Optimizing a linear function over an efficient set |
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Authors: | J G Ecker J H Song |
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Institution: | (1) Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York |
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Abstract: | The problem (P) of optimizing a linear functiond
T
x over the efficient set for a multiple-objective linear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function directiond and the set of domination directionsD, ifd
T
![pgr](/content/k5p04k556r197641/xxlarge960.gif) 0 for all nonzero ![pgr](/content/k5p04k556r197641/xxlarge960.gif) D, then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point
, if there is no adjacent efficient edge yielding an increase ind
T
x, then a cutting plane
is used to obtain a multiple-objective linear program (
) with a reduced feasible set and an efficient set
. To find a better efficient point, we solve the problem (Ii) of maximizingc
i
T
x over the reduced feasible set in (
) sequentially fori. If there is a
that is an optimal solution of (Ii) for somei and
, then we can choosex
i
as a current efficient point. Pivoting on the reduced feasible set allows us to find a better efficient point or to show that the current efficient point
is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.The authors would like to thank an anonymous referee, H. P. Benson, and P. L. Yu for numerous helpful comments on this paper. |
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Keywords: | Multiple-objective linear programming efficient sets domination cones nonconvex optimization |
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