Three methods for postoptimal analysis in integer linear programming

Holm, Søren; Klein, Dieter

doi:10.1007/bfb0121213

Cited by 32 publications

(10 citation statements)

References 10 publications

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“…For integer programming, numerous works are devoted to sensitivity analysis (Geoffrion and Nauss, 1977;Gusfield, 1980;Holm and Klein, 1984;Schrage and Wolsey, 1985;Wagelmans, 1990;Sotskov, Leontev, and Gordeev, 1995;Libura, 1996;Gal and Greenberg, 1997;Greenberg, 1998;Chakravarti and Wagelmans, 1999;Thiongane, Nagih, and Plateau, 2003a) and parametrization (Nauss, 1975;Geoffrion and Nauss, 1977;Gusfield, 1983;Carstensen, 1983;Loukakis and Muhlemann, 1984;Jenkins, 1987bJenkins, , 1990Crema, 2000Crema, , 2002Thiongane, Nagih, and Plateau, 2003b) (an extended bibliography on sensitivity analysis and parametrization can be found at Greenberg's web site 1 ). However, only a few works exist in literature on integer programming reoptimization (Roodman, 1974;Nauss, 1975;Pipers and Zoltners, 1976;Thiongane, Nagih, and Plateau, 2003a) include a state of the art on reoptimization techniques for 0-1 linear problems).…”

Section: Introductionmentioning

confidence: 99%

An Adapted Step Size Algorithm for a 0-1 Biknapsack Lagrangean Dual

2005

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This paper deals with a new algorithm for a 0-1 bidimensional knapsack Lagrangean dual which relaxes one of the two constraints. Classical iterative algorithms generate a sequence of multipliers which converges to an optimal one. In this way, these methods generate a sequence of 0-1 one-dimensional knapsack instances. Generally, the procedure for solving each instance is considered as a black box. We propose to design a new iterative scheme in which the computation of the step size takes into account the algorithmic efficiency of each instance. Our adapted step size iterative algorithm is compared favorably with several other algorithms for the 0-1 biknapsack Lagrangean dual over difficult instances for CPLEX 7.0. Notation(·): linear relaxation of a given integer problem (·), v(·): optimal value of a given problem (·), v(·): lower bound value of a given problem (·), ]a, b[: interval [a, b] without a and b.

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Section: Introductionmentioning

confidence: 99%

An Adapted Step Size Algorithm for a 0-1 Biknapsack Lagrangean Dual

2005

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“…Shortest paths and maximum capacity paths are studied in [14], assignment problems in [12], minimum spanning trees and shortest path trees in [17], and knapsacks in [9]. In addition, right-hand side vector sensitivity in integer and mixed-integer linear programs is studied in [5,10,15]. Related complexity issues are studied in [6,8,13,18].…”

Section: Introductionmentioning

confidence: 99%

Approximating the stability region for binary mixed-integer programs

Kılınç-Karzan

Toriello

Ahmed

et al. 2009

Operations Research Letters

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We consider optimization problems with some binary variables, where the objective function is linear in these variables. The stability region of a given solution of such a problem is the polyhedral set of objective coefficients for which the solution is optimal. A priori knowledge of this set provides valuable information for sensitivity analysis and re-optimization when there is objective coefficient uncertainty. An exact description of the stability region typically requires an exponential number of inequalities. We develop useful polyhedral inner and outer approximations of the stability region using only a linear number of inequalities. Furthermore, when a new objective function is not in the stability region, we produce a list of good solutions that can be used as a quick heuristic or as a warm start for future solves.

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“…The main limitation of most existing methods is that they can only be applied to problems with a single uncertain parameter or several uncertain parameters varying in a single direction. A number of approaches have been developed for parametric integer programming problems that involve a single parameter/scalar variation, basically including implicit enumeration methods (Roodman,1 Piper and Zoltners2), branch and bound methods (Roodman,3 Marsten and Morin,4 Ohtake and Nishida5), and cutting plane methods (Holm and Klein,6 Jenkins and Peters7), and so on. A detailed literature review can be found in Jenkins 8…”

Section: Introductionmentioning

confidence: 99%

Uncertainty analysis on the righthand side for MILP problems

Jia

Ierapetritou

2006

AIChE Journal

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A systematic framework is developed to solve the parametric mixed integer linear programming (pMILP) problems where uncertain parameters are present on the righthand side (RHS) of the constraints. For the case of multiple uncertain parameters, a new algorithm of multiparametric linear programming (mpLP) is proposed, which solves a number of nonlinear problems (NLP) iteratively. At each iteration, a point at which the objective value cannot be represented by the current optimal functions is found, and the new optimal function is included in the next iteration. Given the range of uncertain parameters in a MILP problem, the output of this proposed framework is a set of optimal integer solutions and their corresponding critical regions and optimal functions. A number of examples are presented to illustrate the applicabilities of the proposed approach and comparison with existing techniques.

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Three methods for postoptimal analysis in integer linear programming

Cited by 32 publications

References 10 publications

An Adapted Step Size Algorithm for a 0-1 Biknapsack Lagrangean Dual

An Adapted Step Size Algorithm for a 0-1 Biknapsack Lagrangean Dual

Approximating the stability region for binary mixed-integer programs

Uncertainty analysis on the righthand side for MILP problems

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