随机P矩阵和随机P0矩阵线性互补问题

定义了随机P矩阵和随机P0矩阵,给出了矩阵为随机P矩阵或随机P0矩阵的充要条件.研究了随机线性互补问题(SLCP)的矩阵为随机P矩阵时,期望残差方法(ERM)解集的有界性.得到了期望矩阵为P矩阵时,(ERM)解集非空有界.并且研究离散情形(ERM)与期望值方法(EV)解的关系,给出了(ERM)解唯一的条件.     

Ellipsoids that contain all the solutions of a positive semi-definite linear complementarity problem

This paper deals with the LCP (linear complementarity problem) with a positive semi-definite matrix. Assuming that a strictly positive feasible solution of the LCP is available, we propose ellipsoids each of which contains all the solutions of the LCP. We use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the LCP. We can apply the lower bound to test whether a given variable is positive over the solution set of the LCP. That is, if the lower bound is positive, we know that the variable is positive over the solution set of the LCP; hence, by the complementarity condition, its complement is zero. In this case we can eliminate the variable and its complement from the LCP. We also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path-following algorithm proposed by the authors for the LCP. If the LCP has a unique non-degenerate solution, the lower bound and the upper bound for the solution, computed at each iteration of the path-following algorithm, both converge to the solution of the LCP.Supported by Grant-in-Aids for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.Supported by Grant-in-Aids for Young Scientists (63730014) and for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.     

A Barzilai–Borwein type method for stochastic linear complementarity problems

We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.     

An algorithm based on a sequence of linear complementarity problems applied to a walrasian equilibrium model: An example

The Walrasian equilibrium problem is cast as a complementarity problem, and its solution is computed by solving a sequence of linear complementarity problems (SLCP). Earlier numerical experiments have demonstrated the computational efficiency of this approach. So far, however, there exist few relevant theoretical results that characterize the performance of this algorithm. In the context of a simple example of a Walrasian equilibrium model, we study the iterates of the SLCP algorithm. We show that a particular LCP of this process may have no, one or more complementary solutions. Other LCPs may have both homogeneous and complementary solutions. These features complicate the proof of convergence for the general case. For this particular example, however, we are able to show that Lemke's algorithm computes a solution to an LCP if one exists,and that the iterative process converges globally.     

A continuation method for linear complementarity problems with P 0 matrix

In this article, we propose a new continuation method for solving the linear complementarity problem (LCP). The method solves one system of linear equations and carries out only a one-line search at each iteration. The continuation method is based on a modified smoothing function. The existence and continuity of a smooth path for solving the LCP with a P ₀ matrix are discussed. We investigate the boundedness of the iteration sequence generated by our continuation method under the assumption that the solution set of the LCP is nonempty and bounded. It is shown to converge to an LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution under suitable assumption. In addition, some numerical results are also reported in this article.     

Topological Properties of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

We establish a continuous mapping of the space of the matrix elements of an arbitrary nonempty set of quasi solutions of Lipschitzian quantum stochastic differential inclusion (QSDI) into the space of the matrix elements of its solutions. As a corollary, we furnish a generalization of a previous selection result. In particular, when the coefficients of the inclusion are integrably bounded, we show that the space of the matrix elements of solutions is an absolute retract, contractible, locally and integrally connected in an arbitrary dimension. Permanent address: E.O. Ayoola, Department of Mathematics, University of Ibadan, Ibadan, Nigeria.     

Existence of optimal solutions for general stochastic linear complementarity problems

In this paper we show the solvability of the expected residual minimization (ERM) formulation for the general stochastic linear complementarity problem (SLCP) under mild assumptions. The properties of the ERM formulation are dependent on the choice of NCP functions. We focus on the ERM formulations defined by the “min” NCP function and the penalized FB function, both of which are nonconvex programs on the nonnegative orthant.     

随机线性互补问题的无约束优化再定式

针对随机线性互补问题,提出等价的无约束优化再定式模型,即由D-间隙函数定义的确定性的无约束期望残差极小化问题.通过拟Monte Carlo方法,将样本进行了推广,得到了相关的离散近似问题.在适当的条件下,提出了最优解存在的充分条件,以及探究了离散近似问题的最优解及稳定点的收敛性.另外,在针对一类带有常系数矩阵的随机互补线性问题,研究了解存在的充要条件.     

A multi-space sampling heuristic for the vehicle routing problem with stochastic demands

The vehicle routing problem with stochastic demands consists in designing transportation routes of minimal expected cost to satisfy a set of customers with random demands of known probability distributions. This paper proposes a simple yet effective heuristic approach that uses randomized heuristics for the traveling salesman problem, a tour partitioning procedure, and a set partitioning formulation to sample the solution space and find high-quality solutions for the problem. Computational experiments on benchmark instances from the literature show that the proposed approach is competitive with the state-of-the-art algorithm for the problem in terms of both accuracy and efficiency. In experiments conducted on a set of 40 instances, the proposed approach unveiled four new best-known solutions (BKSs) and matched another 24. For the remaining 12 instances, the heuristic reported average gaps with respect to the BKS ranging from 0.69 to 0.15 % depending on its configuration.     

一类随机线性互补问题的求法

考虑一类随机互线性补问题的求解方法,目的是通过定义NCP函数来使正则化期望残差最小化.通过拟蒙洛包洛方法产生一系列观察值并且证得离散近似问题最小值解的聚点就是相应随机线性互补问题的期望残差最小值ERM,同时得到利用ERM到解为有界的充分条件.进一步证明ERM法能够得到具有稳定性和最小灵敏度的稳健解.