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An Alternating Primal Heuristic for Nonconvex MIQCQP with Dynamic Convexification and Parallel Local Branching

Yongzheng Dai, Chen Chen

Abstract

We develop a novel primal heuristic for nonconvex Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQPs). The method is built around a convex approximation that is dynamically adjusted within a feasibility-pump-style alternating heuristic. Approximations are adjusted based on the structure of the MIQCQP instance. Additionally, parallelized local branching is incorporated to further refine detected solutions. This paper builds upon the second-place finalist submission in the 2025 Land-Doig MIP Computational Competition. Our results are validated with computational experiments on instances from QPLIB, finding feasible solutions for three previously unsolved cases and improving the best-known solutions for fifteen instances within five minutes of runtime.

An Alternating Primal Heuristic for Nonconvex MIQCQP with Dynamic Convexification and Parallel Local Branching

Abstract

We develop a novel primal heuristic for nonconvex Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQPs). The method is built around a convex approximation that is dynamically adjusted within a feasibility-pump-style alternating heuristic. Approximations are adjusted based on the structure of the MIQCQP instance. Additionally, parallelized local branching is incorporated to further refine detected solutions. This paper builds upon the second-place finalist submission in the 2025 Land-Doig MIP Computational Competition. Our results are validated with computational experiments on instances from QPLIB, finding feasible solutions for three previously unsolved cases and improving the best-known solutions for fifteen instances within five minutes of runtime.

Paper Structure

This paper contains 33 sections, 4 theorems, 56 equations, 4 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Organization
  4. Convex Approximations
  5. Established Convex Reformulations
  6. Eigenvalue-based Relaxation
  7. Eigenvalue-based Approximation
  8. Approximation for MI(B)QPs
  9. Redesigning Perturbations
  10. Primal Heuristics
  11. Mixed-Integer Box-Constrained Quadratic Programs
  12. Mixed-Integer Quadratic programming
  13. Case 1
  14. Case 2
  15. Mixed-Integer Quadratically-Constrained Quadratic Programming
  16. ...and 18 more sections

Key Result

Proposition 1

Define $\mathrm{Q}_{\mathrm{MIQP}}:\min_x\ x^TQ^0x + a^Tx$ with $n_\alpha$ binary variables $x_\alpha$, $n_\beta$ continuous variables $x_\beta \geq 0$, and Let $x(\hat{u})=(x_\alpha(\hat{u}), x_\beta(\hat{u}))$ be an optimal solution to $\mathrm{Approx}(\hat{u})$ for $\mathrm{Q}_{\mathrm{MIQP}}$. If $\hat{u}_\beta=2x_\beta(\hat{u})$, then $x_\beta(\hat{u})$ satisfies stationarity of $\mathrm{Q}_

Figures (4)

  • Figure 1: Caption
  • Figure 2: Distribution of the time to first feasible solution, and primal integral over modified and original eigenvalue methods on MIQCQPs.
  • Figure 3: Distribution of the optimality gap, and primal integral over MIQCQP with different parallel settings on MIQCQPs.
  • Figure 4: Distribution of the time to first feasible solution, and primal integral over Gurobi and our proposed method on MIQCQPs.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof