Interior-point algorithms for a generalization of linear programming and weighted centring

Abstract

In this paper, 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 centre 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 utilizing 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.

Keywords:

• linear programming
• weighted analytic centre
• interior-point algorithm
• Fisher equilibrium
• volumetric barrier

AMS Subject Classifications :

• 90C05
• 90C25
• 90C51

Acknowledgements

I am grateful to Eric Denardo for a series of conversations that stimulated my interest in the topic of this paper.

Notes

The bound in [4, Remark 7.2] does not explicitly appear in the published version 5, but is a simple consequence of [5, Lemma 6.2].

In much of the literature on the volumetric barrier, the variables m and n are interchanged, and the constraint matrix A is replaced by A ^T.