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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 169 (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 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.
ConditionBased Complexity Of Convex Optimization In Conic Linear Form Via The Ellipsoid Algorithm
, 1998
"... A convex optimization problem in conic linear form is an optimization problem of the form CP (d) : maximize c T ..."
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Cited by 38 (17 self)
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A convex optimization problem in conic linear form is an optimization problem of the form CP (d) : maximize c T
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.
Building and Solving Largescale Stochastic Programs on an Affordable Distributed Computing System
, 1999
"... We present an integrated procedure to build and solve big stochastic programming models. The individual components of the system the modeling language, the solver and the hardware are easily accessible, or a least affordable to a large audience. The procedure is applied to a simple financial mod ..."
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Cited by 10 (1 self)
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We present an integrated procedure to build and solve big stochastic programming models. The individual components of the system the modeling language, the solver and the hardware are easily accessible, or a least affordable to a large audience. The procedure is applied to a simple financial model, which can be expanded to arbitrarily large sizes by enlarging the number of scenarios. We generated a model with one million scenarios, whose deterministic equivalent linear program has 1,111,112 constraints and 2,555,556 variables. We have been able to solve it on the cluster of ten PCs in less than 3 hours. Key words. Algebraic modeling language, decomposition methods, distributed systems, largescale optimization, stochastic programming. 1 Introduction Practical implementations of stochastic programming involve two big challenges. First, we have to build the model: its size is almost invariably large, if not huge, and this task is in itself a challenge for This research was suppor...
Improved Complexity for Maximum Volume Inscribed Ellipsoids
 SIAM Journal on Optimization
, 2001
"... 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 nu ..."
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Cited by 10 (0 self)
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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
ACCURACY CERTIFICATES FOR COMPUTATIONAL PROBLEMS WITH CONVEX STRUCTURE
, 2007
"... The goal of the current paper is to introduce the notion of certificates which verify the accuracy of solutions of computational problems with convex structure; such problems include minimizing convex functions, variational inequalities with monotone operators, computing saddle points of convexconc ..."
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Cited by 3 (3 self)
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The goal of the current paper is to introduce the notion of certificates which verify the accuracy of solutions of computational problems with convex structure; such problems include minimizing convex functions, variational inequalities with monotone operators, computing saddle points of convexconcave functions and solving convex Nash equilibrium problems. We demonstrate how the implementation of the Ellipsoid method and other cutting plane algorithms can be augmented with the computation of such certificates without essential increase of the computational effort. Further, we show that (computable) certificates exist whenever an algorithm is capable to produce solutions of guaranteed accuracy.
The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases
, 2007
"... ..."
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...
Using the Primal Dual Infeasible Newton Method in the Analytic Center Method for Problems Defined by Deep Cutting Planes.
, 1998
"... The convergence and the complexity of a primaldual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a "deep cut" separation oracle is studied. The primaldual infeasible Newton method is used to generate a primaldual ..."
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Cited by 1 (0 self)
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The convergence and the complexity of a primaldual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a "deep cut" separation oracle is studied. The primaldual infeasible Newton method is used to generate a primaldual updating direction. The number of recentering steps is O(1) for cuts as deep as half way to the deepest cut, where the deepest cut is a cut that is tangent to the primaldual variant of Dikin's ellipsoid. Keywords: Convex feasibility problem, analytic center, column generation, cutting planes, deep cut. AMS subject classification: 90C25, 90C26, 90C60. 1 This research is supported by the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152, by the FCAR of Quebec and by an Obermann fellowship at the Center for Advanced Studies at the University of Iowa. 1 Introduction The convex feasibility problem defined by a separation oracle is: find an interior poin...