A Boundary Point Method to Solve Semidefinite Programs

Povh, Janez; Rendl, Franz; Wiegele, Angelika

doi:10.1007/s00607-006-0182-2

Cited by 94 publications

(110 citation statements)

References 8 publications

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“…The analysis of Sections 3 and 4 holds for any L satisfying (18), and hence for the Lipschitz constant L F ′ of F ′ , which by the definition is the smallest L satisfying (18). Clearly, the best iterationcomplexity bound, and most likely computational performance of the NPE, is obtained with L = L F ′ .…”

Section: Final Remarkmentioning

confidence: 97%

“…In addition, the use of the notion of approximate solution as in Definition 2.3 have allowed us to obtain new block-decomposition methods (see [13]). In particular, one such block-decomposition method for solving conic semidefinite programming has been shown in [10] to outperform the two most competitive codes for large-scale conic semidefinite programs, namely: the boundary point method introduced by Povh et al [18] and the Newton-CG augmented Lagrangian method by Zhao et al [27]. Moreover, the use of the tolerances ε k in the definition of approximate solutions of (6) has played an important role in the analysis of Korpelevich's method given in [11,14] as well as the forward-backward method for convex optimization presented in [12].…”

Section: Endmentioning

confidence: 99%

See 1 more Smart Citation

Iteration-Complexity of a Newton Proximal Extragradient Method for Monotone Variational Inequalities and Inclusion Problems

Monteiro¹,

Svaiter²

2012

SIAM J. Optim.

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In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient (HPE) framework has been used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth variational inequalities and inclusion problems consisting of the sum of a smooth monotone map and a maximal monotone point-to-set operator. Each iteration of the method computes an approximate solution of a proximal subproblem, obtained by linearizing the smooth part of the operator in the corresponding proximal equation for the original problem, which is then used to perform an extragradient step as prescribed by the HPE framework. Both pointwise and ergodic iterationcomplexity results are derived for the aforementioned first-order method using corresponding results obtained here for a subfamily of the HPE framework.

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Section: Final Remarkmentioning

confidence: 97%

Section: Endmentioning

confidence: 99%

Iteration-Complexity of a Newton Proximal Extragradient Method for Monotone Variational Inequalities and Inclusion Problems

Monteiro¹,

Svaiter²

2012

SIAM J. Optim.

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“…There is a long list of quite different algorithmic approaches for solving the SDP problem, see for instance [6,18,101,103,66,72,99,59,75,95,96,31,30,81,55,117,147,116], among many others. The previous list is by far incomplete and we do not intend to describe here all the diverse ideas to deal with the efficient solution of semidefinite programming.…”

Section: Theorem 23 If the Original Problem Optimal Value Dmentioning

confidence: 99%

Linear and Nonlinear Semidefinite Programming

Bofill

Gómez

2014

Pesqui. Oper.

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ABSTRACT. This paper provides a short introduction to optimization problems with semidefinite constraints. Basic duality and optimality conditions are presented. For linear semidefinite programming some advances by dealing with degeneracy and the semidefinite facial reduction are discussed. Two relatively recent areas of application are presented. Finally a short overview of relevant literature on algorithmic approaches for efficiently solving linear and nonlinear semidefinite programming is provided.

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“…The concept of Algorithm 2 is in some sense 'complementary' to the boundary point method of [8]. The latter algorithm generates iterates within the primal-dual cone approaching the set of linear constraints, while the iterates in Algorithm 2 always satisfy the linear constraints and approach the primal-dual cone.…”

Section: Remarkmentioning

confidence: 99%

“…Toh uses an iterative solver for the augmented KKT system, and Kocvara and Stingl apply an iterative solver to a modified barrier problem. The approach presented in the current paper is closely related to the 'boundary point method' from [8] and the regularization approaches in [6].…”

Section: Introductionmentioning

confidence: 99%

An Augmented Primal-Dual Method for Linear Conic Programs

Jarre¹,

Rendl²

2008

SIAM J. Optim.

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Abstract. We propose a new iterative approach for solving linear programs over convex cones. Assuming that Slaters condition is satisfied, the conic problem is transformed to the minimization of a convex differentiable function. This "agumented primal-dual function" or "apd-function" is restricted to an affine set in the primal-dual space. The evaluation of the function and its derivative is cheap if the projection of a given point onto the cone can be computed cheaply, and if the projection of a given point onto the affine subspace defining the primal problem can be computed cheaply. For the special case of a semidefinite program, a certain regularization of the apd-function is analyzed. Numerical examples minimizing the apd-function with a conjugate gradient method illustrate the potential of the approach.

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A Boundary Point Method to Solve Semidefinite Programs

Cited by 94 publications

References 8 publications

Iteration-Complexity of a Newton Proximal Extragradient Method for Monotone Variational Inequalities and Inclusion Problems

Iteration-Complexity of a Newton Proximal Extragradient Method for Monotone Variational Inequalities and Inclusion Problems

Linear and Nonlinear Semidefinite Programming

An Augmented Primal-Dual Method for Linear Conic Programs

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