Random Test Problems and Parallel Methods for

Random Test Problems and Parallel Methods for Quadratic Programs and Quadratic Stochastic Programs

Xiao jun Chen

3

y

Robert S. Womersley

January 1999, Revised June 1999

Abstract This paper proposes a data parallel procedure for randomly generating test problems for two-stage quadratic stochastic programming. Multiple quadratic programs in the second stage are randomly generated in parallel. A solution of the quadratic stochastic program is determined by multiple symmetric linear complementarity problems. The procedure allows the user to specify the size of the problem, the condition numbers of the Hessian matrices of the objective functions and the structure of the feasible regions in the rst and the second stages. These test problems are used to evaluate three parallel algorithms for multiple quadratic programs and a parallel inexact Newton method for quadratic stochastic programming. Numerical experiments on a Thinking Machine CM-5, Silicon Graphics Power Challenge, and DEC Alpha Server are reported.

Parallel computation, quadratic programs, test problem generation, stochastic programming.

Keywords:

Short title:

Quadratic Programming Test Problems

3 Department

of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan. [email protected]. The work of this author was supported in part by the Scienti c Research Grant C11640119 from the Ministry of Education, Science and Culture of Japan, and the Australian Research Council. y School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia. [email protected]. The work of this author was supported by the Australian Research Council

1

1

Introduction

Many scienti c, engineering and economic optimization problems involve data which is uncertain, and at best can be modelled by a discrete or continuous distribution. Explicitly dealing with the stochastic nature of the problem can produce qualitatively and quantitatively dierent solutions, see for example the recent books by Birge and Louveaux [3] and Kall and Wallace [18]. Stochastic programming techniques have also been used to improve the robustness of large-scale optimization problems [24]. One key to the increasing practical use of stochastic programming models has been in the increasing power of high performance computing systems, in particular parallel computing systems, see for example [28, 35]. However the generally available software is still well behind the state of the art programs for linear and nonlinear programming available from, amongst others, ILOG/CPLEX, NAG and Visual Numerics. Approaches to stochastic programming include posing the problem as a very large sparse optimization problem, decomposition approaches [36], and smaller dimensional, but nonsmooth, outer optimization problems which require a huge number of inner optimization problems to approximate the expected values and appropriate derivative information. The inner optimization problems are typically linear programming or quadratic programming problems [7, 8, 15, 25, 32]. When the random variables are discretely distributed values and the inner optimization problems are quadratic programming problems, the model is called a two-stage quadratic stochastic programming problem, which was introduced by Rockafellar and Wets [32]. This model is closely related, and in some cases equivalent, to extended linear-quadratic programming [29, 40]. Stochastic programming has been successfully used in many decision problems in nance, manufacturing, telecommunications and transportation [4]. In particular two-stage nancial planning problems which use quadratic risk measures produce quadratic stochastic programming problems. So far this has been largely restricted to multi-stage stochastic linear programming models, for example [9]. Ferris and Pang [11] survey engineering and economic applications of complementarity problems, and point out their relation to extended linear-quadratic programming problems. In two-stage quadratic stochastic programming problems with r discrete random variables, calculating the objective value involves the solution of multiple quadratic programs (MQPs) : minimize z 2 < n2

1 2

z T Hz + z T Cj

subject to W z  Qj ;

j = 1; 2; : : : ; r

(1.1)

where H 2