Stability analysis for stochastic programs

For stochastic programs with recourse and with (several joint) probabilistic constraints, respectively, we derive quantitative continuity properties of the relevant expectation functionals and constraint set mappings. This leads to qualitative and quantitative stability results for optimal values and optimal solutions with respect to perturbations of the underlying probability distributions. Earlier stability results for stochastic programs with recourse and for those with probabilistic constraints are refined and extended, respectively. Emphasis is placed on equipping sets of probability measures with metrics that one can handle in specific situations. To illustrate the general stability results we present possible consequences when estimating the original probability measure via empirical ones.     

Bounding multi-stage stochastic programs from above

The purpose of this paper is to present general approaches for bounding some multi-stage stochastic programs from above. The results are based on restricting the solution set, such that the remaining multi-stage stochastic program is easy to solve. An example where the methods can be applied is presented.Supported in part by NATO Collaborative Research Grant No. 0785/87.     

A gaussian upper bound for gaussian multi-stage stochastic linear programs

This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to find upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semidefinite program. The algorithm for the multi-stage problem invovles the solution of a quadratically constrained convex programming problem.     

On the implementation of a log-barrier progressive hedging method for multistage stochastic programs

A progressive hedging method incorporated with self-concordant barrier for solving multistage stochastic programs is proposed recently by Zhao [G. Zhao, A Lagrangian dual method with self-concordant barrier for multistage stochastic convex nonlinear programming, Math. Program. 102 (2005) 1-24]. The method relaxes the nonanticipativity constraints by the Lagrangian dual approach and smoothes the Lagrangian dual function by self-concordant barrier functions. The convergence and polynomial-time complexity of the method have been established. Although the analysis is done on stochastic convex programming, the method can be applied to the nonconvex situation. We discuss some details on the implementation of this method in this paper, including when to terminate the solution of unconstrained subproblems with special structure and how to perform a line search procedure for a new dual estimate effectively. In particular, the method is used to solve some multistage stochastic nonlinear test problems. The collection of test problems also contains two practical examples from the literature. We report the results of our preliminary numerical experiments. As a comparison, we also solve all test problems by the well-known progressive hedging method.     

On complexity of multistage stochastic programs

In this paper we derive estimates of the sample sizes required to solve a multistage stochastic programming problem with a given accuracy by the (conditional sampling) sample average approximation method. The presented analysis is self-contained and is based on a relatively elementary, one-dimensional, Cramér's Large Deviations Theorem.     

On some methods of deriving geometric expectation

Formulas presented for the calculation of ∑ _j=1 ⁿ j^k (n, k ∈ N do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the coefficients in turn follow a simple pattern (almost similar to Pascal's Triangle). Although the pattern for calculating numerical coefficients based on forming a table is easy, non-recursive formulas are presented to determine the numerical coefficients.     

Asymptotic analysis of stochastic programs

In this paper we discuss a general approach to studying asymptotic properties of statistical estimators in stochastic programming. The approach is based on an extended delta method and appears to be particularly suitable for deriving asymptotics of the optimal value of stochastic programs. Asymptotic analysis of the optimal value will be presented in detail. Asymptotic properties of the corresponding optimal solutions are briefly discussed.     

Controlled perturbations for quadratically constrained quadratic programs

Consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function. The duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear. However, the objective function is not differentiable over the constraint region. In this paper, we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem. Based on this relationship, we establish a perturbation theory to display that a well-controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an ε-optimal dual solution for an arbitrarily small number ε. A simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding ε-optimal primal solution. Moreover, a numerical example is included to illustrate the potential of this controlled perturbation scheme.     

On augmented Lagrangian decomposition methods for multistage stochastic programs

A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios and by decomposing it into nodes corresponding to stages. Theoretical convergence properties of the two approaches are derived and a computational illustration is presented.     

Sublinear upper bounds for stochastic programs with recourse

Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.This research has been partially supported by the National Science Foundation. The first author's work was also supported in part by Office of Naval Research Grant N00014-86-K-0628 and by the National Research Council under a Research Associateship at the Naval Postgraduate School, Monterey, California.     

The value of the stochastic solution in stochastic linear programs with fixed recourse

Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.This paper is an extension of part of the author's dissertation in the Department of Operations Research, Stanford University, Stanford, California. The research was supported at Stanford by the Department of Energy under Contract DE-AC03-76SF00326, PA#DE-AT03-76ER72018, Office of Naval Research under Contract N00014-75-C-0267 and the National Science Foundation under Grants MCS76-81259, MCS-7926009 and ECS-8012974 (formerly ENG77-06761).     

A note on constraint aggregation and value functions for two-stage stochastic integer programs

We consider a class of two-stage stochastic integer programs and their equivalent reformulation that uses the integer programming value functions in both stages. One class of solution methods in the literature is based on the idea of pre-computing and storing exact value functions, and then exploiting this information within a global branch-and-bound framework. Such methods are known to be very sensitive to the magnitude of feasible right-hand side values. In this note we propose a simple constraint-aggregation based approach that potentially alleviates this limitation.     

On complexity of multistage stochastic programs under heavy tailed distributions

In this paper, the complexity of sample average approximation (SAA) of multistage stochastic programs under heavy tailed distributions is investigated. Specifically, we estimate confidence levels when the accuracy parameter and sample size are given under independently and identically distributed (iid) and non-iid conditional samples, respectively. Different from the existing works, we emphasize the impact of heavy tailed distributions, non-iid conditional sampling and stages dependence of the random process in multistage stochastic programs.     

Stability results for stochastic programming problems

The paper deals with a statistical approach to stability analysis in nonlinear stochastic programming. Firstly the distribution function of the underlying random variable is estimated by the empirical distribution function, and secondly the problem of estimated parameters is considered. In both the cases the probability that the solution set of the approximate problem, is not contained in an l-neighbourhood of the solution set to the original problem is estimated, and under differentiability properties an asymptotic expansion for the density of the (unique) solution to the approximate problem is derived.     

A Mellin transform approach to pricing barrier options under stochastic elasticity of variance

This article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi-closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black–Scholes price for a barrier option with respect to the underlying price and computing the one-dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte-Carlo simulation and find the SEV effect on the Black–Scholes barrier option prices.     

On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables

In this paper, we propose a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having mixed-integer first- and second-stage variables. A modified Benders' decomposition method is developed, where the Benders' subproblems define lower bounding second-stage value functions of the first-stage variables that are derived by constructing a certain partial convex hull representation of the two-stage solution space. This partial convex hull is sequentially generated using a convexification scheme such as the Reformulation-Linearization Technique (RLT) or lift-and-project process, which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first-stage variables. A branch-and-bound algorithm is designed based on a hyperrectangular partitioning process, using the established property that any resulting lower bounding Benders' master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first-stage variable solution is an extreme point of the defining hyperrectangle or the second-stage solution satisfies the binary restrictions. We prove that this algorithm converges to a global optimal solution. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach.     

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.     

On structure and stability in stochastic programs with random technology matrix and complete integer recourse

For two-stage stochastic programs with integrality constraints in the second stage, we study continuity properties of the expected recourse as a function both of the first-stage policy and the integrating probability measure.Sufficient conditions for lower semicontinuity, continuity and Lipschitz continuity with respect to the first-stage policy are presented. Furthermore, joint continuity in the policy and the probability measure is established. This leads to conclusions on the stability of optimal values and optimal solutions to the two-stage stochastic program when subjecting the underlying probability measure to perturbations.This research is supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.The main part of the paper was written while the author was an assistant at the Department of Mathematics at Humboldt University Berlin.     

A tighter variant of Jensen’s lower bound for stochastic programs and separable approximations to recourse functions

In this paper, we propose a new method to compute lower bounds on the optimal objective value of a stochastic program and show how this method can be used to construct separable approximations to the recourse functions. We show that our method yields tighter lower bounds than Jensen’s lower bound and it requires a reasonable amount of computational effort even for large problems. The fundamental idea behind our method is to relax certain constraints by associating dual multipliers with them. This yields a smaller stochastic program that is easier to solve. We particularly focus on the special case where we relax all but one of the constraints. In this case, the recourse functions of the smaller stochastic program are one dimensional functions. We use these one dimensional recourse functions to construct separable approximations to the original recourse functions. Computational experiments indicate that our lower bounds can significantly improve Jensen’s lower bound and our recourse function approximations can provide good solutions.     

Multi-policy improvement in stochastic optimization with forward recursive function criteria

Iwamoto recently established a formal transformation via an invariant imbedding to construct a controlled Markov chain that can be solved in a backward manner, as in backward induction for finite-horizon Markov decision processes (MDPs), for a given controlled Markov chain with non-additive forward recursive objective function criterion. Chang et al. presented formal methods, called “parallel rollout” and “policy switching,” of combining given multiple policies in MDPs and showed that the policies generated by both methods improve all of the policies that the methods combine. This brief paper extends the methods of parallel rollout and policy switching for forward recursive objective function criteria and shows that the similar property holds as in MDPs. We further discuss how to implement these methods via simulation.