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Time-Varying Semidefinite Programs

Author

Listed:
  • Amir Ali Ahmadi

    (Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey 08544)

  • Bachir El Khadir

    (Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey 08544)

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 vary 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 (positive locus theorem) 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 that 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 biobjective semidefinite optimization where the goal is to approximate the Pareto curve in one shot.

Suggested Citation

  • Amir Ali Ahmadi & Bachir El Khadir, 2021. "Time-Varying Semidefinite Programs," Mathematics of Operations Research, INFORMS, vol. 46(3), pages 1054-1080, August.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:3:p:1054-1080
    DOI: 10.1287/moor.2020.1117
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