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constraints

We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a self-contained convergence analysis, that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.

We study the problem of computing a (1 + #)-approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's first-order algorithm. Our analysis leads to a slightly improved complexity bound of O(nd (0, 1). As a byproduct, our algorithm returns a core set with the property that the minimum volume enclosing ellipsoid of provides a good approximation to that of S.

In this paper we study practical solution methods for finding the maximum-volume ellipsoid inscribing a given full-dimensional polytope in ! n defined by a finite set of linear inequalities. Our goal is to design a general-purpose algorithmic framework that is reliable and efficient in practice. To evaluate the merit of a practical algorithm, we consider two key factors: the computational cost per iteration and the typical number of iterations required for convergence. In addition, numerical stability is also an important factor. We investigate some new formulations upon which we build primal-dual type, interior-point algorithms, and we provide theoretical justifications for the proposed formulations and algorithmic framework. Extensive numerical experiments have shown that one of the new algorithms should be the method of choice among the tested algorithms.

A ray-tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a given set of linear separable examples. For randomly distributed examples the billiard estimate of the single Perceptron with best average generalization probability agrees with known analytic results, while for real-life classification problems the generalization probability is consistently enhanced when compared to the maximal stability Perceptron. 1 Introduction Neural networks can be used for both concept learning (classification) and for function interpolation and/or extrapolation. Two basic mathematical methods seem to be particularly adequate for studying neural networks: geometry (especially combinatorial geometry) and probability theory (statistical physics). Geometry is illuminating and probability theory is powerful. In this paper I consider the perhaps simplest neural network, the venerable Perceptron [1]: given a set of examples falling in two classes,...

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a 1 , . . . , am . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30, 000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.

In this report, we consider the problem of finding the maximumvolume ellipsoid inscribing a given full-dimensional polytope in ! n defined by a finite set of affine inequalities. We present several formulations for the problem that may serve as algorithmic frameworks for applying interior-point methods. We propose a practical interior-point algorithm based on one of the formulations and present preliminary numerical results. 1

.<F3.866e+05> The feasibility problem for a system of linear inequalities can be converted into an unconstrained optimization problem by using ideas from the ellipsoid method, which can be viewed as a very simple minimization technique for the resulting nonlinear function. This function is related to the volume of an ellipsoid containing all feasible solutions, which is parametrized by certain weights which we choose to minimize the function. The center of the resulting ellipsoid turns out to be a feasible solution to the inequalities. Using more sophisticated nonlinear minimization algorithms, we develop and investigate more e#cient methods, which lead to two kinds of weighted centers for the feasible set. Using these centers, we develop new algorithms for solving linear programming problems.<F3.977e+05> Key words.<F3.866e+05> weighted center, the ellipsoid method, Newton's method, linear programming<F3.977e+05> AMS subject classifications.<F3.866e+05> 65K, 90C<F5.13e+05> 1. Introduct...

In this paper, we consider the problem of computing a maximum inscribed sphere inside a high dimensional polytope formed by a set of halfspaces (or linear constraints), and present an efficient algorithm for computing a (1 − ǫ)-approximation of the sphere. More specifically, given any bounded polytope P defined by n d-dimensional halfspaces, an interior point O of P, and a constant ǫ> 0, our algorithm computes in O(nd/ǫ 3) time a sphere inside P with a radius no less than (1 − ǫ)Ropt, where Ropt is the radius of a maximum inscribed sphere of P. Our algorithm is based on the core-set concept and a number of interesting geometric observations. Our result settles an open problem posted by Khachiyan and Todd [4] for the case of spheres. 1