Invariant semidefinite programs

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.

51 pages, this article was written for the "Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications", (v2) revision based on suggestions by referee

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• $\mathrm{Aut}({\mathbb{F}}_5)$ has property $(T)$
• Dimension reduction for semidefinite programs via Jordan algebras
• Long-Step Path-Following Algorithm for Quantum Information Theory: Some Numerical Aspects and Applications
• Linear programming with unitary-equivariant constraints
• Jordan symmetry reduction for conic optimization over the doubly nonnegative cone: theory and software
• Bootstrapping SU(3) Lattice Yang-Mills Theory
• Refuting spectral compatibility of quantum marginals
• A recursive theta body for hypergraphs