Quantitative Stability in Stochastic Programming: The Method of Probability Metrics

Quantitative stability of optimal values and solution sets to stochastic programming problems is studied when the underlying probability distribution varies in some metric space of probability measures. We give conditions that imply that a stochastic program behaves stable with respect to a minimal information (m.i.) probability metric that is naturally associated with the data of the program. Canonical metrics bounding the m.i. metric are derived for specific models, namely for linear two-stage, mixed-integer two-stage and chance-constrained models. The corresponding quantitative stability results as well as some consequences for asymptotic properties of empirical approximations extend earlier results in this direction. In particular, rates of convergence in probability are derived under metric entropy conditions. Finally, we study stability properties of stable investment portfolios having minimal risk with respect to the spectral measure and stability index of the underlying stable probability distribution.

[1]  B. Mordukhovich.  Lipschitzian stability of constraint systems and generalized equations , 1994 .

[2]  B. WETSt,et al.  STOCHASTIC PROGRAMS WITH FIXED RECOURSE : THE EQUIVALENT DETERMINISTIC PROGRAM , 2022 .

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

[4]  Related Topics,et al.  Parametric Optimization and Related Topics V , 1987 .

[5]  Vlasta Kaňková.  A note on estimates in stochastic programming , 1994 .

[6]  Roger J.-B. Wets,et al.  Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems , 1995, Ann. Oper. Res..

[7]  R. Fortet,et al.  Convergence de la répartition empirique vers la répartition théorique , 1953 .

[8]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[9]  Jitka Dupačová,et al.  Applications of stochastic programming under incomplete information , 1994 .

[10]  F. Nožička.  Theorie der linearen parametrischen Optimierung , 1974 .

[11]  S. Rachev,et al.  Modeling asset returns with alternative stable distributions , 1993 .

[12]  Werner Römisch,et al.  Obtaining convergence rates for approximations in stochastic programming , 1987 .

[13]  Stephen M. Robinson,et al.  Analysis of Sample-Path Optimization , 1996, Math. Oper. Res..

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

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

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

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

[18]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[19]  S. Rachev,et al.  Stable Paretian Models in Finance , 2000 .

[20]  V. M. Zolotarev,et al.  Addendum: Probability Metrics , 1984 .

[21]  Roger J.-B. Wets,et al.  Quantitative Stability of Variational Systems II. A Framework for Nonlinear Conditioning , 1993, SIAM J. Optim..

[22]  P. L. Hammer,et al.  Mathematical programming in theory and practice : proceedings of the NATO Advanced Study Institute, Figueira da Foz, Portugal, June 12-23, 1972 , 1974 .

[23]  Georg Ch. Pflug,et al.  Stochastic programs and statistical data , 1999, Ann. Oper. Res..

[24]  M. Talagrand.  The Glivenko-Cantelli problem, ten years later , 1996 .

[25]  R. Wets,et al.  Consistency of Minimizers and the SLLN for Stochastic Programs 1 , 1995 .

[26]  Z. Artstein.  Sensitivity with respect to the underlying information in stochastic programs , 1994 .

[27]  B. Mandelbrot.  New Methods in Statistical Economics , 1963, Journal of Political Economy.

[28]  R. Wets,et al.  Epi‐consistency of convex stochastic programs , 1991 .

[29]  École d'été de probabilités de Saint-Flour,et al.  École d'Été de Probabilités de Saint-Flour XII - 1982 , 1984 .

[30]  George L. Nemhauser,et al.  Handbooks in operations research and management science , 1989 .

[31]  René Henrion,et al.  Metric regularity and quantitative stability in stochastic programs with probabilistic constraints , 1999, Math. Program..

[32]  Werner Römisch,et al.  Differential Stability of Two-Stage Stochastic Programs , 2000, SIAM J. Optim..

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

[34]  Rüdiger Schultz,et al.  Strong convexity in stochastic programs with complete recourse , 1994 .

[35]  Svetlozar T. Rachev,et al.  MULTIVARIATE STABLE FUTURES PRICES , 1995 .

[36]  Charles E. Blair,et al.  The value function of a mixed integer program: I , 1977, Discret. Math..

[37]  Peter Kall,et al.  Stochastic Linear Programming , 1975 .

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

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

[40]  J. Dupacová,et al.  ASYMPTOTIC BEHAVIOR OF STATISTICAL ESTIMATORS AND OF OPTIMAL SOLUTIONS OF STOCHASTIC OPTIMIZATION PROBLEMS , 1988 .

[41]  J. Dupacová.  Stability and sensitivity-analysis for stochastic programming , 1991 .

[42]  M. Talagrand.  Sharper Bounds for Gaussian and Empirical Processes , 1994 .

[43]  A. Shapiro.  Simulation-based optimization—convergence analysis and statistical inference , 1996 .

[44]  R. Dudley.  A course on empirical processes , 1984 .

[45]  Nicole Gröwe.  Estimated stochastic programs with chance constraints , 1997 .

[46]  Rüdiger Schultz,et al.  Some Aspects of Stability in Stochastic Programming , 2000, Ann. Oper. Res..

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

[48]  Roger J.-B. Wets,et al.  Chapter VIII Stochastic programming , 1989 .

[49]  Georg Ch. Pflug,et al.  On the Glivenko-Cantelli Problem in Stochastic Programming: Linear Recourse and Extensions , 1996, Math. Oper. Res..

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

[51]  C. Grundmann.  Humboldt‐Universität Berlin , 1950 .

[52]  W. Ziemba.  Choosing investment portfolios when the returns have stable distributions , 1972 .

[53]  Werner Römisch,et al.  Lipschitz Stability for Stochastic Programs with Complete Recourse , 1996, SIAM J. Optim..

[54]  Jim Freeman.  Probability Metrics and the Stability of Stochastic Models , 1991 .

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

[56]  R. Tyrrell Rockafellar,et al.  Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming , 1993, Math. Oper. Res..

[57]  S. Vogel.  A stochastic approach to stability in stochastic programming , 1994 .

[58]  Roger J.-B. Wets,et al.  Lifting projections of convex polyhedra. , 1969 .