Güzin Bayraksan

Assessing solution quality in stochastic programs

Determining whether a solution is of high quality (optimal or near optimal) is fundamental in optimization theory and algorithms. In this paper, we develop Monte Carlo sampling-based procedures for assessing solution quality in stochastic programs. Quality is defined via the optimality gap and our procedures output is a confidence interval on this gap. We review a multiple-replications procedure that requires solution of, say, 30 optimization problems and then, we present a result that justifies a computationally simplified single-replication procedure that only requires solving one optimization problem. Even though the single replication procedure is computationally significantly less demanding, the resulting confidence interval might have low coverage probability for small sample sizes for some problems. We provide variants of this procedure that require two replications instead of one and that perform better empirically. We present computational results for a newsvendor problem and for two-stage stochastic linear programs from the literature. We also discuss when the procedures perform well and when they fail, and we propose using ɛ-optimal solutions to strengthen the performance of our procedures.

A Sequential Sampling Procedure for Stochastic Programming

We develop a sequential sampling procedure for a class of stochastic programs. We assume that a sequence of feasible solutions with an optimal limit point is given as input to our procedure. Such a sequence can be generated by solving a series of sampling problems with increasing sample size, or it can be found by any other viable method. Our procedure estimates the optimality gap of a candidate solution from this sequence. If the point estimate of the optimality gap is sufficiently small according to our termination criterion, then we stop. Otherwise, we repeat with the next candidate solution from the sequence under an increased sample size. We provide conditions under which this procedure (i) terminates with probability one and (ii) terminates with a solution that has a small optimality gap with a prespecified probability.

Scheduling jobs sharing multiple resources under uncertainty: A stochastic programming approach

A two-stage stochastic integer program to determine an optimal schedule for jobs requiring multiple classes of resources under uncertain processing times, due dates, resource consumption and availabilities is formulated. Temporary resource capacity expansion for a penalty is allowed. Potential applications of this model include team scheduling problems that arise in service industries such as engineering consulting and operating room scheduling. An exact solution method is developed based on Benders decomposition for problems with a moderate number of scenarios. Benders decomposition is then embedded within a sampling-based solution method for problems with a large number of scenarios. A sequential sampling procedure is modified to allow for approximate solution of integer programs and its asymptotic validity and finite stopping are proved under this modification. The solution methodologies are compared on a set of test problems. Several algorithmic enhancements are added to improve efficiency.

Simulation optimization: a panel on the state of the art in research and practice

The goal of this panel was to discuss the state of the art in simulation optimization research and practice. The participants included representation from both academia and industry, where the latter was represented by participation from a leading software provider of optimization tools for simulation. This paper begins with a short introduction to simulation optimization, and then presents a list of specific questions that served as a basis for discussion during the panel discussion. Each of the panelists was given an opportunity to provide their preliminary thoughts on simulation optimization for this paper, and the remainder of the paper summarizes those points, ranging from assessments of the field from their perspective to initial reactions to some of the posed questions. Finally, one of the panelists who has worked on an online testbed of simulation optimization problems for the research community was asked to provide an update of the status of the Web site.

A probability metrics approach for reducing the bias of optimality gap estimators in two-stage stochastic linear programming

Monte Carlo sampling-based estimators of optimality gaps for stochastic programs are known to be biased. When bias is a prominent factor, estimates of optimality gaps tend to be large on average even for high-quality solutions. This diminishes our ability to recognize high-quality solutions. In this paper, we present a method for reducing the bias of the optimality gap estimators for two-stage stochastic linear programs with recourse via a probability metrics approach, motivated by stability results in stochastic programming. We apply this method to the Averaged Two-Replication Procedure (A2RP) by partitioning the observations in an effort to reduce bias, which can be done in polynomial time in sample size. We call the resulting procedure the Averaged Two-Replication Procedure with Bias Reduction (A2RP-B). We provide conditions under which A2RP-B produces strongly consistent point estimators and an asymptotically valid confidence interval. We illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small-sample behavior of A2RP-B on a number of two-stage stochastic linear programs from the literature. Our computational results indicate that the procedure effectively reduces bias. We also observe variance reduction in certain circumstances.

A combined deterministic and sampling-based sequential bounding method for stochastic programming

We develop an algorithm for two-stage stochastic programming with a convex second stage program and with uncertainty in the right-hand side. The algorithm draws on techniques from bounding and approximation methods as well as sampling-based approaches. In particular, we sequentially refine a partition of the support of the random vector and, through Jensens inequality, generate deterministically valid lower bounds on the optimal objective function value. An upper bound estimator is formed through a stratified Monte Carlo sampling procedure that includes the use of a control variate variance reduction scheme. The algorithm lends itself to a stopping rule theory that ensures an asymptotically valid confidence interval for the quality of the proposed solution. Computational results illustrate our approach.

Simulation-Based Optimality Tests for Stochastic Programs

Assessing whether a solution is optimal, or near-optimal, is fundamental in optimization. We describe a simple simulation-based procedure for assessing the quality of a candidate solution to a stochastic program. The procedure is easy to implement, widely applicable, and yields point and interval estimators on the candidate solutions optimality gap. Our simplest procedure allows for significant computational improvements. The improvements we detail aim to reduce computational effort through single- and two-replication procedures, reduce bias via a class of generalized jackknife estimators, and reduce variance by using a randomized quasi-Monte Carlo scheme.

Decomposition Algorithms for Risk-Averse Multistage Stochastic Programs with Application to Water Allocation under Uncertainty

We study a risk-averse approach to multistage stochastic linear programming, where the conditional value-at-risk is incorporated into the objective function as the risk measure. We consider five decompositions of the resulting risk-averse model to solve it via the nested L-shaped method. We introduce separate approximations of the mean and the risk measure and also investigate the effectiveness of multiple cuts. As an application, we formulate a water allocation problem by risk-averse multistage programming, which has the advantage of controlling high-risk severe water shortage events. We apply the proposed formulation to the southeastern portion of Tucson, AZ to best use the limited water resources available to that region. In numerical experiments we (1) present a comparative computational study of the risk-averse nested L-shaped variants and (2) analyze the risk-averse approach to the water allocation problem.

Stochastic Constraints and Variance Reduction Techniques

We provide an overview of two select topics in Monte Carlo simulation-based methods for stochastic optimization: problems with stochastic constraints and variance reduction techniques. While Monte Carlo simulation-based methods have been successfully used for stochastic optimization problems with deterministic constraints, there is a growing body of work on its use for problems with stochastic constraints. The presence of stochastic constraints brings new challenges in ensuring and testing optimality, allocating sample sizes, etc., especially due to difficulties in determining feasibility. We review results for general stochastic constraints and also discuss special cases such as probabilistic and stochastic dominance constraints. Next, we review the use of variance reduction techniques (VRT) in a stochastic optimization setting. While this is a well-studied topic in statistics and simulation, the use of VRT in stochastic optimization requires a more thorough analysis. We discuss asymptotic properties of the resulting approximations and their use within Monte Carlo simulation-based solution methods.

Overlapping batches for the assessment of solution quality in stochastic programs

We investigate the use of overlapping batches for assessing solution quality in stochastic programs. Motivated by the original use of overlapping batches in simulation, we present a variant of the multiple replications procedure that reuses data via variably overlapping batches to obtain alternative variance estimators. These estimators have lower variances, where the degree of variance reduction depends on the amount of overlap. We provide several asymptotic properties and present computational results to examine small-sample behavior.