For a compact subset Y of Rm, and for i = 1, ..., n, let fi be a continuous mapping from Y to R1. Also, for any y in Y, let qi(y) be the ith largest of {fi(y)}. We present an efficient algorithm for lexicographically minimizing (q1(y), . . . , qn(y)) over Y, when Y is convex and fi 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 fi is convex for all i, the classical lexicographic minimization of {fi} 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 {qi}.
