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A geometrical analysis of the efficient outcome set in multiple objective convex programs with linear criterion functions

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Abstract

This article performs a geometrical analysis of the efficient outcome setY E of a multiple objective convex program (MLC) with linear criterion functions. The analysis elucidates the facial structure ofY E and of its pre-image, the efficient decision setX E . The results show thatY E often has a significantly-simpler structure thanX E . For instance, although both sets are generally nonconvex and their maximal efficient faces are always in one-to-one correspondence, large numbers of extreme points and faces inX E can map into non-facial subsets of faces inY E , but not vice versa. Simple tests for the efficiency of faces in the decision and outcome sets are derived, and certain types of faces in the decision set are studied that are immune to a common phenomenon called “collapsing”. The results seem to indicate that significant computational benefits may potentially be derived if algorithms for problem (MLC) were to work directly with the outcome set of the problem to find points and faces ofY E , rather than with the decision set.

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  1. Department of Decision and Information Sciences, University of Florida, 351 Business Building, 32611, Gainesville, FL, USA

    Harold P. Benson

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Benson, H.P. A geometrical analysis of the efficient outcome set in multiple objective convex programs with linear criterion functions. J Glob Optim 6, 231–251 (1995). https://doi.org/10.1007/BF01099463

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