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25
Determinant maximization with linear matrix inequality constraints
 SIAM Journal on Matrix Analysis and Applications
, 1998
"... constraints ..."
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 51 (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 25 (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 improved ..."
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Cited by 25 (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.
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 21 (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,...
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 21 (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.
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 10 (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.
A Predictive Controller with Artificial Lyapunov Function for Linear Systems with Input/State Constraints
, 1998
"... This paper copes with the problem of satisfying input and/or state hard constraints in setpoint tracking problems. Stability is guaranteed by synthesizing a Lyapunov quadratic function for the system, and by imposing that the terminal state lies within a level set of the function. Procedures to max ..."
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Cited by 3 (0 self)
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This paper copes with the problem of satisfying input and/or state hard constraints in setpoint tracking problems. Stability is guaranteed by synthesizing a Lyapunov quadratic function for the system, and by imposing that the terminal state lies within a level set of the function. Procedures to maximize the volume of such an ellipsoidal set are provided, and interiorpoint methods to solve online optimization are considered. Key words: Predictive control, Constraints, Lyapunov function, Setpoint control, Optimization problems, Interiorpoint methods, Quadratically constrained quadratic programming. 1 Introduction The necessity of satisfying input/state constraints is a feature that frequently arises in control applications. Constraints are dictated for instance by physical limitations of the actuators or by the necessity to keep some plant variables within safe limits. In recent years, several control techniques have been developed which are able to handle hard constraints, see e....
Solving LP Problems Via Weighted Centers
 J. Global Opt
, 1996
"... . 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 t ..."
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Cited by 2 (2 self)
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.<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...