Abstract not found

constraints

This paper was conceived in part while the author was sponsored by the visiting scientist program at the IBM T.J. Watson Research Center. Special thanks to Mike Shub, Roy Adler and Shmuel Winograd for their generosity. 1 Introduction

Abstract — A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser and Donoho. These measures include the p-norm-like (`(p 1)) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the Affine Scaling Transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the `(p 1) diversity measures are equivalent to a recently developed class of algorithms called FOCal Underdetermined System Solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a well-behaved p =0norm-like optimization algorithm. Computer experiments demonstrate that the p-norm-like and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution. I.

In this paper we present several "infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an ffl-solution for an ffl-perturbation of an initial strictly (primal and dual) feasible problem in O( p ln fflae f ) iterations, where is the parameter of a self-concordant barrier for the cone, ffl is a relative accuracy and ae f is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O( p ln ae \Delta ) iterations, where ae \Delta is a primal or dual infeasibility measure. 1 Introduction Nesterov and Nemirovskii [9] first developed and investigated extensions of several classes of interior-point algorithms for linear programming t...

Abstract. In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an ɛ-optimal solution can be computed efficiently using interior-point algorithms. As applications to this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ɛ-optimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in O(N √ N(log(¯c/ɛ)+ log N)) arithmetic operations where ¯c is the largest pairwise distance among the given points. The previous best-known result on this problem is a graphical algorithm which requires O(N 2) arithmetic operations under certain conditions. Key words. polynomial time, interior-point algorithm, minimizing a sum of Euclidean norms, Euclidean facilities location, shortest networks, Steiner minimum trees

This article provides a synopsis of the major developments in interior point methods for mathematical programming in the last thirteen years, and discusses current and future research directions in interior point methods, with a brief selective guide to the research literature. AMS Subject Classification: 90C, 90C05, 90C60 Keywords: Linear Programming, Newton's Method, Interior Point Methods, Barrier Method, Semidefinite Programming, Self-Concordance, Convex Programming, Condition Numbers 1 An earlier version of this article has previously appeared in OPTIMA -- Mathematical Programming Society Newsletter No. 51, 1996 2 M.I.T. Sloan School of Management, Building E40-149A, Cambridge, MA 02139, USA. email: rfreund@mit.edu 3 The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106 JAPAN. e-mail: mizuno@ism.ac.jp INTERIOR POINT METHODS 1 1 Introduction and Synopsis The purpose of this article is twofold: to provide a synopsis of the major developments in ...

We propose a BFGS primal-dual interior point method for minimizing a convex function on a convex set defined by equality and inequality constraints. The algorithm generates feasible iterates and consists in computing approximate solutions of the optimality conditions perturbed by a sequence of positive parameters µ converging to zero. We prove that it converges q-superlinearly for each fixed µ. We also show that it is globally convergent to the analytic center of the primal-dual optimalset when µ tends to 0 and strict complementarity holds.

We study the problem of minimizing a sum of p-norms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the p-norm problem is not self-dual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous self-concordant barrier functions for this problem. The barrier parameter of the first barrier function does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primal-dual potential reduction algorithm to compute an ffl-optimal solution in polynomial time that is independent of p. Computational experiences of a Matlab implementation are also reported. Key words. Shortest network, Steiner minimum trees, facilities location, po...

Let P = fx j Ax bg, where A is an m \Theta n matrix. We assume that P contains a ball of radius one centered at the origin, and is contained in a ball of radius R centered at the origin. We consider the problem of approximating the maximum volume ellipsoid inscribed in P . Such ellipsoids have a number of interesting applications, including the inscribed ellipsoid method for convex optimization. We reduce the complexity of finding an ellipsoid whose volume is at least a factor e \Gammaffl of the maximum possible to O(m 3:5 ln(mR=ffl)) operations, improving on previous results of Nesterov and Nemirovskii, and Khachiyan and Todd. A further reduction in complexity is obtained by first computing an approximation of the analytic center of P . Keywords: Maximum volume inscribed ellipsoid, inscribed ellisoid method. 1 1