Convexity in semi-algebraic geometry and polynomial optimization

TL;DR

This paper explores the role of convexity in semi-algebraic geometry and polynomial optimization, demonstrating simplified relaxations, new convexity certificates, and extensions of Jensen's inequality.

Contribution

It provides new results on convexity in semi-algebraic sets, including certificates for convexity, finite convergence of relaxations, and extensions of Jensen's inequality.

Findings

Semidefinite relaxations simplify under convexity and have finite convergence.

A numerical certificate for convexity of semi-algebraic sets is proposed.

Extension of Jensen's inequality to certain linear functionals beyond probability measures.

Abstract

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked numerically and also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.