On a variant of lexicographic multi-objective programming

Abstract For a compact subset Y of R m , and for i = 1, …, n , let φ i be a continuous mapping from Y to R 1 . Also, for any y in Y , let qi ( y ) be the i th largest of { φ i ( y )}. We present an efficient algorithm for lexicographically minimizing ( ql ( y ),…, qn ( Y )) over Y , when Y is convex and φ 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 φ i , is convex for all i , the classical lexicographic minimization of { φ 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 { qi }.