Differential Stability of Two-Stage Stochastic Programs

Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function.

[1]  A. Ben-Tal,et al.  Directional derivatives in nonsmooth optimization , 1985 .

[2]  Charles Leake,et al.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1994 .

[3]  J. F. Bonnans,et al.  Perturbed Optimization in Banach spaces I: A General Theory based on a Weak Directional Constraint Qualification , 1996 .

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

[5]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

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

[7]  Jason H. Goodfriend,et al.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1995 .

[8]  R. Wets,et al.  A Lipschitzian characterization of convex polyhedra , 1969 .

[9]  E. Gobet,et al.  Stochastic Linear Programming , 2022 .

[10]  J. Frédéric Bonnans,et al.  Quadratic Growth and Stability in Convex Programming Problems With Multiple Solutions , 1994 .

[11]  Diethard Klatte,et al.  Error bounds for solutions of linear equations and inequalities , 1995, Math. Methods Oper. Res..

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

[13]  B. Mordukhovich.  Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis , 1994 .

[14]  Darinka Dentcheva,et al.  Differentiable selections and Castaing representations of multifunctions , 1998 .

[15]  J. F. Bonnans,et al.  Second Order Necessary and Sufficient Optimality Conditions under Abstract Constraints , 1996 .

[16]  Petr Lachout.  On multifunction transforms of probability measures , 1995, Ann. Oper. Res..

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

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

[19]  A. Shapiro.  On concepts of directional differentiability , 1990 .

[20]  Alexander Shapiro,et al.  On differential stability in stochastic programming , 1990, Math. Program..

[21]  Alexander Shapiro,et al.  Asymptotic analysis of stochastic programs , 1991, Ann. Oper. Res..

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

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

[24]  R. Wets,et al.  On the Convergence in Distribution of Measurable Multifunctions Random Sets Normal Integrands, Stochastic Processes and Stochastic Infima , 1986 .

[25]  Alexander Shapiro,et al.  Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets , 1999, SIAM J. Optim..

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

[27]  Werner Römisch,et al.  Strong Convexity and Directional Derivatives of Marginal Values in Two-Stage Stochastic Programming , 1995 .

[28]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[29]  Jean-Paul Penot,et al.  Differentiability of Relations and Differential Stability of Perturbed Optimization Problems , 1984 .

[30]  R. Cominetti.  Metric regularity, tangent sets, and second-order optimality conditions , 1990 .

[31]  A. Shapiro.  Sensitivity analysis of nonlinear programs and differentiability properties of metric projections , 1988 .

[32]  R. Rockafellar.  Proto-Differentiability of Set-Valued Mappings and its Applications in Optimization☆ , 1989 .

[33]  Jinde Wang,et al.  Distribution sensitivity analysis for stochastic programs with complete recourse , 1985, Math. Program..

[34]  Roberto Cominetti,et al.  A Comparative Study of Multifunction Differentiability with Applications in Mathematical Programming , 1991, Math. Oper. Res..

[35]  Alberto Seeger.  Second Order Directional Derivatives in Parametric Optimization Problems , 1988, Math. Oper. Res..

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

[37]  Alan J. King,et al.  Generalized Delta Theorems for Multivalued Mappings and Measurable Selections , 1989, Math. Oper. Res..

[38]  K. Marti.  Approximationen der Entscheidungsprobleme mit linearer Ergebnisfunktion und positiv homogener, subadditiver Verlustfunktion , 1975 .

[39]  R. Tyrrell Rockafellar,et al.  Sensitivity analysis for nonsmooth generalized equations , 1992, Math. Program..

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

[41]  Alexander Shapiro,et al.  Optimization Problems with Perturbations: A Guided Tour , 1998, SIAM Rev..

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

[43]  J. Dupacová.  Stability in stochastic programming with recourse. Contaminated distributions , 1986 .

[44]  Alexander D. Ioe.  Quadratic Growth and Stability in Convex Programming Problems with Multiple Solutions , 1995 .

[45]  J. Du.  STABILITY IN STOCHASTIC PROGRAMMING WITH RECOURSE. CONTAMINATED DISTRIBUTIONS , 1986 .