Theorem and a D Algorithm for Large Scale Stochastic Integer Programming � Set Convexi cation (original) (raw)
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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.
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.
Computations with disjunctive cuts for two-stage stochastic mixed 0-1 integer programs
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.
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.
2013
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, as well as a new set of test problems, which we refer ...
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.
Algorithms and Reformulations for Large-Scale Integer and Stochastic Integer Programs
2012
In this dissertation, we develop methodologies to solve difficult classes of discrete optimization problems under uncertainty by using techniques from integer programming. First, we consider a class of two-stage stochastic integer programs with binary variables in the first stage, general integer variables in the second stage and random data with finitely many outcomes. We develop an L-shaped decomposition algorithm that iteratively tightens the linear relaxation of the scenario subproblems using Gomory cuts and maintains convex first stage approximations. We show that the algorithm is not only finitely convergent, but also computationally amenable allowing several alternative implementations. We develop a computer implementation of this algorithm and report computational results on the stochastic server location problem instances. With the goal of extending our methods to solve stochastic mixed integer programs, we develop extensions to the cutting plane tree algorithm by integrating Gomory cuts with disjunctive cuts. We report computational results with the Gomory-enhanced simple disjunctive cuts using benchmark test instances. Second, we introduce the concept of service levels into deterministic lot sizing problems and develop a polynomial time algorithm for a single item lot sizing problem with a ready rate service level constraint. Based on the algorithm, we develop compact extended reformulations for this problem and a relaxation. We show that although the extended reformulations are large, they outperform standard formulations of the problem while guaranteeing optimal solutions when they are used to solve capacitated multi-item instances.
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.
Tight Second Stage Formulations in Two-Stage Stochastic Mixed Integer Programs
SIAM Journal on Optimization, 2018
We study two-stage stochastic mixed integer programs (TSS-MIPs) with integer variables in the second stage. We show that under suitable conditions, the second stage MIPs can be convexified by adding parametric cuts a priori. As special cases, we extend the results of Miller and Wolsey (Math Program 98(1):73-88, 2003) to TSS-MIPs. Furthermore, we consider second stage programs that are generalizations of the well-known mixing (and continuous mixing) set, or certain piecewise-linear convex objective integer programs. These results allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the TSS-MIP. We also use four variants of the two-stage stochastic capacitated lot-sizing problems as test problems for computational experiments, and present tight second-stage formulations for these problems. Our computational results show that adding parametric inequalities that a priori convexify the second stage formulation significantly reduces the total solution time taken to solve these problems.