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Time-Varying Semidefinite Programming: Path Following a Burer-Monteiro Factorization

Antonio Bellon, Mareike Dressler, Vyacheslav Kungurtsev, Jakub Marecek, André Uschmajew

TL;DR

Numerical experiments for a time-varying max-cut SDP relaxation demonstrate the computational advantages of the proposed method for tracking TV-SDPs in terms of runtime compared to off-the-shelf interior point methods.

Abstract

We present an online algorithm for time-varying semidefinite programs (TV-SDPs), based on the tracking of the solution trajectory of a low-rank matrix factorization, also known as the Burer-Monteiro factorization, in a path-following procedure. There, a predictor-corrector algorithm solves a sequence of linearized systems. This requires the introduction of a horizontal space constraint to ensure the local injectivity of the low-rank factorization. The method produces a sequence of approximate solutions for the original TV-SDP problem, for which we show that they stay close to the optimal solution path if properly initialized. Numerical experiments for a time-varying max-cut SDP relaxation demonstrate the computational advantages of the proposed method for tracking TV-SDPs in terms of runtime compared to off-the-shelf interior point methods.

Time-Varying Semidefinite Programming: Path Following a Burer-Monteiro Factorization

TL;DR

Numerical experiments for a time-varying max-cut SDP relaxation demonstrate the computational advantages of the proposed method for tracking TV-SDPs in terms of runtime compared to off-the-shelf interior point methods.

Abstract

We present an online algorithm for time-varying semidefinite programs (TV-SDPs), based on the tracking of the solution trajectory of a low-rank matrix factorization, also known as the Burer-Monteiro factorization, in a path-following procedure. There, a predictor-corrector algorithm solves a sequence of linearized systems. This requires the introduction of a horizontal space constraint to ensure the local injectivity of the low-rank factorization. The method produces a sequence of approximate solutions for the original TV-SDP problem, for which we show that they stay close to the optimal solution path if properly initialized. Numerical experiments for a time-varying max-cut SDP relaxation demonstrate the computational advantages of the proposed method for tracking TV-SDPs in terms of runtime compared to off-the-shelf interior point methods.
Paper Structure (11 sections, 12 theorems, 94 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 12 theorems, 94 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and key assumptions
  3. Quotient geometry of positive semidefinite rank-$\bm{r}$ matrices
  4. Horizontal space and unique factorizations
  5. A time interval for the factorized problem
  6. Path following the trajectory of solutions
  7. Linearized KKT conditions and second-order sufficiency
  8. A path-following predictor-corrector algorithm
  9. Error analysis
  10. Numerical experiments on Time-Varying Max Cut
  11. Conclusion

Key Result

Theorem 2.4

(Bellon et al., Bellon2021) Let (P$t$,D$t$) be a primal-dual pair of TV-SDPs parametrized over a time interval $[0,T]$ such that primal-dual strict feasibility holds for any $t\in [0,T]$ and assume that the data $\mathcal{A}_t,b_t,C_t$ are continuously differentiable functions of $t$. Let $t^* \in [

Figures (3)

  • Figure 1: Distribution of the average residuals as a function of the step size using three different methods: an interior point method (IPM), in bordeaux, the splitting conic solver (SCS), in orange, and our path following (PF) algorithm, in green. The data in both plots are the same except that the left plot also shows ten rank changing instances, depicted by light green dots, which were removed in the right plot.
  • Figure 2: Distribution of the runtime as function of the step size.
  • Figure 3: Average runtime of MOSEK IPM and Algorithm \ref{['alg: 1']} for tracking the TV-SDP solutions with the same residual accuracy on a grid, as a function of the number of gridpoints.

Theorems & Definitions (26)

  • Definition 2.1: strict feasibility
  • Definition 2.2: strict complementarity
  • Definition 2.3: nondegeneracy
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 16 more