A two-stage stochastic integer programming approach (original) (raw)
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We present an algorithmic approach for solving two-stage stochastic mixed 0-1 problems. The first stage constraints of the Deterministic Equivalent Model have 0-1 variables and continuous variables. The approach uses the Twin Node Family (TNF) concept within the algorithmic framework so-called Branch-and-Fix Coordination for satisfying the nonanticipativity constraints, jointly with a Benders Decomposition scheme for solving a given LP model at each TNF integer set. As an illustrative case, the structuring of a portfolio of Mortgage-Backed Securities under uncertainty in the interest rate path along a given time horizon is used. Some computational experience is reported.
Computational Management Science, 2009
We present a model for optimizing a mean-risk function of the terminal wealth for a fixed income asset portfolio restructuring with uncertainty in the interest rate path and the liabilities along a given time horizon. Some logical constraints are considered to be satisfied by the assets portfolio. Uncertainty is represented by a scenario tree and is dealt with by a multistage stochastic mixed 0-1 model with complete recourse. The problem is modelled as a splitting variable representation of the Deterministic Equivalent Model for the stochastic model, where the 0-1 variables and the continuous variables appear at any stage. A Branch-and-Fix Coordination approach for the multistage 0-1 program solving is proposed. Some computational experience is reported.
European Journal of Operational Research, 2010
We present an algorithmic framework, so-called BFC-TSMIP, for solving two-stage stochastic mixed 0-1 problems. The constraints in the Deterministic Equivalent Model have 0-1 variables and continuous variables at any stage. The approach uses the Twin Node Family (TNF) concept within an adaptation of the algorithmic framework so-called Branch-and-Fix Coordination for satisfying the nonanticipativity constraints for the first stage 0-1 variables. Jointly we solve the mixed 0-1 submodels defined at each TNF integer set for satisfying the nonanticipativity constraints for the first stage continuous variables. In these submodels the only integer variables are the second stage 0-1 variables. A numerical example and some theoretical and computational results are presented to show the performance of the proposed approach.
A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Software
Algorithms, 2022
This paper presents a tutorial on the state-of-the-art software for the solution of two-stage (mixed-integer) linear stochastic programs and provides a list of software designed for this purpose. The methodologies are classified according to the decomposition alternatives and the types of the variables in the problem. We review the fundamentals of Benders decomposition, dual decomposition and progressive hedging, as well as possible improvements and variants. We also present extensive numerical results to underline the properties and performance of each algorithm using software implementations, including DECIS, FORTSP, PySP, and DSP. Finally, we discuss the strengths and weaknesses of each methodology and propose future research directions.
Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming
Mathematical Programming, 2006
Decomposition has proved to be one of the more effective tools for the solution of large scale problems, especially those arising in stochastic programming. A decomposition method with wide applicability is Benders' decomposition, and has been applied to both stochastic programming, as well as integer programming problems. However, this method of decomposition relies on convexity of the value function of linear programming subproblems. This paper is devoted to a class of problems in which the second stage subproblem(s) may impose integer restrictions on some variables. The value function of such integer subproblem(s) is not convex, and new approaches must be designed. In this paper, we discuss alternative decomposition methods in which the second stage integer subproblems are solved using branch-and-cut methods. One of the main advantages of our decomposition scheme is that Stochastic Mixed-Integer Programming (SMIP) problems can be solved by dividing a large problem into smaller MIP subproblems which can be solved in parallel. This paper lays the foundation for such decomposition methods for two-stage stochastic integer programs.
2019
This paper presents a tutorial on the state-of-the-art methodologies for the solution of two-stage (mixed-integer) linear stochastic programs and provides a list of software designed for this purpose. The methodologies are classified according to the decomposition alternatives and the types of the variables in the problem. We review the fundamentals of Benders Decomposition, Dual Decomposition and Progressive Hedging, as well as possible improvements and variants. We also present extensive numerical results to underline the properties and performance of each algorithm using software implementations including DECIS, FORTSP, PySP, and DSP. Finally, we discuss the strengths and weaknesses of each methodology and propose future research directions.
Partial Benders Decomposition Strategies for Two-Stage Stochastic Integer Programs
2016
Benders decomposition is a broadly used used exact solution method for stochastic programming, enabling such programs to be decomposed according to the realizations of the random events that set the values of their stochastic parameters. This strategy also comes with important drawbacks, however, such as a weak master problem following the relaxation step that confines the dual cuts to the scenario sub-problems. We propose the first comprehensive Partial Benders Decomposition methodology for twostage integer stochastic program, based on the idea of including explicit information from the scenario sub-problems in the master. We propose two scenario-retention strategies that include a subset of the second stage scenario variables in the master, aiming to significantly reduce the number of feasibility cuts generated. We also propose a scenariocreation strategy to improve the lower-bound provided by the master problem, as well as three hybrids obtained by combining these pure strategies...
TOP, 2009
We present an exact algorithmic framework, so-called BFC-MSMIP, for optimizing multistage stochastic mixed 0-1 problems with complete recourse. The uncertainty is represented by using a scenario tree and lies anywhere in the model. The problem is modeled by a splitting variable representation of the Deterministic Equivalent Model of the stochastic problem, where the 0-1 variables and the continuous variables appear at any stage. The approach uses the Twin Node Family concept within the algorithmic framework, so-called Branch-and-Fix Coordination, for satisfying the nonanticipativity constraints in the 0-1 variables. Some blocks of additional strategies are used in order to show the performance of the proposed approach. The blocks are related to the scenario clustering, the starting branching and the branching order strategies, among others. Some computational experience is reported. It shows that the new approach obtains the optimal solution in all instances under consideration.
An Introduction to Two-Stage Stochastic Mixed-Integer Programming
2017
This paper provides an introduction to algorithms for two-stage stochastic mixed-integer programs. Our focus is on methods which decompose the problem by scenarios representing randomness in the problem data. The design of these algorithms depend on where the uncertainty appears (right-hand-side, recourse matrix and/or technology matrix) and where the continuous and discrete decision variables are (first-stage and/or secondstage). In addition we provide computational evidence that, similar to other classes of stochastic programming problems, decomposition methods can provide desirable theoretical properties (such as finite convergence) as well as enhanced computational performance when compared to solving a deterministic equivalent formulation using an advanced commercial MIP solver.
Journal of Global Optimization, 2019
In this paper, we propose a generalized Benders decomposition-based branch and cut algorithm for solving two stage stochastic mixed-integer nonlinear programs (SMINLPs). At a high level, the proposed decomposition algorithm performs spatial branch and bound search on the first stage variables. Each node in the branch and bound search is solved with a Benders-like decomposition algorithm where both Lagrangean cuts and Benders cuts are included in the Benders master problem. The Lagrangean cuts are derived from Lagrangean decomposition. The Benders cuts are derived from the Benders subproblems, which are convexified by cutting planes, such as rankone lift-and-project cuts. We prove that the proposed algorithm has convergence in the limit. We apply the proposed algorithm to a stochastic pooling problem, a crude selection problem, and a storage design problem. The performance of the proposed algorithm is compared with a Lagrangean decomposition-based branch and bound algorithm and solving the corresponding deterministic equivalent with the solvers including BARON, ANTIGONE, and SCIP.