次模盒约束二次规划的Ω(n) SDP松弛间隙研究

中文摘要 AI 辅助

连续次模优化是一类具有全局保证的重要非凸问题，其中次模盒约束二次规划（BCQP）构成了一个基础子类。它自然产生于一般二次可微次模问题的二次子问题，也直接应用于定价等商业和管理场景。本文研究次模BCQP的半定规划（SDP）松弛在有限族有效线性割下的最坏情况松弛间隙。我们证明，在维度n≥4时，对于任意用有限族线性割构建的SDP松弛，总存在一个次模BCQP实例，其松弛间隙严格为正。通过限制到消除问题缩放因子的归一化BCQP实例，我们进一步用显式下界量化了这一局限性。在一般维度n≥4时，当SDP松弛仅用布尔二次多面体（BQP）有效割构建，包括常用的McCormick不等式和三角不等式时，我们建立了Ω(n)的松弛间隙下界；对于任意m个有效线性割的族，我们建立了Ω(n/m²)的下界。这些结果揭示了SDP松弛，尤其是带BQP有效割的SDP松弛的基本局限性，为次模BCQP更有效的凸近似所需的额外约束提供了指导。

英文摘要

Continuous submodular optimization is an important class of nonconvex problems with global guarantees, for which the submodular box-constrained quadratic programming (BCQP) forms a fundamental subclass. It arises naturally as quadratic subproblems of general twice-differentiable submodular problems and also has direct business and management applications such as pricing. This paper studies the worst-case relaxation gap of semidefinite programming (SDP) relaxations with a finite family of valid linear cuts for submodular BCQP. We show that in dimension $n\geq4$, for any SDP relaxation formulated with a finite family of linear cuts, there always exists a submodular BCQP instance with a strictly positive relaxation gap. By restricting to the normalized BCQP instances that remove the problem's scaling factor, we further quantify this limitation with explicit lower bounds. In general dimension $n\geq4$, when the SDP relaxation is formulated with only Boolean-quadric-polytope (BQP) valid cuts, including the commonly used McCormick and triangle inequalities, we establish an $Ω(n)$ relaxation gap lower bound. For an arbitrary family of $m$ valid linear cuts, we establish an $Ω(n/m^2)$ lower bound. These results capture the fundamental limitations of SDP relaxations, especially SDP with BQP valid cuts, and provide guidance on what additional constraints may be needed for more effective convex approximations of submodular BCQP.

发表机构

• National University of Singapore（新加坡国立大学）

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