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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 65 (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
Some Geometric Results in Semidefinite Programming
 Jnl. Glob. Opt
, 1995
"... The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. More specifically, given a point x in a spectrahedron, we derive an expression fo ..."
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Cited by 24 (2 self)
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The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. More specifically, given a point x in a spectrahedron, we derive an expression for the minimal face containing x. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar. 1 Support received under the grants: NSFSTC9119999 (DIMACS) and Air Force grant F496209310041 (RUTCOR). 2 Support from the NSF grant ECS9111548 is acknowledged. 1 Introduction and Motivation Let S n ; P n denote respectively, the space of nbyn real symmetric matrices and the cone of n \Theta n positive semidefinite (PSD) matrices. We let denote the Loewner partial order induced by P n on S n , i.e. A B if A ...
Semidefinite Programming and Combinatorial Optimization
 Appl. Numer. Math
, 1998
"... Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete opti ..."
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Cited by 13 (4 self)
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Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete optimization problems. The duality theory for semidefinite programs is the key to understand algorithms to solve them. The relevant duality results are therefore summarized. The second part of the paper deals with the approximation of integer problems both in a theoretical setting, and from a computational point of view. 1 Introduction The interest in Semidefinite Programming (SDP) has been growing rapidly in the last few years. Here are some possible explanations for this sudden rise of interest. The algorithmic development of interiorpoint methods for Linear Programs indicated the potential of this approach to solve general convex problems. Semidefinite Programs are a natural generaliza...
On The PrimalDual Geometry Of Level Sets in Linear and Conic Optimization
, 2001
"... For a conic optimization problem P: minimize x c ..."
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Presolving for Semidefinite Programs Without Constraint Qualifications
, 1998
"... Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack ..."
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Cited by 4 (0 self)
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Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack of a constraint qualification which can result in both theoretical and numerical difficulties. A nonzero duality gap can exist which nullifies the elegant and powerful duality theory. Small perturbations can result in infeasibility and/or large perturbations in solutions. Such problems fall into the class of illposed problems. It is interesting to note that classes of problems where constraint qualifications fail arise from semidefinite programming relaxations of hard combinatorial problems. We look at several such classes and present two approaches to find regularized solutions. Some preliminary numerical results are included.
SEMIDEFINITE AND LAGRANGIAN RELAXATIONS FOR HARD COMBINATORIAL PROBLEMS
"... Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this ..."
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Cited by 2 (2 self)
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Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this
Preprocessing and Regularization for Degenerate . . .
, 2013
"... This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming r ..."
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This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primaldual interiorpoint, pd ip methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, wellposedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the BorweinWolkowicz facial reduction process to find a finite number, k, of rankrevealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, wellposed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The
On The PrimalDual Geometry Of Level Sets In Linear
"... For a conic optimization problem P : minimizex c we present a geometric relationship between the primal objective function level sets and the dual objective function level sets, which shows that the maximum norms of the primal objective function level sets are nearly inversely propor ..."
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For a conic optimization problem P : minimizex c we present a geometric relationship between the primal objective function level sets and the dual objective function level sets, which shows that the maximum norms of the primal objective function level sets are nearly inversely proportional to the maximum inscribed radii of the dual objective function level sets.
Preprocessing and Regularization for . . .
, 2012
"... This paper presents a backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming ..."
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This paper presents a backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primaldual interiorpoint, pd ip methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, wellposedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the BorweinWolkowicz facial reduction process to find a finite number, k, of rankrevealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, wellposed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The