An Inner Convex Approximation Algorithm for BMI Optimization and Applications in Control

TL;DR

This paper introduces an inner convex approximation algorithm for solving nonconvex BMI optimization problems, with applications in control, providing a new iterative approach with proven convergence.

Contribution

It presents a novel local optimization method using inner convex approximations for nonconvex SDPs, specifically targeting BMI problems in control applications.

Findings

Algorithm converges under mild assumptions

Effective in static output feedback control problems

Numerical tests validate the approach

Abstract

In this work, we propose a new local optimization method to solve a class of nonconvex semidefinite programming (SDP) problems. The basic idea is to approximate the feasible set of the nonconvex SDP problem by inner positive semidefinite convex approximations via a parameterization technique. This leads to an iterative procedure to search a local optimum of the nonconvex problem. The convergence of the algorithm is analyzed under mild assumptions. Applications in static output feedback control are benchmarked and numerical tests are implemented based on the data from the COMPLeib library.