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43
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 201 (18 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
A Spectral Bundle Method for Semidefinite Programming
 SIAM JOURNAL ON OPTIMIZATION
, 1997
"... A central drawback of primaldual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applica ..."
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Cited by 138 (6 self)
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A central drawback of primaldual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order. We present a method that allows to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored for eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completene...
PrimalDual PathFollowing Algorithms for Semidefinite Programming
 SIAM Journal on Optimization
, 1996
"... This paper deals with a class of primaldual interiorpoint algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara [11]. These authors proposed a family of primaldual search directions that generalizes the one used in algorithms for linear programmin ..."
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Cited by 135 (9 self)
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This paper deals with a class of primaldual interiorpoint algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara [11]. These authors proposed a family of primaldual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X 1=2 S \Gamma1=2 . They study three primaldual algorithms based on this family of search directions: a shortstep pathfollowing method, a feasible potentialreduction method and an infeasible potentialreduction method. However, they were not able to provide an algorithm which generalizes the longstep pathfollowing algorithm introduced by Kojima, Mizuno and Yoshise [10]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present: 1) a simplified polynomial convergence proof for one of their shortstep pathfollowing ...
The Complex Structures Singular Value
, 1993
"... A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis on the mathematical aspects of µ. The µbased methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests ..."
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Cited by 120 (10 self)
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A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis on the mathematical aspects of µ. The µbased methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests
A PrimalDual Potential Reduction Method for Problems Involving Matrix Inequalities
 in Protocol Testing and Its Complexity", Information Processing Letters Vol.40
, 1995
"... We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worstcase analysis shows that the number of iterations ..."
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Cited by 84 (21 self)
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We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worstcase analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interiorpoint methods the overall computational effort is therefore dominated by the leastsquares system that must be solved in each iteration. A type of conjugategradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugategradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration.
Method of centers for minimizing generalized eigenvalues
 Linear Algebra Appl
, 1993
"... We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fr ..."
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Cited by 60 (12 self)
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We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a selfconcordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 60 (2 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
On Extending Some PrimalDual InteriorPoint Algorithms From Linear Programming to Semidefinite Programming
 SIAM Journal on Optimization
, 1998
"... This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a search direction originally proposed by HelmbergRendlVanderbeiWolkowicz [5] and KojimaShindohHara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these meth ..."
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Cited by 55 (1 self)
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This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a search direction originally proposed by HelmbergRendlVanderbeiWolkowicz [5] and KojimaShindohHara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [11] and also in [15] through different means and in different forms. In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variablemetric measures of centrality. These results provide convenient tools for deriving polynomiality results for primaldual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples...
On Extending PrimalDual InteriorPoint Algorithms from Linear Programming to Semidefinite Programming
, 1995
"... This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a linearized complementarity equation originally proposed by Kojima, Shindoh and Hara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of ba ..."
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Cited by 47 (0 self)
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This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a linearized complementarity equation originally proposed by Kojima, Shindoh and Hara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [11] and also in [15] through different means and in different forms. In this paper, we give a very short derivation of the key equalities and inequalities along the exact line used in linear programming (LP), producing basic relationships that have highly compact forms almost identical to their counterparts in LP. We also introduce a new definition of the central path and variablemetric measures of centrality. These results provide convenient tools for extending existing polynomiality results for many, if not most, algorithms from LP to SDP with little complication. We present examples of such extensions, including the longstep infeasible...