In this paper, we present approximation algorithms for the problem of cutting out a convex polygon P with n vertices from another convex polygon Q with m vertices by a sequence of guillotine cuts of smallest total length. Specifically, we give an O(n + m) running time, constant factor approximation algorithm, and an O(n+m) running time, O(log n)-factor approximation algorithm for cutting P out ...

We study asymptotic properties of the approximation of a suuciently smooth convex body K in R d by the convex hulls of n points in the boundary of K, for n ! 1. The deviation is measured by the Hausdorr distance. The asymptotic distribution of the vertices of best-approximating polytopes is determined. Further results involve prescribed densities for the vertices and describe the strength of ap...

We show that the saturation order of piecewise constant approximation in Lp norm on convex partitions with N cells is N −2/(d+1), where d is the number of variables. This order is achieved for any f ∈ W 2 p (Ω) on a partition obtained by a simple algorithm involving an anisotropic subdivision of a uniform partition. This improves considerably the approximation order N−1/d achievable on isotropi...

In this paper I explore the concept of the Domains of Best Approximations. These structures are of interest in the study of Diophantine Approximation as they encode the inequalities that govern how efficiently an irrational point can be approximated by a rational point with a given bound on the (lowest common) denominator. The problem of best approximation can be formulated as follows. Given a ...

Recently, Krylov, Barles, and Jakobsen developed the theory for estimating errors of monotone approximation schemes for the Bellman equation (a convex Isaacs equation). In this paper we consider an extension of this theory to a class of non-convex multidimensional Isaacs equations. This is the first result of this kind for non-convex multidimensional fully non-linear problems. To get the error ...

Logistic regression and matrix factorization specifically SVD are both very popular in application areas of computational advertising and recommender system. This project basically can be divided into two parts. The first part is to test different optimization methods to solve sparse 2-class logistic regression. This is a convex non-smooth problem. Particularly, I’ll try to answer the question ...

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi-continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided.

We consider the problem of minimizing the max of two convex functions from both approximation and sensitivity point of view.This lead up to study the epiconvergence of a sequence of level sums of convex functions and the related dual problems.

In an interference limited network, joint power and admission control (JPAC) aims at supporting a maximum number of links at their specified signal to interference plus noise ratio (SINR) targets while using minimum total transmission power. Various convex approximation deflation approaches have been developed for the JPAC problem. In this paper, we propose an effective polynomial time non-conv...

In this paper, we give an overview of some results concerning best and random approximation convex bodies by polytopes. We explain how both are linked see that is almost as good approximation.