Characterization and separation of the $chv\acute{a}tal$ -gomory inequalities = 고모리 부등식의 특성과 절단평면의 생성에 관한 연구

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In this thesis, we consider the $Chv\acute{a}tal$ -Gomory inequalities for integer programming problem. Although the $Chv\acute{a}tal$ -Gomory inequalities motivated the cutting plane approaches for combinatorial optimization problems, researches focusing on the practical aspects of the $Chv\acute{a}tal$ -Gomory inequalities seem to be limited. The goal of the research is to provide efficient separation procedures to get strong $Chv\acute{a}tal$ -Gomory inequalities for even multiple constraints. As a result, we provide several standard forms of strong rank 1 $Chv\acute{a}tal$ -Gomory inequalities and efficient separation heuristics even with complexity O(n). To begin with, we investigate the separation problem for the rank 1 $Chv\acute{a}tal$ -Gomory inequalities for the integer knapsack problem. First, we develop a dominant relation in the elementary closure for the knapsack problem. Then, we explicitly describe necessary conditions for an inequality to be a nondominated rank 1 $Chv\acute{a}tal$ -Gomory inequality for the knapsack problem, which we call maximal inequality. Independently, we show that the separation problem can be seen as a series of combinatorial optimization problems. Finally, we show that the optimal solution of the separation problem also falls into the category of the maximal inequalities. The concept of a cover naturally arises in the description of the maximal inequalities. We develop a relationship between the traditional cover inequalities and maximal inequalities. Furthermore, the relationship is expanded to the integer knapsack problem. In short, cover inequalities belong to a subclass of maximal inequalities. Next, we extend the results to the case of $Chv\acute{a}tal$ -Gomory inequalities for multiple constraints with special structure. We explicitly describe a useful condition to find the most-violated rank 1 $Chv\acute{a}tal$ -Gomory inequality for the generalized assignment problem. Then, we observe that the separation problem ...
Park, Sung-Sooresearcher박성수researcher
한국과학기술원 : 산업및시스템공학과,
Issue Date
466353/325007  / 020005147

학위논문(박사) - 한국과학기술원 : 산업및시스템공학과, 2011.2, [ vii, 88 p. ]


Cutset Inequalities; Maximal Inequalities; O(n) Separation Heuristic; $Chv\acute{a}tal$-Gomory Inequalities; Multiple Constraints; c-MIR 부등식; cutset inequality; $O(n)$ 절단평면생성 알고리즘; 고모리부등식; 범용절단평면

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