Improving the Solution of Indefinite Quadratic Programs and Linear Programs with Complementarity Constraints by a Progressive MIP Method
Xinyao Zhang, Shaoning Han, Jong-Shi Pang
TL;DR
The paper tackles the global optimization of indefinite QPs and LPCCs, presenting Progressive Integer Programming (PIP), a MILP-based method that solves a sequence of reduced mixed-integer subproblems with progressively larger fractions of integer variables, warm-started from an initial LPCC feasible solution obtained via NLP. It establishes a theoretical bridge between LPCC subproblems and the original QP through fixed-point properties and reduced-KKT constructions, showing that, under exact computation, PIP solutions are local minimizers of the underlying QP or LPCC. Empirically, PIP consistently improves NLP-derived solutions and, when possible, matches or closely approximates global MILP solutions with substantially less computational effort, with strong scalability to large problem instances. By integrating NLP and MILP paradigms, the method leverages advances from both realms to efficiently obtain high-quality solutions for nonconvex QCQP and LPCC problems and clarifies how LPCC suboptimal solutions relate to local optimality of the QP.
Abstract
Indefinite quadratic programs (QPs) are known to be very difficult to be solved to global optimality, so are linear programs with linear complementarity constraints. Treating the former as a subclass of the latter, this paper presents a progressive mixed integer linear programming method for solving a general linear program with linear complementarity constraints (LPCC). Instead of solving the LPCC with a full set of integer variables expressing the complementarity conditions, the presented method solves a finite number of mixed integer subprograms by starting with a small fraction of integer variables and progressively increasing this fraction. After describing the PIP (for progressive integer programming) method and its various implementations, we demonstrate, via an extensive set of computational experiments, the superior performance of the progressive approach over the direct solution of the full-integer formulation of the LPCCs. It is also shown that the solution obtained at the termination of the PIP method is a local minimizer of the LPCC, a property that cannot be claimed by any known non-enumerative method for solving this nonconvex program. In all the experiments, the PIP method is initiated at a feasible solution of the LPCC obtained from a nonlinear programming solver, and with high likelihood, can successfully improve it. Thus, the PIP method can improve a stationary solution of an indefinite QP, something that is not likely to be achievable by a nonlinear programming method. Finally, some analysis is presented that provides a better understanding of the roles of the LPCC suboptimal solutions in the local optimality of the indefinite QP.