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Instance-specific linear relaxations of semidefinite optimization problems

Daniel de Roux, Robert Carr, R. Ravi

Abstract

We introduce a generic technique to obtain linear relaxations of semidefinite programs with provable guarantees based on the commutativity of the constraint and the objective matrices. We study conditions under which the optimal value of the SDP and the proposed linear relaxation match, which we then relax to provide a flexible methodology to derive effective linear relaxations. We specialize these results to provide linear programs that approximate well-known semidefinite programs for the max cut problem proposed by Poljak and Rendl, and the Lovasz theta number; we prove that the linear program proposed for max cut certifies a known eigenvalue bound for the maximum cut value and is in fact stronger. Our ideas can be used to warm-start algorithms that solve semidefinite programs by iterative polyhedral approximation of the feasible region. We verify this capability through multiple experiments on the max cut semidefinite program, the Lovasz theta number and on three families of semidefinite programs obtained as convex relaxations of certain quadratically constrained quadratic problems.

Instance-specific linear relaxations of semidefinite optimization problems

Abstract

We introduce a generic technique to obtain linear relaxations of semidefinite programs with provable guarantees based on the commutativity of the constraint and the objective matrices. We study conditions under which the optimal value of the SDP and the proposed linear relaxation match, which we then relax to provide a flexible methodology to derive effective linear relaxations. We specialize these results to provide linear programs that approximate well-known semidefinite programs for the max cut problem proposed by Poljak and Rendl, and the Lovasz theta number; we prove that the linear program proposed for max cut certifies a known eigenvalue bound for the maximum cut value and is in fact stronger. Our ideas can be used to warm-start algorithms that solve semidefinite programs by iterative polyhedral approximation of the feasible region. We verify this capability through multiple experiments on the max cut semidefinite program, the Lovasz theta number and on three families of semidefinite programs obtained as convex relaxations of certain quadratically constrained quadratic problems.
Paper Structure (39 sections, 17 theorems, 76 equations, 14 figures, 7 tables, 1 algorithm)

This paper contains 39 sections, 17 theorems, 76 equations, 14 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Hardness of approximation of the max cut problem
  3. Exact linear relaxations under $\mathcal{O}$
  4. Overview and outline of this paper
  5. Notation
  6. Instance-specific linear relaxations of semidefinite optimization problems
  7. Finding initial sets
  8. Finding commuting matrices
  9. Linear Relaxations of the max cut semidefinite program
  10. Instance-specific linear relaxations.
  11. Hidden basis property and stronger guarantees
  12. Solvability of maxcut under $\mathcal{O}$
  13. Applications to semidefinite programs
  14. Maximum Cut
  15. Lovász theta number
  16. ...and 24 more sections

Key Result

theorem 1

Consider a generic semidefinite optimization problem SDP, with dual given by DSDP. Suppose that the set is a polytope with extreme points $p_1\dots,p_k$, and define $\mathcal{S}:=\bigcup_{i=1}^k \mathcal{E}(C- \mathcal{A}(p_k))$. Then, $L_{\mathcal{S}}$ is a linear program and solves SDP.

Figures (14)

  • Figure 1: Ratio of $\frac{z_\mathcal{S}}{z_{sdp}}$ (Eigen cuts) and $\frac{z_n}{z_{sdp}}$ (Oracle cuts) for instances of max cut where the graph has been sampled according to the Erdős-Rényi random model, for different values of $p$, as $n$ grows.
  • Figure 2: Ratio of $\frac{z_\mathcal{S}}{z_{sdp}}$ (Eigen cuts) and $\frac{z_n}{z_{sdp}}$ (Oracle cuts) for instances of max cut where the graph is a random $d-$regular graph, for different values of $d$, as $n$ grows.
  • Figure 3: Quotients for the Lovász theta number $\frac{z_\mathcal{S}}{z_{sdp}}$ (Eigen cuts) and $\frac{z_n}{z_{sdp}}$ (Oracle cuts) as $n$ grows for Erdős-Rényi random graphs with different values of $p$.
  • Figure 4: Quotients for the Lovász theta number $\frac{z_\mathcal{S}}{z_{sdp}}$ and $\frac{z_n}{z_{sdp}}$ as $n$ grows for random $d-$regular graphs with different values of $d$, as $n$ grows.
  • Figure 5: Quality of the ratios $\frac{z_\mathcal{S}}{z_{sdp}}$ (eigen cuts), $\frac{z_n}{z_{sdp}}$, (oracle cuts), $\frac{Z_0}{z_{sdp}}$ (base cuts) and $\frac{z_{soc}}{z_{sdp}}$ for random QCQP instances with density $0.25$.
  • ...and 9 more figures

Theorems & Definitions (39)

  • definition 1
  • theorem 1
  • proof
  • proof : Proof of Observation \ref{['obs:simulDiag']}. Also see wang2022tightness, Lemma $9$
  • theorem 2
  • proof
  • corollary 1
  • proof
  • lemma 1
  • lemma 2
  • ...and 29 more