Multistage Stochastic Linear Programming Research Articles

Solving Sequences of Refined Multistage Stochastic Linear Programs

Multistage stochastic programs with continuous underlying distributions involve the obstacle of high-dimensional integrals where the integrands' values again are given by solutions of stochastic programs. A common solution technique consists of discretizing the support of the original distributions leading to scenario trees and corresponding LPs which are – up to a certain size – easy to solve. In order to improve the accuracy of approximation, successive refinements of the support result in rapidly expanding scenario trees and associated LPs. Hence, the solvability of the multistage stochastic program is limited by the numerical solvability of sequences of such expanding LPs. This work describes an algorithmic technique for solving the large-scale LP of refinement ν based on the solutions at the previous ν−1 refinements. Numerical results are presented for practical problem statements within financial applications demonstrating significant speedup (depending on the size of the LP instances).

A Comprehensive Input Format for Stochastic Linear Programs

Based on the MPS standard for linear programs, data conventions for the description of multistage stochastic linear programs were described by Birge et al. [3]. This paper proposes extensions to the so-called SMPS standard, in order to address known shortcomings and to extend the range of problems that can be expressed within the standard.

A Dynamic Network Flow Problem with Uncertain arc Capacities: Formulation and Problem Structure

We consider a dynamic network flow problem where the arc capacities are random variables. This gives a multistage stochastic linear program. We describe the randomness using a multi-scenario approach. Because of the multilayered structure, there are several ways to decompose the linear program. We classify different decomposition schemes, and we develop a scheme called compath decomposition, which is derived from path decomposition for network flows. We give a polynomial-time algorithm to find a cheapest compath that can solve the subproblems resulting from compath decomposition. Finally, compath decomposition is extended to multicommodity flow problems.

Inexact Cuts in Benders Decomposition

Benders decomposition is a well-known technique for solving large linear programs with a special structure. In particular, it is a popular technique for solving multistage stochastic linear programming problems. Early termination in the subproblems generated during Benders decomposition (assuming dual feasibility) produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution. We describe an inexact cut algorithm, prove its convergence under easily verifiable assumptions, and discuss a corresponding Dantzig--Wolfe decomposition algorithm. The paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling.

Convergent Cutting-Plane and Partial-Sampling Algorithm for Multistage Stochastic Linear Programs with Recourse

We propose an algorithm for multistage stochastic linear programs with recourse where random quantities in different stages are independent. The algorithm approximates successively expected recourse functions by building up valid cutting planes to support these functions from below. In each iteration, for the expected recourse function in each stage, one cutting plane is generated using the dual extreme points of the next-stage problem that have been found so far. We prove that the algorithm is convergent with probability one.

An Interior Random Vector Algorithm for MultiStage Stochastic Linear Programs

In this paper, an interior point algorithm for linear programs is adapted for solving multistage stochastic linear programs. The algorithm is based on Monteiro and Adler's path-following algorithm for deterministic linear programs. In practice, the complexity of the algorithm is linear with respect to the size of the sample space. The algorithm starts from a feasible solution of the problem and proceeds along a path of random vectors. The cubic polynomial complexity of the algorithm for deterministic linear programs is derived from the calculations of the Newton steps. In the algorithm developed in this paper, the probabilistic structure of the problem is taken into consideration while calculating a Newton step and the size of the sample space appears linearly in the complexity. The development of an algorithm that requires a relatively small number of arithmetic operations, in terms of the sample space size, allows the use of the algorithm for multistage stochastic linear programs with a very large number of scenarios.

Formulation of the Russell-Yasuda Kasai Financial Planning Model

This paper describes the formulation of the Russell-Yasuda Kasai financial planning model, including the motivation for the model. The presentation complements the discussion of the technical details of the financial modeling process and the managerial impact of its use to help allocate the firm's assets over time discussed in Cariño et al. (1994, 1998, respectively). The multistage stochastic linear program incorporates Yasuda Kasai's asset and liability mix over a five-year horizon followed by an infinite horizon steady-state end-effects period. The objective is to maximize expected long-run profits less expected penalty costs from constraint violations over the infinite horizon. Scenarios are used to represent the uncertain parameter distributions. The constraints represent the institutional, cash flow, legal, tax, and other limitations on the asset and liability mix over time.

Concepts, Technical Issues, and Uses of the Russell-Yasuda Kasai Financial Planning Model

This paper discusses technical aspects of the Russell-Yasuda Kasai financial planning model. These include the models for the discrete distribution scenario generation processes for the uncertain parameters of the model, the mathematical approach used to develop the infinite-horizon end-effects part of the model, a comparison of algorithms used in the model's solution, and a comparison of the multistage stochastic linear programming model with the previous technology, static mean-variance analysis. Experience and benefits of the model in Yasuda-Kasai's financial planning process is also discussed.

EVPI-Based Importance Sampling Solution Procedures for Multistage Stochastic Linear Programmes on Parallel MIMD Architectures

Multistage stochastic linear programming has many practical applications for problems whose current decisions have to be made under future uncertainty. There are a variety of methods for solving the deterministic equivalent forms of these dynamic problems, including the simplex and interior point methods and nested Benders decomposition which decomposes the original problem into a set of smaller linear programming problems and has recently been shown to be superior to the alternatives for large problems. The Benders subproblems can be visualised as being attached to the nodes of a tree which is formed from the realisations of the random data vectors determining the uncertainty in the problem. Parallel versions of the nested Benders algorithm involve two obvious techniques for parallelising the associated tree structure for multiprocessors or multicomputers subtree parallelisation or a nodal parallelisation both of which utilise a farming approach. The nodal parallelisation technique is presented in this paper, as it balances load more efficiently than its alternative. Differing structures of the test problems cause differing levels of speedup on a variety of multicomputing platforms: problems with few variables and constraints per node do not gain from this parallelisation. Stage aggregation has been successfully employed for such problems to improve their parallel solution efficiency by increasing the size of the nodes and therefore the time spent calculating relative to the time spent communicating between processors. A parallel version of an importance sampling solution algorithm based on local EVPI information has been developed for extremely large multistage stochastic linear programmes which either have too many data paths to solve directly or a continuous distribution of possible realisations. It utilises the parallel nested Benders algorithm and a parallel version of an algorithm designed to calculate the local EVPI values for the nodes of the tree and achieves near linear speedup.

Parallelization and Aggregation of Nested Benders Decomposition

Dynamic multistage stochastic linear programming has many practical applications for problems whose current decisions have to be made under future uncertainty. There are a variety of methods for solving these problems, including nested Benders decomposition. In this method, recently shown to be superior to the alternatives for large problems, the problem is decomposed into a set of smaller linear programming problems. These problems can be visualised as being attached to the nodes of a tree which is formed from the realizations of the random data vectors determining the uncertainty in the problem. The tree is traversed forwards and backwards, with information from the solutions to each nodal linear programming problem being passed to its immediate descendants by the formation of their right hand sides and to its immediate ancestor in the form of cuts. Problems in the same time period can be solved independently and it is this inherent parallelism that is exploited in our parallel nested Benders algorithm. A parallel version of the MSLiP nested Benders code has been developed and tested on various types of MIMD machines. The differing structures of the test problems cause differing levels of speed-up. Results show that problems with few variables and constraints per node do not gain from this parallelization. Stage aggregation has been successfully explored for such problems to improve their parallel solution efficiency by increasing the size of the nodes and therefore the time spent calculating relative to the time spent communicating between processors.

An enhanced decomposition algorithm for multistage stochastic hydroelectric scheduling

Handling uncertainty in natural inflow is an important part of a hydroelectric scheduling model. In a stochastic programming formulation, natural inflow may be modeled as a random vector with known distribution, but the size of the resulting mathematical program can be formidable. Decomposition-based algorithms take advantage of special structure and provide an attractive approach to such problems. We develop an enhanced Benders decomposition algorithm for solving multistage stochastic linear programs. The enhancements include warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies. Computational results are presented for a collection of stochastic hydroelectric scheduling problems.

A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs

Multistage stochastic linear programs can represent a variety of practical decision problems. Solving a multistage stochastic program can be viewed as solving a large tree of linear programs. A common approach for solving these problems is the nested decomposition algorithm, which moves up down the tree by solving nodes and passing information among nodes. The natural independence of subtrees suggests that much of the computational effort of the nested decomposition algorithm can run in parallel across small numbers of fast processors. This paper explores the advantages of such parallel implementations over serial implementations and compares alternative sequencing protocols for parallel processors. Computational experience on a large test set of practical problems with up to 1.5 million constraints and almost 5 million variables suggests that parallel implementations may indeed work well, but they require careful attention to processor load balancing.

Cut sharing for multistage stochastic linear programs with interstage dependency

Multistage stochastic programs with interstage independent random parameters have recourse functions that do not depend on the state of the system. Decomposition-based algorithms can exploit this structure by sharing cuts (outer-linearizations of the recourse function) among different scenario subproblems at the same stage. The ability to share cuts is necessary in practical implementations of algorithms that incorporate Monte Carlo sampling within the decomposition scheme. In this paper, we provide methodology for sharing cuts in decomposition algorithms for stochastic programs that satisfy certain interstage dependency models. These techniques enable sampling-based algorithms to handle a richer class of multistage problems, and may also be used to accelerate the convergence of exact decomposition algorithms.

Postoptimality for multistage stochastic linear programs

The contamination technique is presented as a flexible and relatively easily tractable tool to postoptimality analysis for scenario based multistage stochastic linear programs. It is promising especially in cases when the influence of additional or out-of-sample scenarios on the already solved problem is to be explored.

Primal-Dual Aggregation and Disaggregation for Stochastic Linear Programs

A multistage stochastic linear program can be approximated by replacing the underlying information structure by a coarser structure. Given the ordinary Lagrangian for the original problem, this amounts to restricting both the primal and dual variables to be adapted to the coarser information structure. The resulting primal problem has the same form as the original, but with aggregated constraints and decisions. Mutual upper and lower bounds on the optimal values of the original and aggregated problems are obtained via partially aggregated problems formed by restricting only the primal variables or only the dual variables to the coarser structure. The introduction of the partially aggregated problems leads to a general disaggregation framework for developing heuristic approaches to solving the original problem.

Multi-stage stochastic linear programs for portfolio optimization

The paper demonstrates how multi-period portfolio optimization problems can be efficiently solved as multi-stage stochastic linear programs. A scheme based on a blending of classical Benders decomposition techniques and a special technique, called importance sampling, is used to solve this general class of multi-stochastic linear programs. We discuss the case where stochastic parameters are dependent within a period as well as between periods. Initial computational results are presented.

Parallel decomposition of multistage stochastic programming problems

A new decomposition method for multistage stochastic linear programming problems is proposed. A multistage stochastic problem is represented in a tree-like form and with each node of the decision tree a certain linear or quadratic subproblem is associated. The subproblems generate proposals for their successors and some backward information for their predecessors. The subproblems can be solved in parallel and exchange information in an asynchronous way through special buffers. After a finite time the method either finds an optimal solution to the problem or discovers its inconsistency. An analytical illustrative example shows that parallelization can speed up computation over every sequential method. Computational experiments indicate that for large problems we can obtain substantial gains in efficiency with moderate numbers of processors.

Sensitivity method for basis inverse representation in multistage stochastic linear programming problems

A version of the simplex method for solving stochastic linear control problems is presented. The method uses a compact basis inverse representation that extensively exploits the original problem data and takes advantage of the supersparse structure of the problem. Computational experience indicates that the method is capable of solving large problems.

Mslip: A computer code for the multistage stochastic linear programming problem

This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.

Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs

Multistage stochastic linear programs model problems in financial planning, dynamic traffic assignment, economic policy analysis, and many other applications. Equivalent representations of such problems as deterministic linear programs are, however, excessively large. This paper develops decomposition and partitioning methods for solving these problems and reports on computational results on a set of practical test problems.