Time-Varying Semidefinite Programs

TL;DR

This paper introduces methods for solving time-varying semidefinite programs with polynomial data, demonstrating that polynomial solutions can be efficiently computed and that bounds converge to the true optimal value under certain conditions.

Contribution

It establishes that polynomial solutions do not reduce optimal value under strict feasibility and provides tractable SDP-based algorithms for finding these solutions and bounds.

Findings

Polynomial solutions preserve optimal value under strict feasibility.

Semidefinite programs can be used to find best polynomial solutions.

Sequence of SDPs converges to the true optimal value under boundedness.

Abstract

We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data varies polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value of the TV-SDP. Moreover, by using a Positivstellensatz on univariate polynomial matrices, we show that the best polynomial solution of a given degree to a TV-SDP can be found by solving a semidefinite program of tractable size. We also provide a sequence of dual problems which can be cast as SDPs and that give upper bounds on the optimal value of a TV-SDP (in maximization form). We prove that under a boundedness assumption, this sequence of upper bounds converges to the optimal value of the TV-SDP. Under the same assumption, we also show that the optimal value of the TV-SDP is attained. We demonstrate the efficacy of our algorithms on a maximum-flow problem with time-varying edge capacities, a wireless coverage problem with time-varying coverage requirements, and on bi-objective semidefinite optimization where the goal is to approximate the Pareto curve in one shot.