Optimizing a linear function over an efficient set

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 ^Tπ≧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 (I_i) of maximizingc ^T _i 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 (I_i) 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.