In [19], a general, inexact, e cient proximal quasi-Newton algorithm for composite optimization problems has been proposed and a sublinear global convergence rate has been established. In this paper, we analyze the convergence properties of this method, both in the exact and inexact setting, in the case when the objective function is strongly convex. We also investigate a practical variant of t...

We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n n) algorithm for the case of normally distributed edge lengths, which is based on quasi-convex maximization. We then prove average and smoothe...

We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of c0 and superreflexivity are discussed. The question of coarse embeddability into uniformly convex Banach spaces became interesting after the recent work of G. Kasparov and G. Yu, who showed the coarse Novi...

We prove some existence results of solutions for a new class of generalized bi-quasivariational inequalities GBQVI for quasi-pseudomonotone type II and strongly quasi-pseudomonotone type II operators defined on noncompact sets in locally convex Hausdorff topological vector spaces. To obtain these results on GBQVI for quasi-pseudomonotone type II and strongly quasipseudomonotone type II operator...

A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points x in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi...

We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x) subject to nonnegative constraints x ≥ 0. The algorithms are based on an entropy-like second-order homogeneous distance function. Under the assumption that the global minimizer set is nonempty and bounded, we prove the full convergence of the sequence generated by the algo...

A rational polytope is the convex hull of a finite set of points in Rd with rational coordinates. Given a rational polytope P ⊆ Rd, Ehrhart proved that, for t ∈ Z>0, the function #(tP ∩ Zd) agrees with a quasi-polynomial LP(t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new ...

Quasi-Newton methods are generally held to be the most efficient minimization methods for small to medium sized problems. From these the symmetric rank one update of Broyden [4] has been disregarded for a long time because of its potential failure. The work of Conn, Gould and Toint [6], Kelley and Sachs [13] and Khalfan, Byrd and Schnabel [14], [15] has renewed the interest in this method. Howe...

The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized.This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this ...

In this paper discrete quasi-copulas (defined on a square grid I n of [0,1]) are studied and it is proved that they can be represented by means of a special class of matrices with entries in [-1,1]. Special considerations are made for the case of irreducible discrete quasi-copulas (those with range In) defined on the finite chain In, showing that they can be represented through Alternating-Sign...