Explicit Solutions to Optimization Problems on the Intersections of the Unit Ball of the $l_1 $ and $l_\infty $ Norms with a Hyperplane

Abstract

In this paper we characterize the extreme points of intersections of the $l_1 $ and $l_\infty $ unit balls in $R^n $ with a hyperplane. Such a characterization enables one to solve maximization problems of convex functions over these sets by enumeration. It turns out that the number of extreme points is $O(n^2 )$ under the $l_1 $ norm, and $O(2^N )$ under the $l_\infty $ norm. In particular, solving corresponding optimization problems by enumeration is efficient only under the $l_1 $ norm but not under the $l_\infty $ norm. Still, we get explicit solutions for maximization problems of linear objectives on the intersection of the $l_\infty $ unit ball with a hyperplane with a computational effort of $O(n)$. Applications of the results for computing bounds on the coefficients of ergodicity of square, nonnegative, irreducible matrices are discussed.