Two-stage Stochastic Optimization Problem Research Articles

Decomposition algorithms for two-stage chance-constrained programs

We study a class of chance-constrained two-stage stochastic optimization problems where second-stage feasible recourse decisions incur additional cost. In addition, we propose a new model, where "recovery" decisions are made for the infeasible scenarios to obtain feasible solutions to a relaxed second-stage problem. We develop decomposition algorithms with specialized optimality and feasibility cuts to solve this class of problems. Computational results on a chance-constrained resource planing problem indicate that our algorithms are highly effective in solving these problems compared to a mixed-integer programming reformulation and a naive decomposition method.

Matching Supply and Demand: Delayed Two-Phase Distribution at Yedioth Group—Models, Algorithms, and Information Technology

This paper details collaboration between the distribution organization within the Yedioth Group, the largest media group in Israel, Technion—Israel Institute of Technology in Haifa, Israel, and the Massachusetts Institute of Technology in Cambridge, Massachusetts. This collaboration has led to fundamental changes in how Yedioth distributes print magazines and newspapers. In our work, we developed and implemented decision support tools that are based on new models and algorithms, which were enabled through an electronic data interchange system, and in the future will be enabled with a specialized radio-frequency identification technological solution. The underlying concept is to use real-time information about the sales at the retail outlets to enable pooling of inventory in the network. In particular, we leverage this information to implement an additional redistribution period during the week, delaying the distribution of some magazines until after we receive partial sales data. We model the system as a two-stage stochastic optimization problem. Moreover, we show that the resulting cost is jointly convex in the decision variables. This approach gives rise to a multidimensional dynamic program. By formulating the second-stage subproblem as a linear program, we develop an innovative stochastic gradient-based optimization algorithm that finds the optimal solution in a matter of seconds. The changes resulting from this collaboration generated substantial cost savings at Yedioth—from both a reduction in magazine production levels and a reduction in the return levels. These savings were achieved while maintaining the same sales levels.

An augmented Lagrangian method for distributed optimization

We propose a novel distributed method for convex optimization problems with a certain separability structure. The method is based on the augmented Lagrangian framework. We analyze its convergence and provide an application to two network models, as well as to a two-stage stochastic optimization problem. The proposed method compares favorably to two augmented Lagrangian decomposition methods known in the literature, as well as to decomposition methods based on the ordinary Lagrangian function.

Modeling and Optimization Techniques for Yield-Aware SRAM Post-Silicon Tuning

SRAM cell design is driven by the need to satisfy several stability and performance criteria for all cells in the array in an energy-efficient manner. Significant randomness of FET threshold voltages makes achieving this difficult and limits both the minimum cell size and minimum array supply voltage. Post-silicon adaptivity in the form of an adaptive-voltage scheme in a partitioned SRAM array can be used to reduce impact of variability despite lack of any spatial correlation in realizations. This paper develops a novel optimization flow for yield-aware cell sizing and voltage selection under variability given the availability of post-silicon voltage tuning. We formulate a two-stage stochastic optimization problem in which the first-stage decision is to select cell size and possible voltage levels, and the second-stage decision is to assign each partition to an optimal voltage after manufacturing. We develop closed-form statistical models of array margin behavior and yield as a function of V dd , cell size, and array size. We solve the problem using dynamic programming that minimizes power while meeting yield constraints on read, write, and static noise margins. The proposed flow allows designs that are on average 8% and up to 17% more power-efficient than the designs in which voltages are selected uniformly. The results also indicate that at high-yield levels power savings can be up to 32% in the active mode and 71% in the standby mode.

Conditioning of linear-quadratic two-stage stochastic optimization problems

In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.

Algorithms for stochastic optimization of multicast content delivery with network coding

The usage of network resources by content providers is commonly governed by Service-Level Agreements (SLA) between the content provider and the network service provider. Resource usage exceeding the limits specified in the SLA incurs the content provider additional charges, usually at a higher cost. Hence, the content provider's goal is to provision adequate resources in the SLA based on forecasts of future demand. We study capacity purchasing strategies when the content provider employs network coded multicast as the media delivery mechanism, with uncertainty in its future customer set explicitly taken into consideration. The latter requires the content provider to make capacity provisioning decisions based on market predictions and historical customer usage patterns. The probabilistic element suggests a stochastic optimization approach. We model this problem as a two-stage stochastic optimization problem with recourse. Such optimizations are #P-hard to solve directly, and we design two approximation algorithms for them. The first is a heuristic algorithm that exploits properties unique to network coding, so that only polynomial-time operations are needed. It performs well in general scenarios, but the gap from the optimal solution is not bounded by any constant in the worst case. This motivates our second approach, a sampling algorithm partly inspired from the work of Gupta et al. [2004a]. We employ techniques from duality theory in linear optimization to prove that the sampling algorithm provides a 3-approximation to the stochastic multicast problem. We conduct extensive simulations to illustrate the efficacy of both algorithms, and show that the performance of both is usually within 10% of the optimal solution in practice.

Stochastic Optimization of Sensor Placement for Diver Detection

A comprehensive framework for diver detection by a hydrophone network in an urban harbor is presented. It includes a signal processing algorithm and a diver detection test and formulates optimal hydrophone placement as a two-stage stochastic optimization problem with respect to different scenarios of underwater noise. The signal processing algorithm identifies sound intensity peaks associated with diver breathing and outputs a diver number measuring the likelihood of diver presence, whereas the diver detection test aggregates the diver numbers obtained from the hydrophones in a linear statistic and optimizes the statistic's coefficients and a detection threshold for each noise scenario. The serial dependence of the diver numbers on a short time scale (several detection periods) is modeled by a hidden Markov chain, and finding the worst-case diver's trajectory for each hydrophone placement and noise scenario is reduced to a linear programming problem. The framework is tested in numerical experiments with real-life data for circular and elliptic hydrophone placements and is shown to be superior to a deterministic energy-based approach.

Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse

We consider two-stage risk-averse stochastic optimization problems with a stochastic ordering constraint on the recourse function. Two new characterizations of the increasing convex order relation are provided. They are based on conditional expectations and on integrated quantile functions: a counterpart of the Lorenz function. We propose two decomposition methods to solve the problems and prove their convergence. Our methods exploit the decomposition structure of the risk-neutral two-stage problems and construct successive approximations of the stochastic ordering constraints. Numerical results confirm the efficiency of the methods.

Sell or Hold: A simple two-stage stochastic combinatorial optimization problem

The sell or hold problem (SHP) is to sell k out of n indivisible assets over two stages, with known first-stage prices and random second-stage prices, to maximize the total expected revenue. We show that SHP is NP-hard when the second-stage prices are realized as a finite set of scenarios. We show that SHP is polynomially solvable when the number of scenarios in the second stage is constant. A max{1/2,k/n}-approximation algorithm is presented for the scenario-based SHP.

Self-concordance and Decomposition-based Interior Point Methods for the Two-stage Stochastic Convex Optimization Problem

We study the two-stage stochastic convex optimization problem whose first- and second-stage feasible regions admit a self-concordant barrier. We show that the barrier recourse functions and the composite barrier functions for this problem form self-concordant families. These results are used to develop prototype primal interior point decomposition algorithms that are more suitable for a heterogeneous distributed computing environment. We show that the worst case iteration complexity of the proposed algorithms is the same as that for the short- and long-step primal interior algorithms applied to the extensive formulation of this problem. The generality of our results allows the possibility of using barriers other than the standard log-barrier in an algorithmic framework.

Expectation and Chance-Constrained Models and Algorithms for Insuring Critical Paths

In this paper, we consider a class of two-stage stochastic optimization problems arising in the protection of vital arcs in a critical path network. A project is completed after a series of dependent tasks are all finished. We analyze a problem in which task finishing times are uncertain but can be insured a priori to mitigate potential delays. A decision maker must trade off costs incurred in insuring arcs with expected penalties associated with late project completion times, where lateness penalties are assumed to be lower semicontinuous nondecreasing functions of completion time. We provide decomposition strategies to solve this problem with respect to either convex or nonconvex penalty functions. In particular, for the nonconvex penalty case, we employ the reformulation-linearization technique to make the problem amenable to solution via Benders decomposition. We also consider a chance-constrained version of this problem, in which the probability of completing a project on time is sufficiently large. We demonstrate the computational efficacy of our approach by testing a set of size-and-complexity diversified problems, using the sample average approximation method to guide our scenario generation.

Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion

We propose a semidefinite optimization (SDP) model for the class of minimax two-stage stochastic linear optimization problems with risk aversion. The distribution of second-stage random variables belongs to a set of multivariate distributions with known first and second moments. For the minimax stochastic problem with random objective, we provide a tight SDP formulation. The problem with random right-hand side is NP-hard in general. In a special case, the problem can be solved in polynomial time. Explicit constructions of the worst-case distributions are provided. Applications in a production-transportation problem and a single facility minimax distance problem are provided to demonstrate our approach. In our experiments, the performance of minimax solutions is close to that of data-driven solutions under the multivariate normal distribution and better under extremal distributions. The minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity.

On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right-hand side of the constraints is uncertain and belongs to a symmetric uncertainty set (such as hypercube, ellipsoid or norm ball) and the probability measure is also symmetric, then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two-stage stochastic problem. Furthermore, we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric. We refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two-stage stochastic problem as the stochasticity gap. We also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive. If both the objective coefficients and right-hand side are uncertain, we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric. However, we prove that the adaptability gap (ratio of optimal cost of the robust problem and the optimal cost of a two-stage fully adaptable problem) is at most four even if both the objective coefficients and the right-hand side of the constraints are uncertain and belong to a symmetric uncertainty set. The bound holds for the class of positive uncertainty sets as well. Moreover, if the uncertainty set is a hypercube (special case of a symmetric set), the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain.

A Linear Decision-Based Approximation Approach to Stochastic Programming

Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse.

Integrated design of power- and resource-saving chemical processes and process control systems: Strategy, methods, and application

A strategy for the integrated design of power-and resource-saving chemical processes and the systems controlling their operating conditions with uncertain input data on physicochemical and process parameters is formulated. A multistep iterative procedure for solving integrated design problems is developed. The procedure includes the generation of alternative chemical processes meeting the “rigid” and/or “soft” flexibility constraints and the choice of operating (control) actions, the synthesis of alternative systems for the automatic control of the operating conditions of the chemical process and the choice of the best control system, the pairwise comparison of feasible automated integrated systems consisting of the chemical engineering process and its control system and the choice of the best integrated system using the criterion based on the power and resource savings and control quality by solving one-and/or two-stage stochastic optimization problems with rigid and/or soft constraints. An example integrated design of the flexible continuous synthesis of azo pigments with an automatic-control system for stabilizing the optimal static conditions is discussed.

Designing the distribution network for an integrated supply chain

We consider an integrated distribution network design problem in which all the retailers face uncertain demand. The risk-pooling benefit is achieved by allowing some of the retailers to operate as distribution centers (DCs) with commitment in service level. The target is to minimize the expected total cost resulted from the DC location, transportation, and inventory. We formulate it as a two-stage nonlinear discrete stochastic optimization problem. The first stage decides which retailers to be selected as DCs and the second stage deals with the costs of DC-retailer assignment, transportation, and inventory. In the literature, the similar models require the demands of all retailers in each scenario to have their variances identically proportional to their means. In this paper, we remove this restriction. We reformulate the problem as a set-covering model and solve it by a column generation approach. With a variable fixing technique, we are able to efficiently solve problems of moderate-size (up to one hundred retailers and nine scenarios).

Design of multiproduct batch plants under demand uncertainty with staged capacity expansions

This paper addresses the staged capacity expansion of multiproduct batch plants operating in single product campaign (SPC) mode and in the presence of product demand uncertainty. This extends previous work by the authors on the design problem. Staged capacity expansions are modeled as additions of extra units of equal size at each stage at the end of each time period. The normality assumption is employed for the uncertain product demands. The problem formulation involves an inner stage which identifies the optimum production levels and an outer stage which establishes the design/capacity expansion decisions by maximizing the net present value (NPV) of the plant. This two-stage stochastic optimization problem for capacity expansion is equivalently recast as a deterministic mixed-integer nonlinear programming (MINLP) problem. Finally, the proposed formulation and solution strategy are illustrated with an example problem.

Implementing bounds-based approximations in convex-concave two-stage stochastic programming

This paper is concerned with implementational issues and computational testing of bounds-based approximations for solving two-stage stochastic programs with fixed recourse. The implemented bounds are those derived by the authors previously, using first and cross moment information of the random parameters and a convex-concave saddle property of the recourse function. The paper first examines these bounds with regard to their tightness, monotonic behavior, convergence properties, and computationally exploitable decomposition structures. Subsequently, the bounds are implemented under various partitioning/refining strategies for the successive approximation. The detailed numerical experiments demonstrate the effectiveness in solving large scenario-based two-stage stochastic optimization problems throughsuccessive scenario clusters induced by refining the approximations.