A more efficient reformulation of complex SDP as real SDP

TL;DR

This paper introduces a new, more efficient way to convert complex semidefinite programs into real SDPs, leading to faster computations in polynomial optimization problems.

Contribution

It presents a novel reformulation of complex SDPs as real SDPs that is more economical and exploits problem structure for improved efficiency.

Findings

Reformulation runs significantly faster than traditional methods.

Numerical examples demonstrate improved computational efficiency.

Exploits inner structure of complex SDP relaxations for further reductions.

Abstract

This note proposes a new reformulation of complex semidefinite programs (SDPs) as real SDPs. As an application, we present an economical reformulation of complex SDP relaxations of complex polynomial optimization problems as real SDPs and derive some further reductions by exploiting inner structure of the complex SDP relaxations. Various numerical examples demonstrate that our new reformulation runs significantly faster than the usual popular reformulation.