Optimizing a linear function over an efficient set
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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 WordsMultiple-objective linear programming efficient sets domination cones nonconvex optimization
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