基于面部约化半定松弛的二元二次规划的精确凸重构
Exact Convex Reformulations of Binary Quadratic Programs from Facially Reduced Semidefinite Relaxations
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中文总结 AI 辅助
该研究开发了MIQCR-FR框架，通过面部约化半定松弛实现二元二次规划的精确凸重构，在431个实例上验证其最优解数量优于MIQCR-CB和直接Gurobi。
中文摘要 AI 辅助
我们开发了MIQCR-FR，这是一种从等式约束二元二次问题的面部约化半定松弛中进行精确凸重构的框架。标准混合整数二次凸重构（MIQCR）使用原始松弛的对偶乘子。面部约化（FR）可降低半定矩阵的阶数，但通常无法保证原始对偶乘子的恢复。对于这类问题，我们确定了保留线性等式及其与每个二元变量的乘积如何实现这种恢复。已建立的对偶恢复公式将约化对偶的每个可行解显式扩展到原始对偶，同时保持其界。对于近似乘子估计，我们给出了恢复对偶可行性的谱偏移，以及在受限乘子族内优化不同界的两种构造。最大化对偶界会产生一个信赖域子问题；最大化连续MIQCR界仅需要一次约化特征值计算。后一种构造直接强制执行凸性，而非完全的对偶可行性。我们使用交替方向乘子法（ADMM）的乘子估计来实现它。在来自六个问题族的431个实例上，该完整方法在共同时间预算下验证了296个最优解，而锥形束方法MIQCR-CB验证了156个，直接Gurobi验证了187个。
英文摘要
We develop MIQCR-FR, a framework for exact convex reformulation from facially reduced semidefinite relaxations of equality-constrained binary quadratic problems. Standard mixed-integer quadratic convex reformulation (MIQCR) uses dual multipliers of the original relaxation. Facial reduction (FR) reduces the semidefinite matrix order, but recovery of the original dual multipliers is not guaranteed in general. For this class of problems, we identify how retaining the linear equalities and their products with each binary variable enables this recovery. An established dual recovery formula gives an explicit extension of every feasible solution of the reduced dual to the original dual, preserving its bound. For approximate multiplier estimates, we give a spectral shift restoring dual feasibility and two constructions optimizing different bounds within a restricted multiplier family. Maximizing the dual bound yields a trust-region subproblem; maximizing the continuous MIQCR bound requires only a reduced eigenvalue calculation. The latter construction enforces convexity directly, rather than full dual feasibility. We implement it using multiplier estimates from the alternating direction method of multipliers (ADMM). On 431 instances from six problem families, the complete method certifies 296 optima, compared with 156 for the conic-bundle method MIQCR-CB and 187 for direct Gurobi under a common time budget.
发表机构
- Clemson University（克莱姆森大学）
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