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arxiv: 2604.04417 · v1 · submitted 2026-04-06 · 🧮 math.OC

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

Yongzheng Dai , Chen Chen This is my paper

Pith reviewed 2026-05-10 20:11 UTC · model grok-4.3

classification 🧮 math.OC
keywords primal heuristicnonconvex MIQCQPdynamic convexificationlocal branchingfeasibility pumpmixed-integer quadratic programmingcomputational optimizationQPLIB benchmarks
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The pith

An alternating heuristic with dynamic convexification and parallel local branching finds feasible solutions for three previously unsolved nonconvex MIQCQPs and improves fifteen known best solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a primal heuristic for nonconvex mixed-integer quadratically constrained quadratic programs that alternates between solving a convex relaxation and fixing integer variables toward feasibility, with the relaxation updated dynamically according to the specific structure of each instance. Parallel local branching is then applied to improve the solutions found. Experiments on QPLIB benchmark instances show the method locating feasible points for three cases with no prior known solutions and producing better objective values than the previous best for fifteen other cases, all within five minutes of runtime. A reader would care because many engineering and operations problems are modeled as nonconvex MIQCQPs where quickly locating any feasible point is a practical bottleneck.

Core claim

The authors develop a novel primal heuristic for nonconvex MIQCQPs 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. Parallelized local branching is incorporated to further refine detected solutions. Computational experiments on instances from QPLIB find feasible solutions for three previously unsolved cases and improve the best-known solutions for fifteen instances within five minutes of runtime.

What carries the argument

The dynamically adjusted convex approximation inside an alternating feasibility-pump loop, which generates candidate solutions that parallel local branching then improves.

If this is right

  • The method can serve as a fast standalone primal finder or be embedded inside branch-and-bound solvers to supply good upper bounds early.
  • Dynamic structure-based convexification can replace static relaxations in other primal heuristics for nonconvex quadratic programs.
  • Parallel local branching reduces the time needed to improve an initial feasible point once one is located.
  • The approach scales to instances where global solvers struggle to find any feasible point in short time limits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same alternating structure with dynamic convexification might transfer to other classes of nonconvex mixed-integer nonlinear programs beyond quadratic constraints.
  • If the dynamic adjustment rule is made instance-independent, the method could be tested for robustness across entirely new problem families not represented in QPLIB.
  • A natural extension would be to combine the heuristic with global solvers so that the feasible points it finds are used to prune the search tree in real time.

Load-bearing premise

The dynamic adjustment of convex approximations based on instance structure will reliably produce high-quality starting points that local branching can then improve, without the adjustments introducing bias or instability.

What would settle it

A collection of MIQCQP instances from QPLIB or similar sets on which the heuristic returns no feasible solution within five minutes while another method does, or on which it produces strictly worse objective values than the previously reported best known solutions.

Figures

Figures reproduced from arXiv: 2604.04417 by Chen Chen, Yongzheng Dai.

Figure 1
Figure 1. Figure 1: Caption 21 [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of the time to first feasible solution, and primal integral [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of the optimality gap, and primal integral over MIQCQP [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of the time to first feasible solution, and primal integral [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a primal heuristic for nonconvex MIQCQPs that alternates between a dynamically adjusted convex relaxation (updated via feasibility-pump-style iterations driven by instance-specific structure) and a parallel local-branching refinement phase. The method is presented as an extension of the authors' second-place entry in the 2025 Land-Doig MIP Computational Competition. Computational validation on selected QPLIB instances is reported to yield feasible solutions for three previously unsolved cases and improved best-known solutions for fifteen instances, all within a five-minute runtime limit.

Significance. If the reported computational outcomes prove robust, the work supplies a practical, implementable heuristic that can quickly locate high-quality feasible points for a difficult class of nonconvex quadratic problems where exact solvers often time out. The combination of structure-aware dynamic convexification with parallel local branching is a concrete algorithmic contribution that could be adopted or hybridized by practitioners. The empirical results on standard QPLIB benchmarks provide falsifiable evidence of utility, though the heuristic character of the approach means it is best viewed as a complement to exact methods rather than a replacement.

major comments (2)
  1. [Computational Experiments] Computational Experiments section: The central performance claims rest on results for QPLIB instances, yet the manuscript provides insufficient detail on instance selection criteria, the total number of instances evaluated, the exact baselines (including other primal heuristics and MIP solvers), parameter-tuning protocol, and any statistical measures (e.g., multiple runs, variability, or success-rate tables). Without these, it is difficult to assess whether the reported improvements on 15 instances and feasibility on 3 unsolved cases generalize or are sensitive to the chosen subset.
  2. [§3] Algorithm 1 / §3 (Dynamic Convexification): The description states that convex approximations are 'adjusted based on the structure of the MIQCQP instance,' but the precise update rules, triggers, and safeguards against instability or bias are not formalized with sufficient rigor (e.g., no explicit pseudocode or convergence argument for the alternating loop). This makes it hard to verify that the dynamic adjustment reliably produces high-quality starting points for the subsequent local-branching phase on unseen instances.
minor comments (2)
  1. [Abstract] The abstract should explicitly state the total number of QPLIB instances tested and the precise runtime limit (five minutes) to allow readers to gauge the scope of the claims immediately.
  2. [§2] Notation for the convexification parameters and the local-branching neighborhood size should be introduced consistently and defined at first use; a small nomenclature table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation of minor revision. The comments identify opportunities to improve clarity and reproducibility, which we will address in the revised version.

read point-by-point responses

  1. Referee: [Computational Experiments] Computational Experiments section: The central performance claims rest on results for QPLIB instances, yet the manuscript provides insufficient detail on instance selection criteria, the total number of instances evaluated, the exact baselines (including other primal heuristics and MIP solvers), parameter-tuning protocol, and any statistical measures (e.g., multiple runs, variability, or success-rate tables). Without these, it is difficult to assess whether the reported improvements on 15 instances and feasibility on 3 unsolved cases generalize or are sensitive to the chosen subset.

    Authors: We agree that the current presentation lacks sufficient detail for full reproducibility and assessment of robustness. In the revised manuscript we will expand the Computational Experiments section to explicitly state the instance selection criteria (instances from QPLIB chosen for the presence of nonconvex quadratic terms where exact solvers frequently time out), report the total number of instances evaluated, list all baselines (including Gurobi, SCIP, and standard primal heuristics such as feasibility-pump variants), describe the parameter-tuning protocol, and add statistical measures including success rates and variability across multiple runs. revision: yes

  2. Referee: [§3] Algorithm 1 / §3 (Dynamic Convexification): The description states that convex approximations are 'adjusted based on the structure of the MIQCQP instance,' but the precise update rules, triggers, and safeguards against instability or bias are not formalized with sufficient rigor (e.g., no explicit pseudocode or convergence argument for the alternating loop). This makes it hard to verify that the dynamic adjustment reliably produces high-quality starting points for the subsequent local-branching phase on unseen instances.

    Authors: We acknowledge that the description of the dynamic convexification in §3 and Algorithm 1 would benefit from greater formalization. We will augment the section with expanded pseudocode detailing the update rules, triggers (based on measured violations of quadratic constraints), and safeguards (such as damping to avoid oscillation). As the method is a primal heuristic without optimality guarantees, a full convergence proof is not feasible, but we will add a short discussion of observed empirical stability and references to related feasibility-pump analyses. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes a primal heuristic algorithm for nonconvex MIQCQPs that combines dynamic convexification within a feasibility-pump framework and parallel local branching. Its central claims consist of empirical computational results on external QPLIB benchmark instances, including new feasible solutions for three unsolved cases and improved best-known solutions for fifteen instances. No equations, predictions, or first-principles derivations are presented that reduce by construction to fitted parameters, self-definitions, or prior self-citations. The reference to building upon a prior competition submission is a minor contextual note and does not serve as load-bearing justification for the reported outcomes, which are measured against independent external benchmarks rather than internal consistency alone. The derivation chain is therefore algorithmic and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The heuristic relies on standard convex relaxation techniques and local branching from prior literature; no new free parameters, axioms, or invented entities are introduced in the abstract.

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