Solving Nonsmooth Nonconvex Compound Stochastic Programs with Applications to Risk Measure Minimization

This paper studies a structured compound stochastic program involving multiple expectations coupled by nonconvex and nonsmooth functions. We present a successive convex-programming based sampling algorithm for solving this problem, establish its subsequential convergence, and discuss a probabilistic stopping rule based on the computable error-bound for the algorithm. We present several risk measure minimization problems that can be formulated as such a compound stochastic program; these include generalized deviation optimization problems based on optimized certainty equivalent and buffered probability of exceedance (bPOE), a distributionally robust bPOE optimization problem, and a multiclass classification problem employing the cost-sensitive error criteria with bPOE risk measure.

[1]  Alexander Shapiro,et al.  Estimation and asymptotics for buffered probability of exceedance , 2018, Eur. J. Oper. Res..

[2]  A. Kleywegt,et al.  Distributionally Robust Stochastic Optimization with Wasserstein Distance , 2016, Math. Oper. Res..

[3]  A. Shapiro,et al.  Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions , 2007 .

[4]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[5]  Katya Scheinberg,et al.  Stochastic optimization using a trust-region method and random models , 2015, Mathematical Programming.

[6]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings , 2009 .

[7]  Feng Ruan,et al.  Stochastic Methods for Composite and Weakly Convex Optimization Problems , 2017, SIAM J. Optim..

[8]  Saeed Ghadimi,et al.  A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization , 2018, SIAM J. Optim..

[9]  Zvi Artstein,et al.  Law of Large Numbers for Random Sets and Allocation Processes , 1981, Math. Oper. Res..

[10]  Jong-Shi Pang,et al.  Computing B-Stationary Points of Nonsmooth DC Programs , 2015, Math. Oper. Res..

[11]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[12]  Houra Mahmoudzadeh,et al.  A robust-CVaR optimization approach with application to breast cancer therapy , 2014, Eur. J. Oper. Res..

[13]  G. Pflug,et al.  The 1/ N investment strategy is optimal under high model ambiguity , 2012 .

[14]  Jong-Shi Pang,et al.  Composite Difference-Max Programs for Modern Statistical Estimation Problems , 2018, SIAM J. Optim..

[15]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[16]  Niao He,et al.  Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization , 2019, SIAM J. Optim..

[17]  Stan Uryasev,et al.  Generalized deviations in risk analysis , 2004, Finance Stochastics.

[18]  Stan Uryasev,et al.  Buffered Probability of Exceedance: Mathematical Properties and Optimization , 2018, SIAM J. Optim..

[19]  A. Ruszczynski,et al.  Statistical estimation of composite risk functionals and risk optimization problems , 2015, 1504.02658.

[20]  Mengdi Wang,et al.  Multilevel Stochastic Gradient Methods for Nested Composition Optimization , 2018, SIAM J. Optim..

[21]  Laurent El Ghaoui,et al.  Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach , 2003, Oper. Res..

[22]  Johannes O. Royset,et al.  On buffered failure probability in design and optimization of structures , 2010, Reliab. Eng. Syst. Saf..

[23]  Dmitriy Drusvyatskiy,et al.  Error Bounds, Quadratic Growth, and Linear Convergence of Proximal Methods , 2016, Math. Oper. Res..

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

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

[26]  Sanjay Mehrotra,et al.  Distributionally Robust Optimization: A Review , 2019, ArXiv.

[27]  Mengdi Wang,et al.  Stochastic compositional gradient descent: algorithms for minimizing compositions of expected-value functions , 2014, Mathematical Programming.

[28]  Stan Uryasev,et al.  Maximization of AUC and Buffered AUC in binary classification , 2018, Mathematical Programming.

[29]  Le Thi Hoai An,et al.  Stochastic Difference-of-Convex Algorithms for Solving nonconvex optimization problems , 2019, ArXiv.

[30]  Tito Homem-de-Mello,et al.  Variable-sample methods for stochastic optimization , 2003, TOMC.

[31]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[32]  Francisco Facchinei,et al.  Exact penalization via dini and hadamard conditional derivatives , 1998 .

[33]  Stavros A. Zenios,et al.  Robust VaR and CVaR Optimization under Joint Ambiguity in Distributions, Means, and Covariances , 2016, Eur. J. Oper. Res..

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

[35]  Jong-Shi Pang,et al.  On the pervasiveness of difference-convexity in optimization and statistics , 2017, Math. Program..

[36]  Yuri M. Ermoliev,et al.  Sample Average Approximation Method for Compound Stochastic Optimization Problems , 2013, SIAM J. Optim..

[37]  E. Polak,et al.  Extensions of Stochastic Optimization Results to Problems with System Failure Probability Functions , 2007 .

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