Ancestral Benders' Cuts and Multi-term Disjunctions for Mixed-Integer Recourse Decisions in Stochastic Programming (original) (raw)
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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.
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Journal of Global Optimization, 2008
Two-stage stochastic mixed-integer programming (SMIP) problems with recourse are generally difficult to solve. This paper presents a first computational study of a disjunctive cutting plane method for stochastic mixed 0-1 programs that uses lift-and-project cuts based on the extensive form of the two-stage SMIP problem. An extension of the method based on where the data uncertainty appears in the problem is made, and it is shown how a valid inequality derived for one scenario can be made valid for other scenarios, potentially reducing solution time. Computational results amply demonstrate the effectiveness of disjunctive cuts in solving several large-scale problem instances from the literature. The results are compared to the computational results of disjunctive cuts based on the subproblem space of the formulation and it is shown that the two methods are equivalently effective on the test instances.
International Journal of Computational Science and Engineering, 2007
This paper presents a branch-and-cut method for two-stage stochastic mixed-integer programming (SMIP) problems with continuous firststage variables. This method is derived based on disjunctive decomposition (D 2) for SMIP, an approach in which disjunctive programming is used to derive valid inequalities for SMIP. The novelty of the proposed method derives from branching on the first-stage continuous domain while the branch-andbound process is guided by the disjunction variables in the second-stage. Finite convergence of the algorithm for mixed-binary second-stage is established and a numerical example to illustrate the new method is given.
Decomposition Algorithms with Parametric Gomory Cuts for Two-Stage Stochastic Integer Programs
2012
Abstract We consider a class of two-stage stochastic integer programs with binary variables in the first stage and general integer variables in the second stage. We develop decomposition algorithms akin to the L-shaped or Benders' methods by utilizing Gomory cuts to obtain iteratively tighter approximations of the second-stage integer programs. We show that the proposed methodology is flexible in that it allows several modes of implementation, all of which lead to finitely convergent algorithms.
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.
Inexact cutting planes for two-stage mixed-integer stochastic programs
2018
We propose a novel way of applying cutting plane techniques to two-stage mixed-integer stochastic programs. Instead of using cutting planes that are always valid, our idea is to apply inexact cutting planes to the second-stage feasible regions that may cut away feasible integer second-stage solutions for some scenarios and may be overly conservative for others. The advantage is that it allows us to use cutting planes that are affine in the first-stage decision variables, so that the approximation is convex, and can be solved efficiently using techniques from convex optimization. We derive performance guarantees for using particular types of inexact cutting planes for simple integer recourse models. Moreover, we show in general that using inexact cutting planes leads to good first-stage solutions if the total variations of the probability density functions of the random variables in the model are small enough.
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...
Integer set reduction for stochastic mixed-integer programming
Computational Optimization and Applications
Two-stage stochastic mixed-integer programming (SMIP) problems with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing the subset of the convex hull of feasible integer points of the second-stage subproblem which can be used for solving the SMIP. The basic idea is to consider a small enough subset of feasible integer points that is necessary for generating a valid inequality for the integer subproblem. An algorithm for obtaining such a subset based on the solution of the subproblem LPrelaxation is then devised and incorporated into the Fenchel decomposition method for SMIP. To demonstrate the performance of the new integer set reduction methodology, a computational study based on randomly generated test instances was performed. The results of the study show that integer set reduction provides significant gains in terms of generating cuts faster leading to better bounds in solving SMIPs than using a direct solver.
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In this paper, we propose a generalized Benders decompositionbased branch and bound algorithm, GBDBAB, to solve two-stage convex 0-1 mixed-integer nonlinear stochastic programs with mixed-integer variables in both first and second stage decisions. In order to construct the convex hull of the MINLP subproblem for each scenario in closed-form, we first represent each MINLP subproblem as a generalized disjunctive program (GDP) in conjunctive normal form (CNF). Second, we apply basic steps to convert the CNF of the MINLP subproblem into disjunctive normal form (DNF) to obtain the convex hull of the MINLP subproblem. We prove that GBD is able to converge for the problems with pure binary variables given that the convex hull of each subproblem is constructed in closed-form. However, for problems with mixed-integer first and second stage variables, we propose an algorithm, GBDBAB, where we may have to branch and bound on the continuous first stage variables to obtain the optimal solution. ...
Mathematical Programming, 2005
This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C 3 (Common Cut Coefficients) Theorem. Based on the C 3 Theorem, we develop a decomposition algorithm which we refer to as Disjunctive Decomposition (D 2). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D 2 algorithm.