Second-stage Problem Research Articles

Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach

This paper presents an asymptotic analysis of a Monte Carlo method, variously known as sample average approximation (SAA) or sample path optimization (SPO), for a general two-stage stochastic minimization problem. We study the case when the second-stage problem may have multiple local optima or stationary points that are not global solutions and SAA is implemented using a general nonlinear programming solver that is only guaranteed to find stationary points. New optimality conditions are developed for both the true problem and its SAA problem to accommodate Karush-Kuhn-Tucker points. Because the optimality conditions are essentially stochastic generalized equations, the asymptotic analysis is carried out for the generalized equations first and then applied to optimality conditions. For this purpose, we analyze piecewise continuous (PC0) stochastic mappings to understand when their expectations are piecewise continuous and thereby derive exponential convergence of SAA. It is shown under moderate conditions that, with probability one, an accumulation point of the SAA stationary points satisfies a relaxed stationary condition for the true problem and further that, with probability approaching one exponentially fast with increasing sample size, a stationary point of SAA converges to the set of relaxed stationary points. These results strengthen or complement existing results where the second-stage problem is often assumed to have a unique solution and the exponential convergence is focused on how fast a solution of the true problem becomes an approximate solution of an SAA problem rather than the other way around.

Dual Toll Pricing for Hazardous Materials Transport with Linear Delay

In this paper, we propose a dual toll pricing method to mitigate risk of hazardous materials (hazmat) transportation. We aim to simultaneously control both regular and hazmat vehicles to reduce the risk. In our model, we incorporate a new risk measure to consider duration-population-frequency of hazmat exposure. We first formulate the model as a Mathematical Program with Equilibrium Constraints (MPEC). Then we decompose the MPEC formulation into first-stage and second-stage problems. Separate methods are developed to solve each stage. A numerical example is provided and possible extensions are discussed.

Continuity and Stability of a Quadratic Mixed-Integer Stochastic Program

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.

A dynamic day-ahead paratransit planning problem

We consider a dynamic planning problem for the transport of elderly and disabled people. The focus is on a decision to make one day ahead: which requests to serve with own vehicles and which ones to assign to subcontractors under uncertainty of late requests that are gradually revealed during the day of operation. We call this problem the dynamic day-ahead paratransit planning problem. The developed model is a nonstandard two-stage recourse model in which ideas from stochastic programming and online optimization are combined: in the first stage. clustered requests are assigned to vehicles, and in the dynamic second-stage problem, an event-driven simulation is used to cluster the late requests once they are revealed and subsequently assign them to vehicles. A genetic algorithm is used to solve the model. Computational results are presented for randomly generated data sets. Furthermore, a comparison is made to a similar problem we studied earlier in which the simplifying but unrealistic assumption has been made that all late requests are revealed at the beginning of the day of operation.

Min-Max Regret Robust Optimization Approach on Interval Data Uncertainty

This paper presents a three-stage optimization algorithm for solving two-stage deviation robust decision making problems under uncertainty. The structure of the first-stage problem is a mixed integer linear program and the structure of the second-stage problem is a linear program. Each uncertain model parameter can independently take its value from a real compact interval with unknown probability distribution. The algorithm coordinates three mathematical programming formulations to iteratively solve the overall problem. This paper provides the application of the algorithm on the robust facility location problem and a counterexample illustrating the insufficiency of the solution obtained by considering only a finite number of scenarios generated by the endpoints of all intervals.

Scenario relaxation algorithm for finite scenario-based min–max regret and min–max relative regret robust optimization

Most practical decision-making problems are compounded in difficulty by the degree of uncertainty and ambiguity surrounding the key model parameters. Decision makers may be confronted with problems in which no sufficient historical information is available to make estimates of the probability distributions for uncertain parameter values. In these situations, decision makers are not able to search for the long-term decision setting with the best long-run average performance. Instead, decision makers are searching for the robust long-term decision setting that performs relatively well across all possible realizations of uncertainty without attempting to assign an assumed probability distribution to any ambiguous parameter. In this paper, we propose an iterative algorithm for solving min–max regret and min–max relative regret robust optimization problems for two-stage decision-making under uncertainty (ambiguity) where the structure of the first-stage problem is a mixed integer (binary) linear programming model and the structure of the second-stage problem is a linear programming model. The algorithm guarantees termination at an optimal robust solution, if one exists. A number of applications of the proposed algorithm are demonstrated. All results illustrate good performance of the proposed algorithm.

Decomposition Algorithms for Stochastic Programming on a Computational Grid

We describe algorithms for two-stage stochastic linear programming with recourse and their implementation on a grid computing platform. In particular, we examine serial and asynchronous versions of the L-shaped method and a trust-region method. The parallel platform of choice is the dynamic, heterogeneous, opportunistic platform provided by the Condor system. The algorithms are of master-worker type (with the workers being used to solve second-stage problems), and the MW runtime support library (which supports master-worker computations) is key to the implementation. Computational results are presented on large sample-average approximations of problems from the literature.

Preprocessing in Stochastic Programming: The Case of Uncapacitated Networks

Preprocessing can speed up the solution procedures for two-stage stochastic programming. We consider the case when the second-stage problem is a pure, uncapacitated network. We describe a number of procedures to reduce the size of the recourse problem. We describe a procedure for generating efficiently the (induced) feasibility cuts, and show that further reductions are possible if more information about the node-types is taken into account. We also investigate network collapsing techniques that would simplify the work required to find both optimality cuts and feasibility cuts, if we had not yet reduced the problem to one with relatively complete recourse. Computational results confirm that substantial savings are possible. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Technical Note—Risk Aversion in Stochastic Programming with Recourse

In stochastic programming with recourse the objective is to maximize the expected net payoff. This assumes implicitly no aversion to risk. With risk aversion, the objective becomes to maximize the expected (concave) utility of the net payoffs. Because of the special structure of the problem with risk aversion, a number of computational short cuts are possible in the mathematical program that results. All the second-stage problems can be solved as linear programs. Unfortunately, whether with or without risk aversion, it is necessary to solve the first-stage problem as a nonlinear program. This note shows that the latest representation of the gradient is but a simple modification of the latest representation of the linear objective function without risk aversion.