Applying the Minimum Risk Criterion in Stochastic Recourse Programs

In the setting of stochastic recourse programs, we consider the problem of minimizing the probability of total costs exceeding a certain threshold value. The problem is referred to as the minimum risk problem and is posed in order to obtain a more adequate description of risk aversion than that of the accustomed expected value problem. We establish continuity properties of the recourse function as a function of the first-stage decision, as well as of the underlying probability distribution of random parameters. This leads to stability results for the optimal solution of the minimum risk problem when the underlying probability distribution is subjected to perturbations. Furthermore, an algorithm for the minimum risk problem is elaborated and we present results of some preliminary computational experiments.

[1]  David L. Woodruff,et al.  Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming , 1996, J. Heuristics.

[2]  R. Schultz,et al.  Solving stochastic programs with integer recourse by enumeration: a framework using Gro¨bner basis reductions , 1998 .

[3]  J. Hoffmann-jorgensen,et al.  Probability with a View Toward Statistics , 1994 .

[4]  B. Bank,et al.  Non-Linear Parametric Optimization , 1983 .

[5]  Leen Stougie,et al.  Approximation in Stochastic integer programming , 2003 .

[6]  Rüdiger Schultz.  Rates of Convergence in Stochastic Programs with Complete Integer Recourse , 1996, SIAM J. Optim..

[7]  Raymond Hemmecke,et al.  Decomposition of test sets in stochastic integer programming , 2003, Math. Program..

[8]  Nikolaos V. Sahinidis,et al.  A finite branch-and-bound algorithm for two-stage stochastic integer programs , 2004, Math. Program..

[9]  R. Wets.  Stochastic Programs with Fixed Recourse: The Equivalent Deterministic Program , 1974 .

[10]  Rüdiger Schultz.  On structure and stability in stochastic programs with random technology matrix and complete integer recourse , 1995, Math. Program..

[11]  Andrey I. Kibzun,et al.  Stochastic Programming Problems with Probability and Quantile Functions , 1996 .

[12]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[13]  Peter Kall,et al.  On approximations and stability in stochastic programming , 1987 .

[14]  Werner Römisch,et al.  Stability analysis for stochastic programs , 1991, Ann. Oper. Res..

[15]  Svetlozar T. Rachev,et al.  Quantitative Stability in Stochastic Programming: The Method of Probability Metrics , 2002, Math. Oper. Res..

[16]  J Figueira,et al.  Stochastic Programming , 1998, J. Oper. Res. Soc..

[17]  Werner Römisch,et al.  Stability of Solutions for Stochastic Programs with Complete Recourse , 1993, Math. Oper. Res..

[18]  J. Dupacová.  Stochastic programming with incomplete information:a surrey of results on postoptimization and sensitivity analysis , 1987 .

[19]  Zvi Artstein,et al.  Stability Results for Stochastic Programs and Sensors, Allowing for Discontinuous Objective Functions , 1994, SIAM J. Optim..

[20]  Werner Römisch,et al.  Distribution sensitivity in stochastic programming , 1991, Math. Program..

[21]  Alexander Shapiro,et al.  Quantitative stability in stochastic programming , 1994, Math. Program..

[22]  Maarten H. van der Vlerk,et al.  On the expected value function of a simple integer recourse problem with random technology matrix , 1994 .

[23]  S. M. Robinson,et al.  Stability in two-stage stochastic programming , 1987 .

[24]  Julia L. Higle,et al.  Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse , 1991, Math. Oper. Res..

[25]  Stephen M. Robinson,et al.  Local epi-continuity and local optimization , 1987, Math. Program..

[26]  Rüdiger Schultz.  Continuity and Stability in Two-Stage Stochastic Integer Programming , 1992 .

[27]  Rüdiger Schultz.  Continuity Properties of Expectation Functions in Stochastic Integer Programming , 1993, Math. Oper. Res..

[28]  Maarten H. van der Vlerk,et al.  Stochastic programming with simple integer recourse , 1993, Math. Program..

[29]  Peter Kall,et al.  Stochastic Programming , 1995 .

[30]  Andrzej Ruszczynski,et al.  A regularized decomposition method for minimizing a sum of polyhedral functions , 1986, Math. Program..

[31]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[32]  R. Wets,et al.  L-SHAPED LINEAR PROGRAMS WITH APPLICATIONS TO OPTIMAL CONTROL AND STOCHASTIC PROGRAMMING. , 1969 .

[33]  Rüdiger Schultz,et al.  Dual decomposition in stochastic integer programming , 1999, Oper. Res. Lett..

[34]  Leen Stougie.  Design and analysis of algorithms for stochastic integer programming , 1987 .