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19
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 247 (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.
Solving semidefinitequadraticlinear programs using SDPT3
 MATHEMATICAL PROGRAMMING
, 2003
"... This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primaldual pathfollowing algorithm ..."
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Cited by 243 (19 self)
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This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primaldual pathfollowing algorithms. The software developed by the authors uses Mehrotratype predictorcorrector variants of interiorpoint methods and two types of search directions: the HKM and NT directions. A discussion of implementation details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.
On implementing a primaldual interiorpoint method for conic quadratic optimization
 MATHEMATICAL PROGRAMMING SER. B
, 2000
"... Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linea ..."
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Cited by 75 (6 self)
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Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic quadratic optimization problems can in theory be solved efficiently using interiorpoint methods. In particular it has been shown by Nesterov and Todd that primaldual interiorpoint methods developed for linear optimization can be generalized to the conic quadratic case while maintaining their efficiency. Therefore, based on the work of Nesterov and Todd, we discuss an implementation of a primaldual interiorpoint method for solution of largescale sparse conic quadratic optimization problems. The main features of the implementation are it is based on a homogeneous and selfdual model, handles the rotated quadratic cone directly, employs a Mehrotra type predictorcorrector
A Convergence Analysis of the Scalinginvariant Primaldual Pathfollowing Algorithms for Secondorder Cone Programming
 Optim. Methods Softw
, 1998
"... This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT di ..."
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Cited by 54 (5 self)
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This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT direction. Specifically, we show that the longstep algorithm using the NT direction has O(n log " 01 ) iterationcomplexity to reduce the duality gap by a factor of ", where n is the number of the secondorder cones. The complexity for the same algorithm using the HRVW/KSH/M direction is improved to O(n 3=2 log " 01 ) from O(n 3 log " 01 ) of the previous analysis. We also show that the short and semilongstep algorithms using the NT direction (and the HRVW/KSH/M direction) have O( p n log " 01 ) and O(n log " 01 ) iterationcomplexities, respectively. keywords: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Introduction...
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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On the implementation and usage of SDPT3  a Matlab software package for semidefinitequadraticlinear programming, version 4.0
, 2006
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Optimization with Semidefinite, Quadratic and Linear Constraints
 RUTCOR, RUTGERS UNIVERSITY
, 1997
"... We consider optimization problems where variables have either linear, or convex quadratic or semidefinite constraints. First, we define and characterize primal and dual nondegeneracy and strict complementarity conditions. Next, we develop primaldual interior point methods for such problems and show ..."
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Cited by 19 (3 self)
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We consider optimization problems where variables have either linear, or convex quadratic or semidefinite constraints. First, we define and characterize primal and dual nondegeneracy and strict complementarity conditions. Next, we develop primaldual interior point methods for such problems and show that in the absence of degeneracy these algorithms are numerically stable. Finally we describe an implementation of our method and present numerical experiments with both degenerate and nondegenerate problems.
A New Secondorder Cone Programming Relaxation for MAXCUT Problems
 Journal of Operations Research of Japan
, 2001
"... We propose a new relaxation scheme for the MAXCUT problem using secondorder cone programming. We construct relaxation problems to reflect the structure of the original graph. Numerical experiments show that our relaxation gives better bounds than those based on the spectral decomposition proposed ..."
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Cited by 10 (1 self)
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We propose a new relaxation scheme for the MAXCUT problem using secondorder cone programming. We construct relaxation problems to reflect the structure of the original graph. Numerical experiments show that our relaxation gives better bounds than those based on the spectral decomposition proposed by Kim and Kojima [16], and that the efficiency of the branchandbound method using our relaxation is comparable to that using semidefinite relaxation in some cases.
Associative Algebras, Symmetric Cones and Polynomial Time Interior Point Algorithms
, 1998
"... We present a generalization of Monteiro's polynomiality proof of a large class of primaldual interior point algorithms for semidefinite programs. We show that this proof, essentially verbatim, applies to all optimization problems over almost all symmetric cones, that is those cones that can be ..."
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Cited by 10 (0 self)
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We present a generalization of Monteiro's polynomiality proof of a large class of primaldual interior point algorithms for semidefinite programs. We show that this proof, essentially verbatim, applies to all optimization problems over almost all symmetric cones, that is those cones that can be derived from classes of normed associative algebras and certain Jordan algebras obtained from them. As a consequence, we show that Monteiro's proof can be extended to convex quadratically constrained quadratic optimization problems. We also extend the notion of Zhang family of algorithms, and show that it can be applied to all symmetric cones, in particular the quadratic cone.
On Commutative Class of Search Directions for Linear Programming over Symmetric Cones
 Journal of Optimization Theory and Applications
, 2000
"... The Commutative Class of search directions for semidefinite programming is first proposed by Monteiro and Zhang [13]. In this paper, we investigate the corresponding class of search directions for linear programming over symmetric cones, which is a class of convex optimization problems including lin ..."
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Cited by 9 (2 self)
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The Commutative Class of search directions for semidefinite programming is first proposed by Monteiro and Zhang [13]. In this paper, we investigate the corresponding class of search directions for linear programming over symmetric cones, which is a class of convex optimization problems including linear programming, secondorder cone programming, and semidefinite programming as special cases. Complexity results are established for short, semilong, and long step algorithms. We then propose a subclass of Commutative Class of search directions which has polynomial complexity even in semilong and long step settings. The last subclass still contains the NT and HRVW/KSH/M directions. An explicit formula to calculate any member of the class is also given. Key words: Symmetric Cone, Primaldual InteriorPoint Method, Jordan Algebra, Polynomial Complexity A#liation: Department of Computer Science, The University of ElectroCommunications 1 1. Introduction In this paper, we consider linear ...