A second-order cone representable class of nonconvex quadratic programs

• 成果类型: Article; Early Access
• 署名作者: Dey, Santanu S.; Khajavirad, Aida
• 署名单位: Georgia Institute of Technology; University System of Georgia; Georgia Institute of Technology; Lehigh University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02364-y
• 发表日期: 2026-05-26
• 关键词: Nonconvex quadratic programming; convex hull; Second-order cone representable; Boolean quadric polytope; Polynomial-size extended formulation; convex-hull; optimization; relaxations; hierarchy; polytope

摘要

We consider the problem of minimizing a sparse nonconvex quadratic function over the unit hypercube. By developing an extension of the Reformulation-Linearization Technique (RLT) to continuous quadratic sets, we propose a novel second-order cone (SOC) representable relaxation for this problem. By exploiting the sparsity of the quadratic function, we establish a sufficient condition under which the convex hull of the feasible region of the lifted quadratic program is SOC-representable. While the proposed formulation may be of exponential size in general, we identify additional structural conditions that guarantee the existence of a polynomial-size SOC-representable formulation, which can be constructed in polynomial time. Under these conditions, the optimal value of the nonconvex quadratic program coincides with that of a polynomial-size second-order cone program. Our results serve as a starting point for bridging the gap between the Boolean quadric polytope of sparse problems and its continuous counterpart.