Duality theory for the matrix linear programming problem

Abstract

Multiple objective linear programming, also known as the linear vector maximization problem, has been studied by a number of authors. In particular, its duality theory has been considered by Philip [9], Isermann [7], Gray and Sutherland [5], Brumelle [ 11, and Ponstein [lo]. Neither the symmetry nor all the relationships of the duality theory of conventional linear programming, however, have been obtained in previous work. Such results are developed in this paper, which is most directly related to [7], by regarding the variables to be matrices. Previous work on matrix linear programming [3] has been restricted to a scalar objective function and differs substantially from that presented here. The notation R,,, will denote the real inner product space of all real m x n matrices A = [ai,], and as usual R, will be written for R, x 1. R Lx,, will be the set of matrices in R,,, with all nonnegative elements and @E Rnlxn the matrix consisting entirely of zeros with its order apparent from context. For A, B E Rmxn wewriteA A)ifB-AER i,,\{ 0). The relation < is obviously a partial order on R, x n. Let S be a subset of R m x n. The point Z* E S is said to be a maximal element (or upper efficient point) of S if there does not exist Z E S for which Z* < Z. Similarly Z* E S is a minimal element (or lower efficient point) of S if there does not exist Z E S for which Z < Z*. The set of all maximal elements of S is denoted by max S and minimal elements by min S.