Universal Barrier Is <i>n</i>-Self-Concordant
نویسندگان
چکیده
This paper shows that the self-concordance parameter of universal barrier on any n-dimensional proper convex domain is upper bounded by n. bound tight and improves previous O(n) Nesterov Nemirovski. The key to our main result a pair new, sharp moment inequalities for s-concave distributions, which could be independent interest.
منابع مشابه
The entropic barrier: a simple and optimal universal self-concordant barrier
We prove that the Fenchel dual of the log-Laplace transform of the uniform measure on a convex body in Rn is a (1 + o(1))n-self-concordant barrier. This gives the first construction of a universal barrier for convex bodies with optimal self-concordance parameter. The proof is based on basic geometry of log-concave distributions, and elementary duality in exponential families.
متن کاملOn the complexity of the primal self-concordant barrier method
In his Introductory Lectures on Convex Programming Nesterov has given an algorithm to nd the analytic centre x F for a given -self-concordant barrier F with bounded domain and a given interior point of this domain. The intended use of this algorithm is as an auxiliary phase in a primal short-step path-following method for solving convex programming problems. For the number of iterations in this...
متن کاملComposite self-concordant minimization
We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size ...
متن کاملOn the optimal parameter of a self-concordant barrier over a symmetric cone
The properties of the barrier F (x) = −log(det(x)), defined over the cone of squares of a Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, in a more direct and suitable way th...
متن کاملA Hybrid Proximal Extragradient Self-Concordant Primal Barrier Method for Monotone Variational Inequalities
In this paper we present a primal interior-point hybrid proximal extragradient (HPE) method for solving a monotone variational inequality over a closed convex set endowed with a selfconcordant barrier and whose underlying map has Lipschitz continuous derivative. In contrast to the method of [7] in which each iteration required an approximate solution of a linearized variational inequality over ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematics of Operations Research
سال: 2021
ISSN: ['0364-765X', '1526-5471']
DOI: https://doi.org/10.1287/moor.2020.1113