Pdf dantzigwolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear. The result is an efficient approach for computing tight lower bounds on the optimal objective value to the original discrete program. The reformulation gives rise to an integer master problem, whose typically large number of variables is dealt with implicitly by using an integer programming column. To solve this model using the generalized benders decomposition gbd method, we derive a convex reformulation of the secondstage problem to overcome algorithmic challenges embedded in the nonconvexity, nonlinearity, and nonseparability of first and. A decomposition technique for solving integer programming. The twostage model is equivalent to a nonconvex mixed integer nonlinear program minlp.
The reformulation gives rise to an integer master problem, whose typically large number of variables is dealt with implicitly by using an integer programming column generation procedure, also known as branch and price algorithm. Citeseerx reformulation and decomposition of integer programs. Projectionbased reformulation and decomposition algorithm for a class of mixed integer bilevel linear programs. Dantzigwolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear programming relaxation bound. Mixed integer programming with decomposition to solve a. Solving bilevel mixed integer program by reformulations and decomposition june, 2014 abstract in this paper, we study bilevel mixed integer programming mip problem and present a novel computing scheme based on reformulations and decomposition strategy. A projectionbased reformulation and decomposition algorithm. Choose an initial restricted set of columns while there is a column with negative reduced cost do. Partial decomposition strategies for twostage stochastic integer programs teodor gabriel crainic1, mike hewitt2, walter rei1, 1 interuniversity research centre on enterprise networks, logistics and transportation cirrelt and department of management and technology, universite du quebec a montreal, p. Integer program, mathematical reformulation, mathematical derivation, cus. There is a large class of integer programs that are well suited for this solution technique. This problem is called the linear integer programming problem.
The dual decomposition algorithm solves a lagrangian relaxation based on relaxing the nonanticipativity constraints, and then embeds this within a branchandbound. We propose an extended variant of the reformulation and decomposition algorithm for solving a special class of mixed integer bilevel linear programs miblps where continuous and integer variables are involved in both upper and lowerlevel problems. In this survey we examine ways to reformulate integer and mixed integer programs. Optimal design of superstructures for placing units and.
Reformulation, linearization, and decomposition techniques for mixed integer nonlinear programming models with an application to balanced distributed operating room scheduling. Their success relies on exploiting problem structures in an integer program via a decomposition or reformulation to which standard branchand. Polyhedral approximations of quadratic semiassignment. We develop linearization techniques for the mixed integer nonlinear model. Despite the growing applications of miblps in various areas, there is no efficient solution algorithm for largescale miblps. Integer programming formulations and benders decomposition.
Given a compact formulation and a relaxation, the framework takes care of all. Reformulating a mixed integer program see nemhauser and wolsey, section ii. This dissertation presents a solution method for solving largescale stochastic mixed integer programs that integrates two scenario decomposition algorithms. Automatic dantzigwolfe reformulation of mixed integer. T1 projectionbased reformulation and decomposition algorithm for a class of mixed integer bilevel linear programs. In this paper, we study bilevel mixed integer programming mip problem and present a novel computing scheme based on reformulations and decomposition strategy. The example in this chapter is based on multicommodity distribution system references design by benders decomposition ge74 by geo. R is a free software environment for statistical computing and graphics. We examine ways to reformulate integer and mixed integer programs.
It is also a valuable reference for industrial users of integer programming and researchers who would like to keep up with advances in the field. It was originally developed by george dantzig and philip wolfe and initially published in 1960. A modification of benders decomposition algorithm for. On dantzigwolfe decomposition in integer programming and ways. It is said to be a mixed integer program when some, but not all, variables are restricted to be integer, and is called a pure integer program when all decision variables must be integers. However, the method is not implemented in any stateoftheart mip solver as it is considered to require structural problem knowledge and tailoring to this structure.
Production planning by mixed integer programming yves. Optimization over the integers, 600 pages, hardcover, ca. However, the method is not implemented in any stateoftheart mip solver as it. Decomposition and reformulation in mixed integer programming ima new directions short course on mathematical optimization jim luedtke department of industrial and systems engineering university of wisconsinmadison august 11, 2016 jim luedtke uwmadison decomposition methods lecture notes 1 47. The dual decomposition algorithm solves a lagrangian relaxation based on relaxing the nonanticipativity constraints, and then embeds this within a branch and bound. Pdf dantzigwolfe decomposition or reformulation is wellknown to provide strong dual bounds for specially structured mixed integer programs mips. Gomory transformed the field of integer programming when he published a paper that described a cuttingplane algorithm for pure integer programs and announced that the method could be refined to give a finite algorithm for integer programming. Reformulation and decomposition of integer programs francois vanderbeck1 and laurence a.
Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for. N2 we propose an efficient algorithm for solving mixed integer bilevel linear programs miblps. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branchandbound based algorithm. Decomposition and reformulation of integer linear programming problems article pdf available in 4or quarterly journal of the belgian, french and italian operations research societies 102. Combinatorial benders cuts for mixedinteger linear programming. Ddsip is a c implementation of a number of scenario decomposition algorithms for solving twostage stochastic mixed integer programs with mixed integer recourse. In this integrated method, fast progress in early iterations of ph speeds up the convergence of dd to an exact solution. Reformulation and decomposition of integer programs franc.
Stochastic programming resources stochastic programming society. By converting bilevel mip into a constrained mathematical program, we present its singlelevel. Solving this problem enables the use of perspective reformulation techniques for obtaining strong lower bounds for micqps with semicontinuous variables but a nonseparable objective. In this paper, we modify benders decomposition method by using concepts from the reformulation linearization technique rlt and liftandproject cuts in order to develop an approch for solving discrete optimization problems that yield integral subproblems, such as those that arise in the case of twostage stochastic programs with integer recourse. In this paper, we develop an algorithm for solving large scale integer program relying on column generation method.
Automaticdantzigwolfe reformulation of mixed integer. Lagrangian decomposition and mixedinteger quadratic. A schematic overview of the suggested decomposition. Stochastic programming resources stochastic programming. Solution methods and bounds for twostage riskneutral and. Reformulation and decomposition of integer programs 2009. Reformulation and decomposition of integer programs citeseerx. Arturo castillosalazar school of computer science, asap research group, the university of nottingham, nottingham, u. In this paper, we propose to base the dantzigwolfe decomposition of an integer program on the discretization of the integer polyhedron associated with a subsystem of constraints as opposed to its convexification. Partial decomposition strategies for twostage stochastic. Dantzigwolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear programming. Solving bilevel mixed integer program by reformulations and decomposition. Automatic dantzigwolfe reformulation of mixed integer programs.
In many settings the term refers to integer linear programming ilp, in which the objective function and the constraints other than the integer constraints are linear integer programming is npcomplete. Reformulation, linearization, and decomposition techniques. In this lecture, we present two related methods, lagrangian relaxation and dantzigwolfe reformulation for exploiting structure of mixed integer programming models to obtain better relaxations or solve largescale instances. A lagrangean based branchandcut algorithm for global. It is well known that a stochastic mixed integer program is difficult to solve due to its nonconvexity. Cutting planes in integer and mixed integer programming. Reformulation and decomposition of integer programs in this survey we examine ways to reformulate integer and mixed integer programs. Benders decomposition reduces a mixed integer optimization problem with p continuous variables and n integer variables to one with just one continuous variable, and still n integer. Many texts on linear programming have sections dedicated to discussing this decomposition algorithm dantzigwolfe decomposition relies on delayed column generation for improving the.
Mccarl andrea gray authors are, respectively, assistant professor in the department of agricultural and resource economics, oregon state university, and professor and programmer, department of agricultural economics, texas am university. We focus on the common threads among three traditional methods for generating approximations to the convex hull of feasible solutions to an milp. Decomposition of integer programs with matchability structure. Decompositions of semidefinite matrices and the perspective. Reformulation and decomposition of integer programs core. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branchandbound based alg. Pdf reformulation, linearization, and decomposition. Mixed integer programming with decomposition to solve a workforce scheduling and routing problem wasakorn laesanklang, dario landasilva and j. Scenario grouping and decomposition algorithms for chance.
Reformulation and decomposition of integer programs the book contains reprints of key historical articles together with new introductions and historical perspectives by the authors. It is used at mit and other places for teaching integer programming at the graduate level. An integer programming and decomposition approach to general chanceconstrained mathematical programs. An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. We develop uni and bilevel logicbased benders decompositions for the linearized. The proposed reformulation and decomposition are useful to apply an optimization algorithm for the master problem that guarantees fixed integer values for the external variables in the primal subproblems.
This textbook provides a comprehensive modeling, reformulation and optimization approach for solving production planning and related supply chain planning problems, covering topics from a basic introduction to planning systems, mixed integer programming mip models and algorithms through the advanced description of mathematical results in polyhedral combinatorics required to solve these. Reformulation and decomposition of integer programs springerlink. Solving this problem enables the use of perspective reformulation techniques for obtaining strong lower bounds for micqps with semicontinuous variables but a nonseparable objective function. Solving bilevel mixed integer program by reformulations and decomposition, optimization online 2014. This dissertation presents an integrated method for solving stochastic mixed integer programs, develops a lower bounding approach for multistage riskaverse stochastic mixed integer programs, and proposes an optimization formulation for mixedmodel assembly line sequencing mmals problems. In practice, however, one would use heuristics to find a good decomposition.
In particular, we consider miblps with upperlevel constraints that involve lowerlevel variables. We then give a mixed integer quadratic programming miqp reformulation of the pcqp and show that the continuous relaxation of the miqp is exactly the socp relaxation. First we cover reformulations based on decomposition, such as lagrangean relaxation, the dantzigwolfe reformulation and the resulting column generation and branchandprice algorithms. New update will be made to edxu96matrixoptim, which is the aggregation of robust optimization and decomposition. Dantzigwolfe decomposition or reformulation is wellknown to provide strong dual bounds for specially structured mixed integer programs mips. Aug 16, 2012 read lagrangian decomposition and mixed integer quadratic programming reformulations for probabilistically constrained quadratic programs, european journal of operational research on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Milton stewart school of industrial and systems engineering, georgia institute of technology, atlanta, ga, usa 30332 august 23, 2016 abstract we consider integer programming ip problems.
These computations are done with a branchandcut algorithm based on polyhedral investigations of the matrix decomposition problem. Generic dantzigwolfe reformulation of mixed integer programs. A number theoretic reformulation and decomposition method for integer programming. Experiments with a generic dantzigwolfe decomposition for. A benders decomposition code for the solution of nonlinear. Also the coinor initiative hosts a generic decomposition code, called dip 12 formerly known as decomp, which is. A benders decomposition code for the solution of nonlinear mixed integer programs gregory m. Mathoc mathematics mathoptimization and control math. We develop linearization techniques for the mixedinteger nonlinear model. These include a method employing an outer approximation, the cuttingplane. Read lagrangian decomposition and mixed integer quadratic programming reformulations for probabilistically constrained quadratic programs, european journal of operational research on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Imposing integrality constraints on the columns in the master program will not.
Based on this singlelevel reformulation, a decomposition algorithm is developed that converges to the global optimal solution in finite iterations. Reformulation and decomposition of integer programs dial. In this paper, we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the bigm coefficients. Decomposition and reformulation in mixedinteger programming. We provide a computational proofofconcept that the reformulation can be. E, an induced matching is a matching such that no two edges. Specifically, a hamiltonian decomposition of a graph interpretation is devised to motivate a lagrangian dual whose subproblems consist of families of separable acyclic minimumcost network flows. Lagrangean relaxation techniques have been used by takriti et al. Projectionbased reformulation and decomposition algorithm. Abstract dantzigwolfe decomposition or reformulation is wellknown to pro vide strong dual bounds for specially structured mixed integer programs mips.
A new dantzigwolfe reformulation and branchandprice. Benders decomposition rensselaer polytechnic institute. Egon balas, michel balinski, jack edmonds, ralph e. From the early years to the stateoftheart on free shipping on qualified orders. For general chanceconstrained programs, we propose a branch and cut algorithm to optimize the bilevel program, and for chanceconstrained linear programs, a mixed integer linearprogramming reformulation is derived. Dip decomposition for integer programming is an opensource extensible. Dantzigwolfe series of decomposition and reformulation algorithm to solve milp. It compiles and runs on a wide variety of unix platforms, windows and macos. Integer programming, benders decomposition, maximum induced matching, distance2 matching, strong matching history.
In this paper, we formulate a mixed integer bilevel program that optimally groups scenarios to tighten the quantile bounds. Mixed integer programs mips involving logical implications modeled through bigm coefficients are notoriously among the hardest to solve. This is followed by an examination of benders type algorithms based on projection. Decomposition and reformulation in mixed integer programming ima new directions short course on mathematical optimization jim luedtke department of industrial and systems engineering university of wisconsinmadison august 11, 2016 jim luedtke uwmadison decomposition methods lecture. First we cover reformulations based on decomposition, such as lagrangean relaxation, the dantzigwolfe. We first derive secondorder cone programming socp relaxation and semidefinite programming sdp relaxation for the problem via a new lagrangian decomposition scheme. Highlightswe develop a mixed integer nonlinear model for balanced distributed operating room scheduling. Wolsey, title reformulation and decomposition of integer programs, year 2009. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch and bound based algorithm. Although the textbook dantzigwolfe decomposition reformulation for the capacitated lotsizing problem, as already proposed by manne manne, a. Dantzigwolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear programming relaxation bound due to the nonconvexity of an integer problem. Solving bilevel mixed integer program by reformulations. Pdf reformulation and decomposition of integer programs. Column generation solve linear programming relaxation using column generation.
Solution methods and bounds for twostage riskneutral. Decomposition, reformulation, and diving in university course. In this paper, we modify benders decomposition method by using concepts from the reformulationlinearization technique rlt and liftandproject cuts in order to develop an approch for solving discrete optimization problems that yield integral subproblems, such as those that arise in the case of twostage stochastic programs with integer. Solving bilevel mixed integer program by reformulations and. A master class on computational techniques for combinatorial optimization on june 5, and the main conference on june 6 june 8, 2017. A multiobjective perspective merve bodur, shabbir ahmed, natashia boland, and george l. Cpaior 2017, the fourteenth international conference on integration of artificial intelligence and operations research techniques in constraint programming, will be held in padova, italy, june 5 june 8, 2017. Reformulation and decomposition of integer programs. On dantzigwolfe decomposition in integer programming and.