On the Nesterov--Todd Direction in Semidefinite Programming

Todd, Michael J.; Toh, Kim-Chuan; Tütüncü, Reha

doi:10.1137/s105262349630060x

Cited by 216 publications

(203 citation statements)

References 14 publications

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“…The method used here differs in that we should guarantee the feasibility at each iteration. In addition, the Nesterov-Todd symmetrization scheme [48] is employed.…”

Section: Numerical Experimentsmentioning

confidence: 99%

An inexact spectral bundle method for convex quadratic semidefinite programming

Lin

2011

Comput Optim Appl

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Abstract. We present an inexact spectral bundle method for solving convex quadratic semidefinite optimization problems. This method is a first-order method, hence requires much less computational cost each iteration than second-order approaches such as interior-point methods. In each iteration of our method, we solve an eigenvalue minimization problem inexactly, and solve a small convex quadratic semidefinite programming as a subproblem. We give a proof of the global convergence of this method using techniques from the analysis of the standard bundle method, and provide a global error bound under a Slater type condition for the problem in question. Numerical experiments with matrices of order up to 3000 are performed and the computational results establish the effectiveness of this method.

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“…The method used here differs in that we should guarantee the feasibility at each iteration. In addition, the Nesterov-Todd symmetrization scheme [48] is employed.…”

Section: Numerical Experimentsmentioning

confidence: 99%

An inexact spectral bundle method for convex quadratic semidefinite programming

Lin

2011

Comput Optim Appl

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“…As in [30], we only consider P such that P XY P −1 is symmetric. We also assume P is an analytic function of X, Y 0.…”

Section: Definition and Basic Properties Of Off-central Pathsmentioning

confidence: 99%

“…Similar to [30], it can be shown that the matrix in (12) is invertible for all X, Y 0 under Assumption 2.2(b).…”

Section: Qedmentioning

confidence: 99%

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Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

Sim¹,

Zhao

2006

Math. Program.

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An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameter µ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of √ µ and have first derivatives which are unbounded as a function of µ at µ = 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at µ = 0. These "nice" paths are characterized by some algebraic equations.

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“…Also, it is shown in Kojima, Shida and Shindoh 466 that the NT direction is a member of the KSH family the corresponding H is not so simple to describe. On the other hand, the AHO direction is not in the KSH family since, as opposed to the directions of the KSH family, the AHO direction does not necessarily exist at every point X;y;S 2 P + IR m P + see the discussion before Theorem 3.2 of Todd et al 831 .…”

Section: Endmentioning

confidence: 99%

Handbook of Semidefinite Programming

Wolkowicz¹,

Saigal²,

Vandenberghe³

2000

International Series in Operations Research &Amp; Management Science

454

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PrefaceSemide nite 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 w as spurred by the discovery of important applications in combinatorial optimization and control theory, the development of e cient i n terior-point 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 semide nite programming. Written by the leading experts on the subject, it o ers an advanced and broad overview of the current state of the eld. 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 nish the book in a timely fashion, we h a ve had to abandon hopes for separate chapters on some important topics such as a discussion of SDP algorithms in the framework of general convex programming using the theory of self-concordant barriers, or an overview of numerical implementation aspects of interior-point methods for SDP. We apologize for any important gaps in the contents, and hope that the historical notes at the end of the book provide a useful guide to the literature on the topics that are not adequately covered in this handbook.We w ould like to thank all the authors for their outstanding contributions, their editorial help, and for their patience during the many revisions of the handbook. In addition, we thank Mike T odd for his valuable editorial advice on many occasions. We w ould also like to thank Erna Unrau for her help in combining some of the bibliographies into one. 11.5 -function, n K = 8 , n A = 10, = 1 0 ,5 , time limit one hour. 334 11.6 PLDI, n 500 where the variable is X 2 S n , the space of real symmetric n n matrices. The vector b 2 IR m , and the matrices A i 2 S n and C 2 S n are given problem parameters. The inequality X 0 means X is positive semide nite. More generally, throughout this book, the notation A B will mean B,A is positive semide nite. The notation C X stands for the inner product of the symmetric matrices C and X:Some authors write the inner product as C X = T r CXwhere Tr CXis the trace of the matrix CX. In other words, C X is a linear function of the elements X ij . In 1.1.1 we minimize a linear function of the matrix variable X, subject to m linear equality constraints A i X = b i , and the positive semide nitess constraint X 0. We refer to problem 1.1.1 as a semide nite program or SDP. W e can think of an SDP as a generalization of the standard form linear programming problem minimize c T x subject to a T i x = b i for all i = 1 ; : : : ; m x 0 1.1.2 1 2 HANDBOOK OF SEMIDEFINITE PROGRAMMING in which the elementwise nonnegativity constraint x 0 is replaced by a generalized inequality with re...

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On the Nesterov--Todd Direction in Semidefinite Programming

Cited by 216 publications

References 14 publications

An inexact spectral bundle method for convex quadratic semidefinite programming

An inexact spectral bundle method for convex quadratic semidefinite programming

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

Handbook of Semidefinite Programming

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