We consider an extension of ordinary linear programming (LP) that adds weighted logarithmic barrier terms for some variables. The resulting problem generalizes both LP and the problem of finding the weighted analytic center of a polytope. We show that the problem has a dual of the same form and give complexity results for several different interior-point algorithms. We obtain an improved complexity result for certain cases by utilyzing a combination of the volumetric and logarithmic barriers. As an application we consider the complexity of solving the Eisenberg-Gale formulation of a Fisher equilibrium problem with linear utility functions.
Dept. of Management Sciences, University of Iowa, February 2011