Andrzej Ruszczyński

Lectures on Stochastic Programming: Modeling and Theory

“SPbook”2009/5/4page iiiiiiiiiiDarinka DentchevaDepartment of Mathematical SciencesStevens Institute of TechnologyHoboken, NJ 07030, USAAndrzej Ruszczynski´Department of Management Science and Information SystemsRutgers UniversityPiscataway, NJ 08854, USAAlexander ShapiroSchool of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlanta, GA 30332, USAThe authors dedicate this book:to Julia, Benjamin, Daniel, Natan and Yael;to Tsonka, Konstatin and Marek;and to the Memory of Feliks, Maria, and Dentcho.

Dual Stochastic Dominance and Related Mean-Risk Models

We consider the problem of constructing mean-risk models which are consistent with the second degree stochastic dominance relation. By exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions. This allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation. We also provide stochastic linear programming formulations of these models.

From Stochastic Dominance to Mean-Risk Models: Semideviations as Risk Measures

Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible trade-off analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean-variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second degree stochastic dominance, provided that the trade-off coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome-Risk diagram, which appears to be particularly useful for comparing uncertain outcomes.

A New Scenario Decomposition Method for Large-Scale Stochastic Optimization

A novel parallel decomposition algorithm is developed for large, multistage stochastic optimization problems. The method decomposes the problem into subproblems that correspond to scenarios. The subproblems are modified by separable quadratic terms to coordinate the scenario solutions. Convergence of the coordination procedure is proven for linear programs. Subproblems are solved using a nonlinear interior point algorithm. The approach adjusts the degree of decomposition to fit the available hardware environment. Initial testing on a distributed network of workstations shows that an optimal number of computers depends upon the work per subproblem and its relation to the communication capacities. The algorithm has promise for solving stochastic programs that lie outside current capabilities.

A branch and bound method for stochastic global optimization

A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem.

A regularized decomposition method for minimizing a sum of polyhedral functions

A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.

Optimization with stochastic dominance constraints

We introduce stochastic optimization problems involving stochastic dominance constraints. We develop necessary and sufficient conditions of optimality and duality theory for these models and show that the Lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions. The models and results are illustrated on a portfolio optimization problem.

On consistency of stochastic dominance and mean–semideviation models

Abstract.We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically.

Risk-averse dynamic programming for Markov decision processes

We introduce the concept of a Markov risk measure and we use it to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model. For both models we derive risk-averse dynamic programming equations and a value iteration method. For the infinite horizon problem we develop a risk-averse policy iteration method and we prove its convergence. We also propose a version of the Newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence. Finally, we discuss relations to min–max Markov decision models.

Decomposition methods in stochastic programming

Stochastic programming problems have very large dimension and characteristic structures which are tractable by decomposition. We review basic ideas of cutting plane methods, augmented Lagrangian and splitting methods, and stochastic decomposition methods for convex polyhedral multi-stage stochastic programming problems.