用渐进式 MIP 方法改进带互补约束的不定二次程序和线性程序的求解方法

Xinyao Zhang, Shaoning Han, Jong-Shi Pang
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引用次数: 0

摘要

众所周知,无限二次型程序(QPs)很难求解到全局最优,具有线性互补约束的线性程序也是如此。本文将前者视为后者的一个子类,提出了一种渐进式混合整数线性规划方法,用于求解具有线性互补约束的一般线性规划(LPCC)。本文提出的方法不是用一整套表示互补条件的整数变量来求解 LPCC,而是从一小部分整数变量开始,逐步增加这部分变量,从而求解有限数量的混合整数子程序。在介绍了 PIP(渐进整数编程)方法及其各种实现方法后,我们通过大量的计算实验证明,渐进方法比直接求解 LPCC 的全整数公式性能更优越。实验还表明,在 PIP 方法结束时得到的解是 LPCC 的局部最小值,这是任何已知的非数值方法都无法解决的非凸程序。在所有实验中,PIP 方法都是在非线性编程求解器获得的 LPCC 可行解上启动的,并且很有可能成功改进 LPCC。因此,PIP 方法可以改进不定 QP 的静态解,而这是非线性编程方法不可能实现的。最后,本文通过分析,更好地理解了 LPCC 次优解在不定 QP 局部最优中的作用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Improving the Solution of Indefinite Quadratic Programs and Linear Programs with Complementarity Constraints by a Progressive MIP Method
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.
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