2}^{*} : {div}_{y} v^{*} (x, y) = 0, in Ω} .

Here we have denoted

G_{1}^{*} (v^{*}) = sup_{v \in Y_{2}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x - G_{1} (v)\},

where

Y_{2} = L^{p} (Ω \times Ω; R^{N \times n}),

Y_{2}^{*} = L^{q} (Ω \times Ω; R^{N \times n}),

and where

\frac{1}{p} + \frac{1}{q} = 1 .

Furthermore,

\begin{matrix} F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) - \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \cdot u_{0} d Γ \\ = & sup_{(v, ϕ) \in V \times V_{0}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x - F (u) + {〈 u, f 〉}_{L^{2}}\}, \end{matrix}

(20)

Therefore, denoting

J_{3}^{*} : Y_{2}^{*} \to R

J_{3}^{*} (v^{*}) = - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\int_{Ω} v^{*} (x, y) d y) + f) + \frac{1}{| Ω |} \int_{\partial Ω} Ω (\int_{Ω} v^{*} (x, y) d y) \cdot u_{0} d Γ,

we have got

inf_{u \in V} J_{2} (u) \geq sup_{v^{*} \in A^{*}} J_{3}^{*} (v^{*}) .

Finally, we highlight such a dual functional

J_{3}^{*}

is convex (in fact concave).

5. Conclusion

In this article we have developed convex dual and primal 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. The main duality principle and a concerning convex (in fact concave) dual formulation

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

4. A duality principle for a related relaxed formulation concerning the vectorial approach in the calculus of variations

5. Conclusion

References

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