Integer Programming Post-Optimal Analysis with Cutting Planes

Klein, Dieter; Holm, Søren

doi:10.1287/mnsc.25.1.64

Cited by 42 publications

(8 citation statements)

References 8 publications

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“…In the sequel we will use the following naming convention: Let Naïve be the naïve algorithm using Equations (9)-(10) and (12)- (13). Let ExactDP be the algorithm based on dynamic programming described in Section 5.1 which makes use of overlapping subproblems.…”

Section: Resultsmentioning

confidence: 99%

“…Greenberg [7] gives a quite recent bibliography for post-optimal analysis in combinatorial optimization, and mentions a number of papers on knapsack problems [4,8,14,22]. Klein and Holm [13] presented a general cutting-plane framework for post-optimal analysis of combinatorial problems and gave sufficient conditions for preserving the same optimal solution when the right-hand side or an objective coefficient is altered.…”

Section: Introductionmentioning

confidence: 99%

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Tolerance analysis for 0–1 knapsack problems

Pisinger

Saidi

2017

European Journal of Operational Research

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Post-optimal analysis is the task of understanding the behavior of the solution of a problem due to changes in the data. Frequently, post-optimal analysis is as important as obtaining the optimal solution itself. Post-optimal analysis for linear programming problems is well established and widely used. However, for integer programming problems the task is much more computationally demanding, and various approaches based on branch-and-bound or cutting planes have been presented. In the present paper we study how much coefficients in the original problem can vary without changing the optimal solution vector, the so-called tolerance analysis. We show how to perform exact tolerance analysis for the 0-1 knapsack problem with integer coefficients in amortized time O(c log n) for each item, where n is the number of items, and c is the capacity of the knapsack. Amortized running times report the time used for each item, when calculating tolerance limits of all items. Exact tolerance limits are the widest possible intervals, while approximate tolerance limits may be suboptimal. We show how various upper bounds can be used to determine approximate tolerance limits in time O(log n) or O(1) per item using the Dantzig bound and Dembo-Hammer bound, respectively. The running times and quality of the tolerance limits of all exact and approximate algorithms are experimentally compared, showing that all tolerance limits can be found in less than a second. The approximate bounds are of good quality for large-sized instances, while it is worth using the exact approach for smaller instances.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tolerance analysis for 0–1 knapsack problems

Pisinger

Saidi

2017

European Journal of Operational Research

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“…Klein et Holm [26] ontétudié l'analyse de sensibilitéà partir de ces conditions. Considéronsà nouveau le problème (P ) dans lequel…”

Section: Conditions Basées Sur La Programmation Linéaireunclassified

Analyse de sensibilité pour les problèmes linéaires en variables 0-1

2003

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“…For linear models without integer variables, the solution is given by linear parametric profiles which are obtained by extending the simplex algorithm to incorporate uncertain parameters, , and for nonlinear programs without integer variables, the solution is obtained by bounding the nonlinear function by linear profiles by using continuity and convexity properties of the objective function. − The presence of integer variables in the model results is an extra complication due to the discrete nature of the solution space. For linear models involving integer variables, the solution techniques are based upon either relaxing the integrality condition or introducing constraints which are obtained from the solution for fixed-integer variables. − For the case when the model is nonlinear and involves integer variables, the solution approaches have been limited to the case when only a single uncertain parameter is present in the model. ,,− In this work, we propose algorithms for the solution of multiparametric mixed-integer nonlinear programming problems which are based upon (i) the systematic characterization of the space of uncertain parameters and (ii) decomposition principles.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems

Dua

Pistikopoulos

1999

Ind. Eng. Chem. Res.

115

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In this paper we present novel theoretical and algorithmic developments for the solution of mixed-integer optimization problems involving uncertainty, which can be posed as multiparametric mixed-integer optimization models, where uncertainty is described by a set of parameters bounded between lower and upper bounds. In particular, we address convex nonlinear formulations involving (i) 0−1 integer variables and (ii) uncertain parameters appearing linearly and separately and present on the right-hand side of the constraints. The developments reported in this work are based upon decomposition principles where the problem is decomposed into two iteratively converging subproblems: (i) a primal and (ii) a master subproblem, representing valid parametric upper and lower bounds on the final solution, respectively. The primal subproblem is formulated by fixing the integer variables which results in a multiparametric nonlinear programming (mp-NLP) problem, which is solved by outer-approximating the nonlinear functions at a number of points in the space of uncertain parameters to derive linear profiles. The aim of the master subproblem is then to propose another set of integer solutions which are better than the integer solutions that have already been analyzed in the primal subproblem. This can be achieved by (i) introducing cuts, (ii) employing outer-approximation (OA) principles, or (iii) using generalized benders decomposition (GBD) fundamentals. The algorithm terminates when the master problem cannot identify a better integer solution.

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Integer Programming Post-Optimal Analysis with Cutting Planes

Cited by 42 publications

References 8 publications

Tolerance analysis for 0–1 knapsack problems

Tolerance analysis for 0–1 knapsack problems

Analyse de sensibilité pour les problèmes linéaires en variables 0-1

Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems

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