The Split Feasibility Problem with Polynomials

TL;DR

This paper introduces a semidefinite relaxation approach to solve the split feasibility problem involving polynomial-defined semi-algebraic sets, providing solutions or infeasibility certificates.

Contribution

It develops a novel semidefinite relaxation method for polynomial split feasibility problems, including feasibility detection and infeasibility certification.

Findings

Semidefinite relaxations effectively represent polynomial set intersections.

The algorithm can find feasible points when solutions exist.

The method provides certificates for infeasibility.

Abstract

This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations for representing the intersection of the sets. Properties of the semidefinite relaxations are studied. Based on that, a semidefinite relaxation algorithm is given for solving the split feasibility problem. Under a general condition, we prove that: if the split feasibility problem is feasible, we can get a feasible point; if it is infeasible, we can obtain a certificate for the infeasibility. Some numerical examples are given.