Pareto group structure in a multicriteria optimization problem

Kozin, I. V.

doi:10.1007/s10559-010-9261-7

Cybern Syst Anal

2010

DOI: 10.1007/s10559-010-9261-7

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Pareto group structure in a multicriteria optimization problem

I. V. Kozin

Abstract: A multicriteria choice problem is considered. It is proposed to solve this problem by requiring the solution to be invariant under a certain group of transformations. Groups of transformations of a linear space that preserve the Pareto order in this space are investigated. The maximum group of such transformations is calculated up to group isomorphism. The most interesting discrete and continuous subgroups of the Pareto group are considered.

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2011

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An approach to estimating the average-case complexity of postoptimality analysis of discrete optimization problems

Mikhailyuk

2011

Cybern Syst Anal

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It is shown that an algorithm polynomial on average with respect to m and that determines an optimal solution to a set cover problem that differs from the initial problem in one position of the constraint matrix does not exist if the optimal solution of the original problem is known and DistNP is not a subset of Average-P. A similar result takes place for the knapsack problem.Keywords: postoptimality analysis, polynomiality on average with respect to m.In theoretical research in informatics, the emphasis is, as a rule, on questions of problems of worst-case complexity analysis. A more natural (and practical) way to the definition of complexity of problems is perhaps average-case complexity analysis. As is well known, many important problems are NP-hard, and there is only a small probability of their efficient solution in the worst case. Then algorithms would be expedient that would solve these problems efficiently on average. It is the main motivation of the theory of average-case complexity.In considering average-case complexity, distributions should be specified for instances of a problem. The same problem can be efficiently solved on average for some distributions, but it can be computationally hard for other distributions.In solving a concrete optimization problem with a fixed collection of input data, a considerable amount of information is usually obtained, but only its part can be used for the solution of this problem, and the other part of obtained information is lost. This circumstance poses the problem of expedient use of surplus information to solve other optimization problems "close" to the initial problem in a sense.The performance of postoptimality analysis of discrete optimization problems [1-4] provides for the solution of the following issues.-How does the optimal solution of a concrete problem change if values of its coefficients are somewhat changed? -How can the information obtained in solving a problem by a concrete method be used for the investigation of the problem changed?-What a minimal amount of additional information should be accumulated in solving the initial problem with a view to efficiently solving the changed problem?Along with the worst-case complexity of a problem [5,6], of interest is the application of the theory of average-case complexity to the postoptimality analysis of discrete optimization problems.

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An approach to estimating the average-case complexity of postoptimality analysis of discrete optimization problems

Mikhailyuk

2011

Cybern Syst Anal

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Pareto group structure in a multicriteria optimization problem

Cited by 1 publication

References 3 publications

An approach to estimating the average-case complexity of postoptimality analysis of discrete optimization problems

An approach to estimating the average-case complexity of postoptimality analysis of discrete optimization problems

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