Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion

We propose a semidefinite optimization (SDP) model for the class of minimax two-stage stochastic linear optimization problems with risk aversion. The distribution of second-stage random variables belongs to a set of multivariate distributions with known first and second moments. For the minimax stochastic problem with random objective, we provide a tight SDP formulation. The problem with random right-hand side is NP-hard in general. In a special case, the problem can be solved in polynomial time. Explicit constructions of the worst-case distributions are provided. Applications in a productiontransportation problem and a single facility minimax distance problem are provided to demonstrate our approach. In our experiments, the performance of minimax solutions is close to that of data-driven solutions under the multivariate normal distribution and better under extremal distributions. The minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity.

[1]  Alexander Shapiro,et al.  On the Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs , 2000, SIAM J. Optim..

[2]  Alexander Shapiro,et al.  Minimax analysis of stochastic problems , 2002, Optim. Methods Softw..

[3]  R. Wets,et al.  Stochastic programming , 1989 .

[4]  Kathrin Klamroth,et al.  Biconvex sets and optimization with biconvex functions: a survey and extensions , 2007, Math. Methods Oper. Res..

[5]  Shabbir Ahmed,et al.  Convexity and decomposition of mean-risk stochastic programs , 2006, Math. Program..

[6]  Melvyn Sim,et al.  TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION , 2009 .

[7]  Werner Römisch,et al.  Polyhedral Risk Measures in Stochastic Programming , 2005, SIAM J. Optim..

[8]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[9]  David P. Rutenberg,et al.  Technical Note - Risk Aversion in Stochastic Programming with Recourse , 1973, Oper. Res..

[10]  J. Dupacová The minimax approach to stochastic programming and an illustrative application , 1987 .

[11]  A. Mehrez,et al.  Note—The Single Facility Minimax Distance Problem Under Stochastic Location of Demand , 1980 .

[12]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[13]  J. Jensen Sur les fonctions convexes et les inégalités entre les valeurs moyennes , 1906 .

[14]  M. Teboulle,et al.  Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming , 1986 .

[15]  C. Floudas,et al.  A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—I. Theory , 1990 .

[16]  Alexander Shapiro,et al.  On a Class of Minimax Stochastic Programs , 2004, SIAM J. Optim..

[17]  D. Rutenberg RISK AVERSION IN STOCHASTIC PROGRAMMING WITH RECOURSE. , 1968 .

[18]  David P. Morton,et al.  Second-Order Lower Bounds on the Expectation of a Convex Function , 2005, Math. Oper. Res..

[19]  C. HuangC.,et al.  Bounds on the Expectation of a Convex Function of a Random Variable , 1977 .

[20]  K. Isii On sharpness of tchebycheff-type inequalities , 1962 .

[21]  Alexander Shapiro,et al.  The Sample Average Approximation Method for Stochastic Discrete Optimization , 2002, SIAM J. Optim..

[22]  A. Madansky Bounds on the Expectation of a Convex Function of a Multivariate Random Variable , 1959 .

[23]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[24]  Michèle Breton,et al.  Algorithms for the solution of stochastic dynamic minimax problems , 1995, Comput. Optim. Appl..

[25]  M. Teboulle,et al.  AN OLD‐NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT , 2007 .

[26]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[27]  M. Sion On general minimax theorems , 1958 .

[28]  Richard E. Wendell,et al.  Minimization of a Non-Separable Objective Function Subject to Disjoint Constraints , 1976, Oper. Res..

[29]  Jitka Dupačová,et al.  Stress Testing via Contamination , 2004, Coping with Uncertainty.

[30]  Yinyu Ye,et al.  Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems , 2010, Oper. Res..

[31]  Morten Riis,et al.  Applying the minimax criterion in stochastic recourse programs , 2005, Eur. J. Oper. Res..

[32]  O. Mangasarian,et al.  A variable-complexity norm maximization problem , 1986 .