On Self-Concordant Convex-Concave Functions
نویسنده
چکیده
In this paper, we introduce the notion of a self-concordant convex-concave function, establish basic properties of these functions and develop a path-following interior point method for approximating saddle points of “good enough” convex-concave functions – those which admit natural self-concordant convex-concave regularizations. The approach is illustrated by its applications to developing an exterior penalty polynomial time method for Semidefinite Programming and to the problem of inscribing the largest volume ellipsoid into a given polytope.
منابع مشابه
Self-concordant barriers for hyperbolic means
The geometric mean and the function (det(·))1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t → p(x+ td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and ...
متن کاملThe 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 کاملLocal Self-concordance of Barrier Functions Based on Kernel-functions
Many efficient interior-point methods (IPMs) are based on the use of a self-concordant barrier function for the domain of the problem that has to be solved. Recently, a wide class of new barrier functions has been introduced in which the functions are not self-concordant, but despite this fact give rise to efficient IPMs. Here, we introduce the notion of locally self-concordant barrier functio...
متن کاملGeneralized Self-Concordant Functions: A Recipe for Newton-Type Methods
We study the smooth structure of convex functions by generalizing a powerful concept so-called self-concordance introduced by Nesterov and Nemirovskii in the early 1990s to a broader class of convex functions, which we call generalized self-concordant functions. This notion allows us to develop a unified framework for designing Newton-type methods to solve convex optimization problems. The prop...
متن کاملExistence and multiplicity of nontrivial solutions for $p$-Laplacian system with nonlinearities of concave-convex type and sign-changing weight functions
This paper is concerned with the existence of multiple positive solutions for a quasilinear elliptic system involving concave-convex nonlinearities and sign-changing weight functions. With the help of the Nehari manifold and Palais-Smale condition, we prove that the system has at least two nontrivial positive solutions, when the pair of parameters $(lambda,mu)$ belongs to a c...
متن کامل