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DETERMINANT MAXIMIZATION WITH LINEAR MATRIX INEQUALITY CONSTRAINTS
"... The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the s ..."
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Cited by 200 (18 self)
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The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the semidefinite programming problem. We give an overview of the applications of the determinant maximization problem, pointing out simple cases where specialized algorithms or analytical solutions are known. We then describe an interiorpoint method, with a simplified analysis of the worstcase complexity and numerical results that indicate that the method is very efficient, both in theory and in practice. Compared to existing specialized algorithms (where they are available), the interiorpoint method will generally be slower; the advantage is that it handles a much wider variety of problems.
Convex Nondifferentiable Optimization: A Survey Focussed On The Analytic Center Cutting Plane Method.
, 1999
"... We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct pr ..."
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Cited by 69 (2 self)
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We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant 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.
On Numerical Solution of the Maximum Volume Ellipsoid Problem
 SIAM JOURNAL ON OPTIMIZATION
, 2001
"... In this paper we study practical solution methods for finding the maximumvolume ellipsoid inscribing a given fulldimensional polytope in ! n defined by a finite set of linear inequalities. Our goal is to design a generalpurpose algorithmic framework that is reliable and efficient in practice. To ..."
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Cited by 32 (1 self)
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In this paper we study practical solution methods for finding the maximumvolume ellipsoid inscribing a given fulldimensional polytope in ! n defined by a finite set of linear inequalities. Our goal is to design a generalpurpose 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 primaldual type, interiorpoint 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.
MinimumVolume Enclosing Ellipsoids and Core Sets
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 2005
"... 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 firstorder algorithm. Our analysis leads to a slightly imp ..."
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Cited by 31 (4 self)
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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 firstorder 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.
Computation of Minimum Volume Covering Ellipsoids
 Operations Research
, 2003
"... We present a practical algorithm for computing the minimum volume ndimensional 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 structur ..."
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Cited by 29 (0 self)
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We present a practical algorithm for computing the minimum volume ndimensional 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 interiorpoint methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interiorpoint and activeset 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.
Playing Billiard in Version Space
, 1997
"... A raytracing 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 know ..."
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Cited by 25 (0 self)
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A raytracing 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 reallife 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,...
An InteriorPoint Algorithm for the MaximumVolume Ellipsoid Problem
 RICE UNIVERSITY
, 1999
"... In this report, we consider the problem of finding the maximumvolume ellipsoid inscribing a given fulldimensional polytope in R^n defined by a finite set of affine inequalities. We present several formulations for the problem that may serve as algorithmic frameworks for applying interiorpoint met ..."
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Cited by 11 (1 self)
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In this report, we consider the problem of finding the maximumvolume ellipsoid inscribing a given fulldimensional polytope in R^n defined by a finite set of affine inequalities. We present several formulations for the problem that may serve as algorithmic frameworks for applying interiorpoint methods. We propose a practical interiorpoint algorithm based on one of the formulations and present preliminary numerical results.
Approximate Minimum Volume Enclosing Ellipsoids Using Core Sets
, 2003
"... We study the problem of computing the minimum volume enclosing ellipsoid containing a given point set S = {p1, p2,..., pn} ⊆ R d. Using “core sets ” and a column generation approach, we develop a (1 + ɛ)approximation algorithm. We prove the existence of a core set X ⊆ S of size at most X  = α = ..."
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Cited by 7 (0 self)
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We study the problem of computing the minimum volume enclosing ellipsoid containing a given point set S = {p1, p2,..., pn} ⊆ R d. Using “core sets ” and a column generation approach, we develop a (1 + ɛ)approximation algorithm. We prove the existence of a core set X ⊆ S of size at most X  = α = O ( d ( log d + 1. We describe an algorithm that computes the set X and a (1 + ɛ)approximation to ɛ) operations by using Khachiyan’s the minimum volume enclosing ellipsoid of S in O(nd 2 α + α 4.5 log α ɛ algorithm to solve each subproblem. This result is an improvement over the previously known algorithms especially for input sets with n ≫ d and reasonably small values of ɛ. We also discuss extensions to the cases in which the input set consists of balls or ellipsoids.
Computing large convex regions of obstaclefree space through semidefinite programming
 in Workshop on the Algorithmic Foundations of Robotics (WAFR
, 2014
"... Abstract. This paper presents iris (Iterative Regional Inflation by Semidefinite programming), a new method for quickly computing large polytopic and ellipsoidal regions of obstaclefree space through a series of convex optimizations. These regions can be used, for example, to efficiently optimiz ..."
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Cited by 6 (6 self)
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Abstract. This paper presents iris (Iterative Regional Inflation by Semidefinite programming), a new method for quickly computing large polytopic and ellipsoidal regions of obstaclefree space through a series of convex optimizations. These regions can be used, for example, to efficiently optimize an objective over collisionfree positions in space for a robot manipulator. The algorithm alternates between two convex optimizations: (1) a quadratic program that generates a set of hyperplanes to separate a convex region of space from the set of obstacles and (2) a semidefinite program that finds a maximumvolume ellipsoid inside the polytope intersection of the obstaclefree halfspaces defined by those hyperplanes. Both the hyperplanes and the ellipsoid are refined over several iterations to monotonically increase the volume of the inscribed ellipsoid, resulting in a large polytope and ellipsoid of obstaclefree space. Practical applications of the algorithm are presented in 2D and 3D, and extensions to Ndimensional configuration spaces are discussed. Experiments demonstrate that the algorithm has a computation time which is linear in the number of obstacles, and our matlab [18] implementation converges in seconds for environments with millions of obstacles. 1