Journal of Optimization Theory and Applications

, Volume 83, Issue 3, pp 541–563

# Optimizing a linear function over an efficient set

• J. G. Ecker
• J. H. Song
Contributed Papers

## 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 π≧0 for all nonzero π∈D, then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point$$\hat x$$, if there is no adjacent efficient edge yielding an increase ind T x, then a cutting plane$$d^T x = d^T \hat x$$ is used to obtain a multiple-objective linear program ($$\bar M$$) with a reduced feasible set and an efficient set$$\bar E$$. To find a better efficient point, we solve the problem (Ii) of maximizingc i T x over the reduced feasible set in ($$\bar M$$) sequentially fori. If there is a$$x^i \in \bar E$$ that is an optimal solution of (Ii) for somei and$$d^T x^i > d^T \hat x$$, 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$$\hat x$$ is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.

## Key Words

Multiple-objective linear programming efficient sets domination cones nonconvex optimization

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