A Scalable Bounding Method for Multi-Stage Stochastic Integer Programs

Stochastic programming offers a flexible modeling framework for optimal decision-making problems under uncertainty. Most practical stochastic programming instances, however, quickly grow too large to solve on a single computer, especially due to memory limitations. This chapter reviews recent developments in solving large-scale stochastic programs, possibly with multiple stages and mixed-integer decision variables, and focuses on a scenario decomposition-based bounding method, which is broadly applicable as it does not rely on special problem structure and stands out as a natural candidate for implementation in a distributed fashion. In addition to discussing the method theoretically, this chapter examines issues related to a distributed implementation of the method on a modern computing grid. Using large-scale instances from the literature, this chapter demonstrates the potential of the method in obtaining high quality solutions to very large-scale stochastic programming instances within a reasonable time frame.

An increasing number of practical scenarios continue to experience problems associated with multistage stochastic programming. This study proposes a decomposition method through which they can be a successful solution to multi-stage stochastic nonlinear programs. The proposed method entails the scenario analysis method. The proposed method also performs its role via search direction generation in such a way that sets of quadratic programming sub-issues are solved in a parallel way, especially when the size is significant lower, compared to the case involving original problems at the respective iterations. Relative to the dual multiplier derivation, which focuses on non-anticipativity constraints, the proposed system advocates for the introduction of generalized reduced gradient approaches.

We present a distributed asynchronous algorithm for solving two-stage stochastic mixed-integer programs (SMIP) using scenario decomposition, aimed at industrial-scale instances of the stochastic unit commitment (SUC) problem. The algorithm is motivated by large differences in run times observed among scenario subproblems of SUC instances, which can result in inefficient use of distributed computing resources by synchronous parallel algorithms. Our algorithm performs dual iterations asynchronously using a block-coordinate subgradient descent method which allows performing block-coordinate updates using delayed information, while candidate primal solutions are recovered from the solutions of scenario subproblems using heuristics. We present a high performance computing implementation of the asynchronous algorithm, detailing the operations performed by each parallel process and the communication mechanisms among them. We conduct numerical experiments using SUC instances of the Western Electricity Coordinating Council system with up to 1000 scenarios and of the Central Western European system with up to 120 scenarios. We also conduct numerical experiments on generic SMIP instances from the SIPLIB library (DCAP and SSLP). The results demonstrate the general applicability of the proposed algorithm and its ability to solve industrial-scale SUC instances within operationally acceptable time frames. Moreover, we find that an equivalent synchronous parallel algorithm would leave cores idle up to 80.4% of the time on our realistic test instances, an observation which underscores the need for designing asynchronous optimization schemes in order to fully exploit distributed computing on real world applications.

We describe computational procedures for solving a wide-ranging class of stochastic programs with chance constraints where the random components of the problem are discretely distributed. Our procedures are based on a combination of Lagrangian relaxation and scenario decomposition, which we solve using a novel variant of Rockafellar and Wets' progressive hedging algorithm [Rockafellar, R. T., R. J.-B. Wets. 1991. Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16(1) 119–147]. Experiments demonstrate the ability of the proposed algorithm to quickly find near-optimal solutions—where verifiable—to both difficult and very large chance-constrained stochastic programs, both with and without integer decision variables. The algorithm exhibits strong scalability in terms of both run time required and final solution quality on large-scale instances. There is a Video Overview associated with this paper. Click here to view the Video Overview . To save the file, right click and choose “Save Link As” from the menu.

This study focuses on a multi-product, multi-period, multi-site supply chain planning problem under demand uncertainty. A multi-stage stochastic linear programming model is proposed to maximize the expected profit. The decisions to be made comprise the production amount, the inventory and backorder sizes as well as the quantity of products to be transported between upstream and downstream plants and customers in each period. A numerical example is presented in order to illustrate the effectiveness of the proposed model. Results indicate that the solution of the multi-stage stochastic model outperforms the deterministic and the two-stage stochastic models.

The growth in computation complexity of multistage stochastic programs (MSSPs) with problem size often prevents its application to real‐world size problems. We present two variants of branch‐and‐bound algorithm, which reduce the resource requirements for the generation and solution of large‐scale MSSPs with endogenous uncertainty. Both variants use Knapsack‐problem based Decomposition Algorithm (Christian and Cremaschi, Comput Chem Eng. 2015;74:34–47) to generate feasible solutions and primal bounds. First variant (PH‐KDA) uses a progressive hedging dual‐bounding approach; the second (OSS‐KDA) solves the MSSP removing all nonanticipativity constraints. Both variants were used to solve several instances of the pharmaceutical clinical trial planning problem. The first iteration of both algorithms provides a feasible solution, and a primal bound and a dual bound for the problem. Although the dual‐bounds of OSS‐KDA were generally weaker than PH‐KDA, they are generated considerably faster. For the seven‐product case the OSS‐KDA generated a solution with a gap of 9.92% in 115 CPU seconds. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1262–1271, 2018

We consider the problem of tracking a target portfolio or index under uncertainty. Due to an embedded NP-hard subproblem, many of the current index tracking models only consider a small number of important portfolio elements such as transaction costs, number of securities to hold, rebalancing, etc. We formulate a tracking portfolio model that includes a comprehensive set of real-world portfolio elements, one of which involves uncertainty. An index tracking model is defined in a Stochastic Mixed-Integer Programming (SMIP) framework. Due to the size and complexity of the stochastic problem, the SMIP model is decomposed into subproblems and an iterative algorithm is developed that exploits the decomposition. A two-stage SMIP is solved and the results are compared with actual index values. We also provide single-scenario dynamic comparisons to illustrate the performance and strengths of the method.

Multistage stochastic programming (SP) with both endogenous and exogenous uncertainties is a novel problem in which some uncertain parameters are decision-dependent and others are independent of decisions. The main difficulty of this problem is that nonanticipativity constraints (NACs) make up a significantly large constraint set, growing very fast with the number of scenarios and leading to an intractable model. Usually, a lot of these constraints are redundant and hence, identification and elimination of redundant NACs can cause a significant reduction in the problem size. Recently, a polynomial time algorithm has been proposed in the literature which is able to identify all redundant NACs in an SP problem with only endogenous uncertainty. In this paper, however, we extend the algorithm proposed in the literature and present a new method which is able to make the upper most possible reduction in the number of NACs in any SP with both exogenous and endogenous uncertain parameters. Proving the validity of this method is another innovation of this study. Computational results confirm that the proposed approach can significantly reduce the problem size within a reasonable computation time.

The purpose of this paper is to apply multistage stochastic programming to the transmission line expansion planning problem, especially when uncertain demand scenarios exist. Since the problem of transmission line expansion planning requires an intensive computational load, dual decomposition is used to decompose the problem into smaller problems. Following this, progressive hedging and proximal bundle methods are used to restore the decomposed solutions to the original problems. Mixed-integer linear programming is involved in the problem to decide where new transmission lines should be constructed or reinforced. However, integer variables in multistage stochastic programming (MSSP) are intractable since integer variables are not restored. Therefore, the branch-and-bound algorithm is applied to multistage stochastic programming methods to force convergence of integer variables.In addition, this paper suggests combining progressive hedging and dual decomposition in stochastic integer programming by sharing penalty parameters. The simulation results tested on the IEEE 30-bus system verify that our combined model sped up the computation and achieved higher accuracy by achieving the minimised cost.

Home Care Routing and Appointment Scheduling with Stochastic Service Durations

This paper focuses on solving two-stage stochastic mixed integer programs (SMIPs) with general mixed integer decision variables in both stages. We develop a decomposition algorithm in which the first stage approximation is solved using a branch-and-bound tree with nodes inheriting Benders' cuts that are valid for their ancestor nodes. In addition, we develop two closely related convexification schemes which use multi-term disjunctive cuts to obtain approximations of the second stage mixed-integer programs. We prove that the proposed methods are finitely convergent. One of the main advantages of our decomposition scheme is that we use a Benders-based branch-and-cut approach in which linear programming approximations are strengthened sequentially. Moreover as in many decomposition schemes, these subproblems can be solved in parallel. We also illustrate these algorithms using several variants of an SMIP example from the literature, and present preliminary evidence of the scalability of these algorithms as the number of scenarios increases.

The large demand of bandwidth due to new and future applications from services being implemented such as smart grid (SG), smart cities (SC), and the Internet of Things has triggered a focus of interest from the scientific community on issues related to the planning and scalability of the communications infrastructure. In this paper, we present three new and important contributions regarding the deployment of FiWi networks that support users who benefit from the services provided by SG and SC. We first propose a planning model, and then formulate a mixed linear programming optimization model using multistage stochastic programming, and finally propose an algorithm, called MOA–FiWi, which has the capacity to handle medium and large instances of the problem and give feasible solutions for the optimization process, where scalability indices between 85%–95% are reached with respect to the projected stochastic values. The proposed model considers the scalability and capital and operating expense, as well as the uncertainty management in the different stages of time–space through multistage stochastic programming. The proposed model presents flexibility in decision making as the time stages progress. This allows for the planning of green fields, as well as the updating of networks that already have communication infrastructure.

Multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events. In order to ensure computational tractability, continuous-state stochastic programs are usually discretized; and frequently, the curse of dimensionality dictates that decision stages must be aggregated. In this article we construct two discrete, stage-aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem. The approximate problems involve finitely many decisions and constraints, thus principally allowing for numerical solution.

Existing compartmental models in epidemiology are limited in terms of optimizing the resource allocation to control an epidemic outbreak under disease growth uncertainty. In this study, we address this core limitation by presenting a multi-stage stochastic programming compartmental model, which integrates the uncertain disease progression and resource allocation to control an infectious disease outbreak. The proposed multi-stage stochastic program involves various disease growth scenarios and optimizes the distribution of treatment centers and resources while minimizing the total expected number of new infections and funerals. We define two new equity metrics, namely infection and capacity equity, and explicitly consider equity for allocating treatment funds and facilities over multiple time stages. We also study the multi-stage value of the stochastic solution (VSS), which demonstrates the superiority of the proposed stochastic programming model over its deterministic counterpart. We apply the proposed formulation to control the Ebola Virus Disease (EVD) in Guinea, Sierra Leone, and Liberia of West Africa to determine the optimal and fair resource-allocation strategies. Our model balances the proportion of infections over all regions, even without including the infection equity or prevalence equity constraints. Model results also show that allocating treatment resources proportional to population is sub-optimal, and enforcing such a resource allocation policy might adversely impact the total number of infections and deaths, and thus resulting in a high cost that we have to pay for the fairness. Our multi-stage stochastic epidemic-logistics model is practical and can be adapted to control other infectious diseases in meta-populations and dynamically evolving situations.

Farmland management and irrigation scheduling are vital to a productive agricultural economy. A multistage stochastic programming model is proposed to maximize farmers’ annual profit under uncertainty. The uncertainties considered include crop prices, irrigation water availability, and precipitation. During the first stage, pre-season decisions including seed type and plant density are made, while determinations of when to irrigate and how much water to be used for each irrigation are made in the later stages. The presented case study, based on a farm in Nebraska, U.S.A., showed that a 10% profit increase could be achieved by taking the corn price and irrigation water availability uncertainties into consideration using two-stage stochastic programming. An additional 13% profit increase could be achieved by taking precipitation uncertainty into consideration using multistage stochastic programming. The stochastic model outperforms the deterministic model, especially when there are limited water supplies. These results indicate that multistage stochastic programming is a promising method for farm-scale irrigation management and can increase farm profitability.