Characterization of the barrier parameter of homogeneous convex cones

We characterize the smallest (best) barrier parameter of self-concordant barriers for homogeneous convex cones. In particular, we prove that this parameter is the same as the rank of the cone which is the number of steps in a recursive construction of the cone (Siegel domain construction). We also provide lower bounds on the barrier parameter in terms of the Carathéodory number of the cone. The bounds are tight for homogeneous self-dual cones. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by an operating grant from NSERC of Canada.Research supported in part by the National Science Foundation under grant DMS-9306318.     

On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

A sufficient condition for self-concordance,with application to some classes of structured convex programming problems

Recently a number of papers were written that present low-complexity interior-point methods for different classes of convex programs. The goal of this article is to show that the logarithmic barrier function associated with these programs is self-concordant. Hence the polynomial complexity results for these convex programs can be derived from the theory of Nesterov and Nemirovsky on self-concordant barrier functions. We also show that the approach can be applied to some other known classes of convex programs.This author's research was supported by a research grant from SHELL.On leave from the Eötvös University, Budapest, Hungary. This author's research was partially supported by OTKA No. 2116.     

Interior-point methods for convex programming

This work is concerned with generalized convex programming problems, where the objective function and also the constraints belong to a certain class of convex functions. It examines the relationship of two basic conditions used in interior-point methods for generalized convex programming—self-concordance and a relative Lipschitz condition—and gives a short and simple complexity analysis of an interior-point method for generalized convex programming. In generalizing ellipsoidal approximations for the feasible set, it also allows a geometrical interpretation of the analysis.This work was supported by a research grant from the Deutsche Forschungsgemeinschaft, and in part by the U.S. National Science Foundation Grant DDM-8715153 and the Office of Naval Research Grant N00014-90-J-1242.On leave from the Institut für Angewandte Mathematik, University of Würzburg, Am Hubland, W-8700 Würzburg, Federal Republic of Germany.