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15
SecondOrder Cone Programming
 MATHEMATICAL PROGRAMMING
, 2001
"... In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic struc ..."
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Cited by 231 (11 self)
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In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic structure that is connected to SOCP. This algebra is a special case of a Euclidean Jordan algebra. After presenting duality theory, complementary slackness conditions, and definitions and algebraic characterizations of primal and dual nondegeneracy and strict complementarity we review the logarithmic barrier function for the SOCP problem and survey the pathfollowing interior point algorithms for it. Next we examine numerically stable methods for solving the interior point methods and study ways that sparsity in the input data can be exploited. Finally we give some current and future research direction in SOCP.
CSDP, a C library for semidefinite programming.
, 1997
"... this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. ..."
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Cited by 209 (1 self)
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this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. Finally, we present results from the solution of a number of test problems. 2 The SDP Problem We consider semidefinite programming problems of the form max tr (CX)
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 89 (3 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
Semidefinite programming for discrete optimization and matrix completion problems
 Discrete Appl. Math
, 2002
"... Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y ..."
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Cited by 14 (5 self)
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Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y
PrimalDual Interior Point Methods For Semidefinite Programming In Finite Precision
 SIAM J. Optimization
, 1997
"... . Recently, a number of primaldual interiorpoint methods for semidefinite programming have been developed. To reduce the number of floating point operations, each iteration of these methods typically performs block Gaussian elimination with block pivots that are close to singular near the optimal ..."
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Cited by 8 (0 self)
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. Recently, a number of primaldual interiorpoint methods for semidefinite programming have been developed. To reduce the number of floating point operations, each iteration of these methods typically performs block Gaussian elimination with block pivots that are close to singular near the optimal solution. As a result, these methods often exhibit complex numerical properties in practice. We consider numerical issues related to some of these methods. Our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are wellconditioned, but are unstable otherwise. With this result, we explain why one particular method, the one introduced by Alizadeh, Haeberly and Overton is in general more stable than others. We also explain why the so called least squares variation, introduced for some of these methods, does not yield more numerical accuracy in general. Finally, we present results from our numerical experiments ...
Algorithms and Software for LMI Problems in Control
 IEEE Control Systems Magazine
, 1997
"... this article is to provide an overview of the state of the art of ..."
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Cited by 7 (0 self)
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this article is to provide an overview of the state of the art of
Mixed semidefinitequadraticlinear programs
 in Recent Advances in LMI Methods for Control
, 2000
"... ..."
Remarks On Nondegeneracy In Mixed SemidefiniteQuadratic Programming
, 1998
"... We consider the definitions of nondegeneracy and strict complementarity given in [5] for semidefinite programming (SDP) and their obvious extensions to mixed semidefinitequadratic programming (SDQP). We show that a solution to SDQP satisfies strict complementarity and primal and dual nondegeneracy ..."
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Cited by 2 (0 self)
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We consider the definitions of nondegeneracy and strict complementarity given in [5] for semidefinite programming (SDP) and their obvious extensions to mixed semidefinitequadratic programming (SDQP). We show that a solution to SDQP satisfies strict complementarity and primal and dual nondegeneracy if and only if the Jacobian of the Newton system determined by the optimality conditions is nonsingular at the solution.