On the Asymptotic Average Number of Efficient Vertices in Multiple Objective Linear Programming

Let a1 , ..., am , c1 , ..., ck be independent random points in R that are identically distributed spherically symmetrical in R and let X :=[x # R|ai x 1, i=1, ..., m] be the associated random polyhedron for m n 2. We consider multiple objective linear programming problems maxx # X c1 x, maxx # X c T 2 x, ..., maxx # X c T k x with 1 k n. For distributions with algebraically decreasing tail in the unit ball, we investigate the asymptotic expected number of vertices in the efficient frontier of X with respect to c1 , ..., ck for fixed n, k and m . This expected number of efficient vertices is the most significant indicator for the average-case complexity of the multiple objective linear programming problem. 1998 Academic Press