On a variant of lexicographic multi-objective programming

For a compact subset Y of R-m, and for i = l,..., n, let f(i) be a continuous mapping from Y to R-1. Also, for any y in Y, let q(i)(y) be the ith largest of {f(i)(y)}. We present an efficient algorithm for lexicographically minimizing (q(1)(y),...,q(n)(y)) over Y, when Y is convex and f(i) is convex for all i. The algorithm has applications in multi-criterial and group decision-making, co-operative game theory, and Rawlsian social welfare. The algorithm requires us to solve n convex programs, each of which has O(n) additional variables and O(n) additional constraints. When Y is convex and f(i) is convex for all i, the classical lexicographic minimization of {f(i)} over Y itself requires us to solve n convex programs, each with O(n) additional constraints. Therefore, it is probably not possible to improve upon our algorithm for lexicographically minimizing {q(i)}. (C) 1998 Elsevier Science B.V. All rights reserved.