Duality suitable for a class of non-convex optimization problems

TL;DR

This paper develops a duality principle applicable to many non-convex optimization problems, establishing a relation between primal and dual critical points and proving no duality gap locally.

Contribution

It introduces a duality framework for non-convex problems using convex analysis, ensuring primal-dual critical point correspondence and gap absence.

Findings

Established a duality relation for non-convex optimization

Proved no duality gap exists locally

Linked critical points of primal and dual problems

Abstract

In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the critical points of the primal and dual formulations and formally prove there is no duality gap between such formulations, in a local extremal context.

Equations

$$J(x)=\frac{1}{2}{x}^{T}Ax+\sum _{j=1}^N\frac{{\gamma }_j}{2}{\left(\frac{x^T{B}_{j}x}{2}+c_j\right)}^2-f^{T}x,\forall x\in {\mathbb{R}}^n$$

$$x^{T}Ax>0,\forall x\in {\mathbb{R}}^n,\text{ such that }x\ne 0.$$

$$J(x)=\frac{1}{2}{x}^{T}Ax+\sum _{j=1}^N\frac{{\gamma }_j}{2}{\left(\frac{x^T{B}_{j}x}{2}+c_j\right)}^2-f^{T}x,\forall x\in {\mathbb{R}}^n,$$

$$J_1(x,v_0^{\ast })$$

$$J_2(x,v^{\ast },v_0^{\ast })$$

$$B^{\ast }=\left\{v_0^{\ast }\in {\mathbb{R}}^N:\sum _{j=1}^N(v_0^{\ast }{)}_j{B}_j+K{I}_d>\frac{K{I}_d}{2}\right\}.$$

$$({\hat{v}}_0^{\ast }{)}_j={\gamma }_j\left(\frac{1}{2}{x}_0^T{B}_j{x}_0+c_j\right),\forall j\in \{1,\cdots ,N\}$$

$${\hat{v}}^{\ast }=-\sum _{j=1}^N(v_0^{\ast }{)}_j{B}_j{x}_0+K{x}_0.$$

$$J^{\ast }(v^{\ast },v_0^{\ast })$$

$$\delta J^{\ast }(\hat{v},{\hat{v}}_0^{\ast })=0$$

$$J(x_0)=J^{\ast }({\hat{v}}^{\ast },{\hat{v}}_0^{\ast }).$$

$$J(x_0)$$

$$A_{+}^{\ast }=\left\{v_0^{\ast }\in {\mathbb{R}}^N:\sum _{j=1}^N(v_0^{\ast }{)}_j{B}_j+A>0\right\},$$

$$J(x_0)$$

$$A_{-}^{\ast }=\left\{v_0^{\ast }\in {\mathbb{R}}^N:\sum _{j=1}^N(v_0^{\ast }{)}_j{B}_j+A<0\right\},$$

$$J(x_0)$$

$$J(x)$$

$$J^{\ast }({\hat{v}}^{\ast },{\hat{v}}_0^{\ast })$$

$$\frac{{\partial }^2{J}^{\ast }(v^{\ast },v_0^{\ast })}{\partial (v^{\ast }{)}^2}>0,\forall v^{\ast }\in {\mathbb{R}}^n,v_0^{\ast }\in E^{\ast }$$

$$J(x_0)=J^{\ast }({\hat{v}}^{\ast },{\hat{v}}_0^{\ast })=\underset{v^{\ast }\in {\mathbb{R}}^n}{\inf }\left\{\underset{v_0^{\ast }\in B_{r_2}({\hat{v}}_0^{\ast })\cap E^{\ast }}{\sup }{J}^{\ast }(v^{\ast },v_0^{\ast })\right\}.$$

$$J(x_0)$$

$$\frac{{\partial }^2{J}^{\ast }(v^{\ast },v_0^{\ast })}{\partial (v^{\ast }{)}^2}<0,\forall v^{\ast }\in {\mathbb{R}}^n,v_0^{\ast }\in B_{r_2}({\hat{v}}_0^{\ast }).$$

$$J(x_0)=\underset{x\in B_r(x_0)}{\sup }J(x),$$

$$\underset{x\in B_r(x_0)}{\sup }{J}_1(x,v_0^{\ast })=\underset{v^{\ast }\in B_{r_1}({\hat{v}}^{\ast })}{\sup }{J}^{\ast }(v^{\ast },v_0^{\ast }),\forall v_0^{\ast }\in B_{r_2}({\hat{v}}_0^{\ast }).$$

$$J^{\ast }({\hat{v}}^{\ast },{\hat{v}}_0^{\ast })$$

$$J(x_0)$$

Full text

1 Introduction

This short letter develops duality for a class of problems in $\mathbb{R}^{n}$. We consider the problem of minimizing the functional $J:\mathbb{R}^{n}\rightarrow\mathbb{R}$ where

[TABLE]

where $A$ is a symmetric $n\times n$ matrix, $B_{j}$ is a symmetric $n\times n$ matrix, $\;\forall j\in\{1,\cdots,N\}$ and $c_{j},\;\gamma_{j}\in\mathbb{R}$, $\gamma_{j}>0,$ $\forall j\in\{1,\cdots,N\}.$ Moreover $f\in\mathbb{R}^{n}$ is a fixed vector.

In this case we do not assume $n=N$ and the results are valid even for the case $n\neq N$, $\;\forall n,N\in\mathbb{N}.$

Remark 1.1**.**

About the notation for a generic $n\times n$ real matrix $A$ we denote $A>\mathbf{0}$ if

[TABLE]

Similarly, we denote $A>B$, if $A-B>\mathbf{0}.$ Moreover $x^{T}$ and $A^{T}$ denotes the transpose of a vector in $\mathbb{R}^{n}$ and for a $n\times n$ matrix, respectively. Finally, $I_{d}$ denotes the $n\times n$ identity matrix.

.

Remark 1.2**.**

About the references, we must emphasize our work is a kind of extension and continuation of the original works of Bielski and Telega [1, 2] combined with the work of Toland [7]. The technical details follow in some extent the results in [3]. Anyway, we highlight once more our work in some sense complements the results in [1, 2] but now applied to a $\mathbb{R}^{n}$ simpler context.

Similar problems have been addressed in [5, 6], among others.

2 The main duality principle

Our main result is summarized by the following theorem.

Theorem 2.1**.**

Consider the main functional $J:\mathbb{R}^{n}\rightarrow\mathbb{R}$ where

[TABLE]

with the assumptions about matrices, vectors and real constants stated in the last section.

Define also $J_{1}:\mathbb{R}^{n}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ and $J_{2}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times B^{*}\rightarrow\mathbb{R}$ by

[TABLE]

and

[TABLE]

where

[TABLE]

Assume $x_{0}\in\mathbb{R}^{n}$ is such that $\delta J(x_{0})=\mathbf{0}$ and define

[TABLE]

and

[TABLE]

Define also $J^{*}:\mathbb{R}^{n}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ by

[TABLE]

Under such hypotheses

[TABLE]

and

[TABLE]

Moreover, for $K>0$ sufficiently big,

if $\delta^{2}J(x_{0})>\mathbf{0}$, then there exist $r,r_{1},r_{2}>0$ such that

[TABLE] 2. 2.

If $\hat{v}_{0}^{*}\in E^{*}=A_{+}^{*}\cap B^{*}$, where

[TABLE]

then there exists $r_{2}>0$ such that

[TABLE] 3. 3.

If $\delta^{2}J(x_{0})<\mathbf{0}$ so that $\hat{v}_{0}^{*}\in A^{*}_{-}$, where

[TABLE]

then there exist $r,r_{1},r_{2}>0$ such that

[TABLE]

Proof.

The proof that $\delta J^{*}(\hat{v}^{*},\hat{v}_{0}^{*})=\mathbf{0}$ and $J(x_{0})=J^{*}(\hat{v}^{*},\hat{v}_{0}^{*})$ results directly from the Legendre transform standard properties.

Now suppose $\delta^{2}J(x_{0})>\mathbf{0}.$ For $K>0$ sufficiently big $J^{*}(v^{*},v_{0}^{*})$ is concave in $v_{0}^{*}$ in a neighborhood of $\hat{v}_{0}^{*}$, so that from the min-max theorem we may obtain $r,r_{1},r_{2}>0$ such that

[TABLE]

Hence for some not relabeled $r>0$, we have

[TABLE]

The proof of item 1 is complete. Suppose now $\hat{v}_{0}^{*}\in E^{*}$.

First observe that

[TABLE]

From the min-max theorem, for $K>0$ sufficiently big, we may find $r_{2}>0$ such that

[TABLE]

In particular,

[TABLE]

$\forall x\in\mathbb{R}^{n},\;v^{*}\in\mathbb{R}^{n}.$

From this we get

[TABLE]

Summarizing these last results, we have obtained,

[TABLE]

Finally assume $\delta^{2}J(x_{0})<\mathbf{0},$ so that $\hat{v}_{0}^{*}\in A_{-}^{*}.$

Since $A_{-}^{*}$ is open, we may obtain $r_{2}>0$ such that

[TABLE]

From such assumptions and results, we may obtain $r,r_{1},r_{2}>0$ such that

[TABLE]

and

[TABLE]

From these results and assumptions, for a not relabeled $r>0$ we have

[TABLE]

Summarizing these last results, we have obtained

[TABLE]

The proof is complete. ∎

3 Conclusion

In this article we have developed a duality principle suitable for a large class of optimization problems in $\mathbb{R}^{n}$. We highlight the min-max theorem has a fundamental role for the proofs of the main results.

We believe these results may be extended to more complex variational models such as non-linear models of plates and shells.