Real ideal and the duality of semidefinite programming for polynomial optimization

TL;DR

This paper introduces an algorithm to compute generators of ideals vanishing on semialgebraic sets, enabling polynomial optimization problems to be reformulated for duality gap-free semidefinite programming relaxations.

Contribution

It proposes a novel algorithm based on cylindrical algebraic decomposition to compute ideal generators, improving the duality properties in polynomial optimization.

Findings

Algorithm effectively computes ideal generators.

Reformulation eliminates duality gaps in SDP relaxations.

Provides criteria for the reality of ideals.

Abstract

We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality constraints can be modified equivalently so that the associated semidefinite programming relaxation problems have no duality gap. Elementary proofs for some criteria on reality of ideals are also given.