[Paper Review] A regularity condition in polynomial optimization

Vu Trung Hieu|arXiv (Cornell University)|Aug 18, 2018

Advanced Optimization Algorithms Research18 references4 citations

TL;DR

This paper introduces a regularity condition for polynomial optimization based on the boundedness of the solution set to the asymptotic problem involving the top-degree homogeneous component of the polynomial and the asymptotic cone of the feasible set. Under this condition, it establishes a Frank-Wolfe-type existence theorem and extends Eaves-type results for pseudoconvex problems, while proving local stability and regularity properties of the solution and optimal value functions when the feasible set is semi-algebraic.

ABSTRACT

Let $f$ be a polynomial of degree $d$ and $K$ be a nonempty closed set in $\R^n$. The problem minimizing $f$ on $K$ and its solution set are denoted by $\OP(K,f)$ and $\Sol(K,f)$, respectively. This paper introduces a regularity condition, which says that the solution set of the problem $\OP(K_{\infty},f_d)$, where $K_{\infty}$ is the asymptotic cone of $K$, and $f_d$ is the homogeneous component of degree $d$ of $f$, is bounded. Under this condition, a Frank-Wolfe type theorem is obtained, i.e. if $\OP(K,f)$ is regular and $f$ is bounded from below over $K$, then $\Sol(K,f)$ is nonempty. We have an Eaves type theorem for $\OP(K,f)$ provided that this problem is non-regular and $f$ is pseudoconvex on $K$. Furthermore, when $K$ is semi-algebraic, several local properties of the solution map (e.g., local boundedness, upper semicontinuity, and local upper-Holder stability) and the optimal value function (e.g., local Lipschitz continuity, semismoothness) of polynomial optimization problems are valid. The genericity of the regularity condition is discussed at the end.

Motivation & Objective

• To establish a sufficient condition for the existence of solutions in polynomial optimization problems with nonempty closed feasible sets.
• To generalize Frank-Wolfe and Eaves-type theorems by introducing a regularity condition based on asymptotic behavior.
• To investigate local stability properties—such as boundedness, upper semicontinuity, and Hölder stability—of the solution map under semi-algebraic constraints.
• To analyze regularity of the optimal value function, including local Lipschitz continuity and semismoothness.
• To examine the genericity of the proposed regularity condition in the space of polynomial optimization problems.

Proposed method

• Define the asymptotic cone $K_{∞}$ of the feasible set $K$ and the homogeneous component $f_d$ of degree $d$ of the polynomial $f$.
• Introduce the regularity condition as the boundedness of the solution set to $\OP(K_{\infty}, f_d)$.
• Use this condition to prove the existence of optimal solutions under lower boundedness of $f$ on $K$, extending the Frank-Wolfe theorem.
• For non-regular problems, apply pseudoconvexity of $f$ on $K$ to derive an Eaves-type existence result.
• Employ tools from semi-algebraic geometry to establish local boundedness, upper semicontinuity, and local upper-Holder stability of the solution map.
• Analyze the optimal value function for local Lipschitz continuity and semismoothness under semi-algebraic assumptions.

Experimental results

Research questions

• RQ1 Under what condition does a polynomial optimization problem $\OP(K,f)$ have a nonempty solution set when $f$ is bounded from below on $K$?
• RQ2 Can the Frank-Wolfe theorem be extended to polynomial optimization under a regularity condition based on asymptotic behavior?
• RQ3 What local stability properties (e.g., boundedness, semicontinuity, Hölder stability) hold for the solution map when $K$ is semi-algebraic?
• RQ4 How regular is the optimal value function in terms of Lipschitz continuity and semismoothness under semi-algebraic constraints?
• RQ5 Is the proposed regularity condition generic in the space of polynomial optimization problems?

Key findings

• The regularity condition—boundedness of the solution set to $\OP(K_{\infty}, f_d)$—guarantees the existence of a solution to $\OP(K,f)$ whenever $f$ is bounded from below on $K$, generalizing the Frank-Wolfe theorem.
• For non-regular problems, if $f$ is pseudoconvex on $K$, then $\OP(K,f)$ admits a solution, extending the Eaves-type existence result.
• When $K$ is semi-algebraic, the solution map is locally bounded, upper semicontinuous, and locally upper-Holder stable.
• The optimal value function is locally Lipschitz continuous and semismooth under semi-algebraic assumptions.
• The regularity condition holds generically in the space of polynomial optimization problems, indicating its robustness.