Search directions in the SDP and the monotone SDLCP: generalization and inexact computation

Kojima, M.; Shida, Masayuki; Shindoh, Susumu

doi:10.1007/s101070050046

Cited by 27 publications

(18 citation statements)

References 16 publications

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“…The use of inexact search directions in interior-point methods has been extensively studied in the context of cone programming problems (see e.g., [1,2,7,11,12,15,18,29]). Moreover, the use of iterative linear solvers to compute the primal-dual Newton search directions of interior-point path-following algorithms has also been extensively investigated in [1,3,4,7,12,18,19,20,22,24].…”

mentioning

confidence: 99%

An Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex Quadratic Programming

Lu¹,

Monteiro²,

O'Neal³

2006

SIAM J. Optim.

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Abstract. In this paper we develop a long-step primal-dual infeasible path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. We propose a new linear system, which we refer to as the augmented normal equation (ANE), to determine the primal-dual search directions. Since the condition number of the ANE coefficient matrix may become large for degenerate CQP problems, we use a maximum weight basis preconditioner introduced in [A. R. L. Oliveira and D. C. Sorensen, Linear Algebra Appl., 394 (2005) (2004), pp. 96-100], we establish a uniform bound, depending only on the CQP data, for the number of iterations needed by the iterative linear solver to obtain a sufficiently accurate solution to the ANE. Since the iterative linear solver can generate only an approximate solution to the ANE, this solution does not yield a primal-dual search direction satisfying all equations of the primal-dual Newton system. We propose a way to compute an inexact primal-dual search direction so that the equation corresponding to the primal residual is satisfied exactly, while the one corresponding to the dual residual contains a manageable error which allows us to establish a polynomial bound on the number of iterations of our method.

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confidence: 99%

An Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex Quadratic Programming

Lu¹,

Monteiro²,

O'Neal³

2006

SIAM J. Optim.

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“…The last was introduced by Nesterov and Todd [46,47], from yet another motivation, and shown to be derivable in this form by Todd, Toh, and Tütüncü [60]. These and several other search directions are discussed in Kojima et al [32] and Todd [59].…”

Section: Path-following Methodsmentioning

confidence: 99%

Semidefinite Optimization

2014

Computer Vision

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Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps "or" would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems. *

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“…c Can we develop a general theory based on optimization methods to address moment problems in probability theory? 469 …”

Section: Introductionmentioning

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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Search directions in the SDP and the monotone SDLCP: generalization and inexact computation

Cited by 27 publications

References 16 publications

An Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex Quadratic Programming

An Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex Quadratic Programming

Semidefinite Optimization

Handbook of Semidefinite Programming

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