Assessing Solution Quality in Stochastic Programs

Determining whether a solution is of high quality (optimal or near optimal) is a fundamental question 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 an when they fail and provide preliminary guidelines for selecting a candidate solution

Residuals-based distributionally robust optimization with covariate information

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of learning setups and DRO ambiguity sets. We investigate the asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified

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

Abstract We formulate 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. We allow temporary resource capacity expansion for a penalty. Potential applications of this model include team scheduling problems that arise in service industries such as engineering consulting and operating room scheduling. We develop an exact solution method based on Benders decomposition for problems with a moderate number of scenarios. Then we embed Benders decomposition within a sampling-based solution method for problems with a large number of scenarios. We modify a sequential sampling procedure to allow for approximate solution of integer programs and prove desired properties. We compare the solution methodologies on a set of test problems. Several algorithmic enhancements are added to improve efficiency

SEQUENTIAL SAMPLING FOR SOLVING STOCHASTIC PROGRAMS

We develop a sequential sampling procedure for solving a class of stochastic programs. A sequence of feasible solutions, with at least one optimal limit point, is given as input to our procedure. Our procedure estimates the optimality gap of a candidate solution from this sequence, and if that point estimate is sufficiently small then we stop. Otherwise, we repeat with the next candidate solution from the sequence with a larger sample size. We provide conditions under which this procedure: (i) terminates with probability one and (ii) terminates with a solution which has a small optimality gap with a prespecified probability

Assessing solution quality in stochastic programs via sampling

Abstract 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 ε-optimal solutions to strengthen the performance of our procedures

Reformulation linearization technique based branch-and-reduce approach applied to regional water supply system planning

<div><p>A regional water supply system design problem that determines pipe and pump design parameters and water flows over a multi-year planning horizon is considered. A non-convex nonlinear model is formulated and solved by a branch-and-reduce global optimization approach. The lower bounding problem is constructed via a three-pronged effort that involves transforming the space of certain decision variables, polyhedral outer approximations, and the Reformulation Linearization Technique (RLT). Range reduction techniques are employed systematically to speed up convergence. Computational results demonstrate the efficiency of the proposed algorithm; in particular, the critical role range reduction techniques could play in RLT based branch-and-bound methods. Results also indicate using reclaimed water not only saves freshwater sources but is also a cost-effective non-potable water source in arid regions. Supplemental data for this article can be accessed at <a href="http://dx.doi.org/10.1080/0305215X.2015.1016508" target="_blank">http://dx.doi.org/10.1080/0305215X.2015.1016508</a>.</p></div