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Beyond binarity: Semidefinite programming for ternary quadratic problems

Frank de Meijer, Veronica Piccialli, Renata Sotirov, Antonio M. Sudoso

Abstract

We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in $\{0, \pm 1\}$. While semidefinite programming (SDP) techniques are well established for $\{0,1\}$- and $\{\pm 1\}$-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and $k$-gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.

Beyond binarity: Semidefinite programming for ternary quadratic problems

Abstract

We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in . While semidefinite programming (SDP) techniques are well established for - and -valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and -gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.

Paper Structure

This paper contains 20 sections, 14 theorems, 51 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Characterizations of PSD $\mathbf{\{0,\pm 1\}}-$matrices and related polyhedra
  3. ISDP formulation for the ternary quadratic problem
  4. SDP relaxation and valid inequalities
  5. Special cases of the TQP
  6. The quadratic unconstrained ternary optimization problem
  7. Ternary quadratic problem with one linear constraint
  8. Ternary quadratic problem with a ratio objective
  9. SDP-based branch-and-bound algorithm for the TQP
  10. Primal heuristic
  11. VNS for QUTO
  12. VNS for TQP-Linear
  13. VNS for TQP-Ratio
  14. Cut separation
  15. Branching strategy
  16. ...and 5 more sections

Key Result

Proposition 1

Let $X \in \{0, \pm 1\}^{n \times n}$ be symmetric. Then, $X \succeq \mathbf{0}$ if and only if $X = \sum_{i = 1}^r x_ix_i^\top$ for some $x_i \in \{0,\pm 1\}^n$, $i \in [r]$.

Figures (1)

  • Figure 1: Survival plots comparing SDP-B&B with GUROBI on QUTO, TQP-Linear and TQP-Ratio instances of size $n \in \{60, 70, 80 ,90\}$. The plot reports the percentage of problems solved over time, where the time is represented on a logarithmic scale.

Theorems & Definitions (23)

  • Proposition 1: DeMeijerSotirov23
  • Corollary 2
  • Proposition 3: DeMeijerSotirov23
  • Proposition 4: DezaLaurent1997
  • Proposition 5: Letchford2008BinaryPS
  • Theorem 6
  • Proof 1
  • Remark 1
  • Proposition 7
  • Proof 2
  • ...and 13 more