Invariant semidefinite programs

TL;DR

This paper reviews the theory and applications of invariant semidefinite programs, highlighting recent advances in understanding SDPs with symmetry across various mathematical and engineering fields.

Contribution

It provides a comprehensive background on invariant SDPs, including theoretical foundations and practical applications in coding theory, combinatorics, geometry, and polynomial optimization.

Findings

Summarizes key theoretical results on invariant SDPs.

Illustrates applications in multiple mathematical disciplines.

Provides foundational knowledge for further research.

Abstract

In the last years many results in the area of semidefinite programming were obtained for invariant (finite dimensional, or infinite dimensional) semidefinite programs - SDPs which have symmetry. This was done for a variety of problems and applications. The purpose of this handbook chapter is to give the reader the necessary background for dealing with semidefinite programs which have symmetry. Here the basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.