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Sparse Cuts for the Positive Semidefinite Cone

Oktay Günlük, Paul Jünger, Jeff Linderoth, Andrea Lodi, James Luedtke

TL;DR

In a computational study, it is found that the sparse LP relaxations defined by these inequalities can accelerate branch-and-bound methods for globally solving nonconvex optimization problems.

Abstract

We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite cone and are sparse in the sense that they are supported only on the variables corresponding to products of variables present in quadratic functions. We show that these sparse linear inequalities yield an LP relaxation that gives the same bound as the SDP relaxation. We demonstrate how to identify these inequalities via a separation procedure that involves solving a structured ``projection'' SDP. In a computational study, we find that the sparse LP relaxations defined by these inequalities can accelerate branch-and-bound methods for globally solving nonconvex optimization problems.

Sparse Cuts for the Positive Semidefinite Cone

TL;DR

In a computational study, it is found that the sparse LP relaxations defined by these inequalities can accelerate branch-and-bound methods for globally solving nonconvex optimization problems.

Abstract

We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite cone and are sparse in the sense that they are supported only on the variables corresponding to products of variables present in quadratic functions. We show that these sparse linear inequalities yield an LP relaxation that gives the same bound as the SDP relaxation. We demonstrate how to identify these inequalities via a separation procedure that involves solving a structured ``projection'' SDP. In a computational study, we find that the sparse LP relaxations defined by these inequalities can accelerate branch-and-bound methods for globally solving nonconvex optimization problems.
Paper Structure (19 sections, 9 theorems, 45 equations, 3 figures, 7 tables)

This paper contains 19 sections, 9 theorems, 45 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Sparse relaxations
  3. A sparse reformulation of the SDP relaxation
  4. Replacing PSD constraints with linear inequalities via duality
  5. Sparse doubly nonnegative relaxation
  6. Solving the E-SDP and E-DNN relaxations
  7. The separation problem
  8. Exploiting nonnegativity
  9. Accelerating the cutting plane procedure
  10. Computational experiments
  11. Dual bound quality
  12. Sparse cutting planes with branch and bound
  13. Benchmark Instances
  14. Computational Environment
  15. Solved BoxQCQP instances
  16. ...and 4 more sections

Key Result

Lemma 1

It holds that $\mathop{\mathrm{proj}}\nolimits_{\mathbb{R}^E}(\mathcal{S}^+) = \mathcal{S}^+_E$.

Figures (3)

  • Figure 1: Bound progression for BoxQCQP instance 125-025-1_10qc. Each dot represents an iteration. Note that the LP time is shown on a logarithmic scale.
  • Figure 2: Computing time of the 25 Solved BoxQCQP instances in Table \ref{['tab:Gurobi_aggregated']}.
  • Figure 3: QPLIB instances solved by at least one algorithm within the time limit of 10h.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 8 more