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354
Review of nonlinear mixedinteger and disjunctive programming techniques
 Optimization and Engineering
, 2002
"... This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are ex ..."
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Cited by 95 (21 self)
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This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.
Column Generation
 CONTRIBUTED TO THE WILEY ENCYCLOPEDIA OF OPERATIONS RESEARCH AND MANAGEMENT SCIENCE (EORMS)
, 2010
"... Column generation is an indispensable tool in computational optimization to solve a mathematical program by iteratively adding the variables of the model. Even though the method is simple in theory there are many algorithmic choices and we discuss the most common ones. Particular emphasis in put on ..."
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Cited by 78 (3 self)
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Column generation is an indispensable tool in computational optimization to solve a mathematical program by iteratively adding the variables of the model. Even though the method is simple in theory there are many algorithmic choices and we discuss the most common ones. Particular emphasis in put on the dual interpretation, relating column generation to Langrangian relaxation and cutting plane algorithms, which revealed several critical issues like the need for dual variable stabilization techniques. We conclude with some advise for computer implementations.
Clearing algorithms for barter exchange markets: Enabling nationwide kidney exchanges
 In Proceedings of the 8th ACM Conference on Electronic commerce (EC
, 2007
"... In barterexchange markets, agents seek to swap their items with one another. These swaps consist of cycles of agents, with each agent receiving the item of the next agent in the cycle. We focus mainly on the upcoming national kidneyexchange market, where patients with kidney disease can obtain com ..."
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Cited by 74 (8 self)
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In barterexchange markets, agents seek to swap their items with one another. These swaps consist of cycles of agents, with each agent receiving the item of the next agent in the cycle. We focus mainly on the upcoming national kidneyexchange market, where patients with kidney disease can obtain compatible donors by swapping their own willing but incompatible donors. With almost 70,000 patients already waiting for a cadaver kidney in the US, this market is seen as the only ethical way to significantly reduce the 4,000 deaths per year attributed to kidney disease. The clearing problem involves finding a social welfare maximizing exchange when the maximum length of a cycle is fixed. Long cycles are forbidden, since, for incentive reasons, all transplants in a cycle must be performed simultaneously. Also, in barterexchanges generally, more agents are affected if one drops out of a longer cycle. We prove that the clearing problem is NPhard. Solving it exactly is one of the main challenges in establishing a national kidney exchange. We present the first algorithm capable of clearing these markets on a nationwide scale. The key is incremental problem formulation. We adapt two paradigms for the task: constraint generation and column generation. For each, we develop techniques that dramatically improve both runtime and memory usage. We conclude that column generation scales drastically better than constraint generation.
Robust branchandcutandprice for the capacitated vehicle routing problem
 IN PROCEEDINGS OF THE INTERNATIONAL NETWORK OPTIMIZATION CONFERENCE
, 2003
"... During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branchandcut algorithms giving bett ..."
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Cited by 60 (15 self)
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During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branchandcut algorithms giving better results. However, several instances in the range of 50–80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over qroutes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branchandcutandprice could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.
CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions
, 2005
"... Combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is NPcomplete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and ..."
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Cited by 55 (6 self)
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Combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is NPcomplete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and lower bounding (also across components), elaborate and dynamically chosen bidordering heuristics, and a host of structural observations. CABOB attempts to capture structure in any instance without making assumptions about the instance distribution. Experiments against the fastest prior algorithm, CPLEX 8.0, show that CABOB is often faster, seldom drastically slower, and in many cases drastically faster—especially in cases with structure. CABOB’s search runs in linear space and has significantly better anytime performance than CPLEX. We also uncover interesting aspects of the problem itself. First, problems with short bids, which were hard for the first generation of specialized algorithms, are easy. Second, almost all of the CATS distributions are easy, and the run time is virtually unaffected by the number of goods. Third, we test several random restart strategies, showing that they do not help on this problem—the runtime distribution does not have a heavy tail.
A Joint LocationInventory Model
, 2003
"... We consider a joint locationinventory problem involving a single supplier and multiple retailers. Associated with each retailer is some variable demand. Due to this variability, some amount of safety stock must be maintained to achieve suitable service levels. However, riskpooling benefits may be ..."
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Cited by 48 (9 self)
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We consider a joint locationinventory problem involving a single supplier and multiple retailers. Associated with each retailer is some variable demand. Due to this variability, some amount of safety stock must be maintained to achieve suitable service levels. However, riskpooling benefits may be achieved by allowing some retailers to serve as distribution centers (and therefore inventory storage locations) for other retailers. The problem is to determine which retailers should serve as distribution centers and how to allocate the other retailers to the distribution centers. We formulate this problem as a nonlinear integerprogramming model. We then restructure this model into a setcovering integerprogramming model. The pricing problem that must be solved as part of the column generation algorithm for the setcovering model involves a nonlinear term in the retailerdistributioncenter allocation terms. We show that this pricing problem can (theoretically) be solved efficiently, in general, and we show how to solve it practically in two important cases. We present computational results on several instances of sizes ranging from 33 to 150 retailers. In all cases, the lower bound from the linearprogramming relaxation to the setcovering model gives the optimal solution.
Polyhedral approaches to machine scheduling
, 1996
"... We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition ..."
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Cited by 41 (8 self)
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We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.
A CostRegular based Hybrid Column Generation Approach
, 2006
"... Constraint Programming (CP) offers a rich modeling language of constraints embedding efficient algorithms to handle complex and heterogeneous combinatorial problems. To solve hard combinatorial optimization problems using CP alone or hybrid CPILP decomposition methods, costs also have to be taken ..."
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Cited by 40 (5 self)
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Constraint Programming (CP) offers a rich modeling language of constraints embedding efficient algorithms to handle complex and heterogeneous combinatorial problems. To solve hard combinatorial optimization problems using CP alone or hybrid CPILP decomposition methods, costs also have to be taken into account within the propagation process. Optimization constraints, with their costbased filtering algorithms, aim to apply inference based on optimality rather than feasibility. This paper introduces a new optimization constraint, costregular. Its filtering algorithm is based on the computation of shortest and longest paths in a layered directed graph. The support information is also used to guide the search for solutions. We believe this constraint to be particularly useful in modelling and solving Column Generation subproblems and evaluate its behaviour on complex Employee Timetabling Problems through a flexible CPbased column generation approach. Computational results on generated benchmark sets and on a complex realworld instance are given.
Retrospective on Optimization
 25 TH YEAR ISSUE ON COMPUTERS AND CHEMICAL ENGINEERING
"... In this paper we provide a general classification of mathematical optimization problems, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering. We then provide a review of solution methods of the major types of op ..."
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Cited by 38 (1 self)
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In this paper we provide a general classification of mathematical optimization problems, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering. We then provide a review of solution methods of the major types of optimization problems for continuous and discrete variable optimization, particularly nonlinear and mixedinteger nonlinear programming. We also review their extensions to dynamic optimization and optimization under uncertainty. While these areas are still subject to significant research efforts, the emphasis in this paper is on major developments that have taken place over the last twenty five years.