A Global Algorithm for Nonlinear Semidefinite Programming

Corrêa, Rafael; Ramírez, Héctor

doi:10.1137/s1052623402417298

Cited by 87 publications

(58 citation statements)

References 21 publications

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“…In this case, the right-hand sides, "0", of the first three rows in (28) are replaced by weaker, penalized right-hand sides. Moreover, the convergence analysis of the method proposed in [10,12] yields results that are comparable to the ones for standard SQP methods.…”

Section: An Ssp Methods For Nonlinear Semidefinite Programsmentioning

confidence: 75%

“…However, in contrast to this paper, the algorithm [12] is not derived from a comparison with interior-point algorithms, and the proof of convergence does not use any differentiability properties of the optimal solutions. In [10], Correa and Ramírez present a proof of global convergence of a modifi-cation of the method proposed in [12]. The modification employs certain merit functions to control the step lengths of the SQP algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

2006

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We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefinite programming (SSP) method, which is a generalization of the well-known SQP method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.

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Section: An Ssp Methods For Nonlinear Semidefinite Programsmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

2006

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“…Consequently, {Pᾱ(x k )} is monotonically nonincreasing for sufficiently large k. Meanwhile, by (6) and the positive definiteness of M k , we have…”

Section: Theorem 1 Suppose That Assumptions (A1) and (mentioning

confidence: 95%

“…An interior point method has been proposed for NSOCP in [17]. Globally convergent SQP-type method and successive linearization method have been developed for NSDP in [6] and [8], respectively.…”

Section: Introductionmentioning

confidence: 99%

An SQP-type algorithm for nonlinear second-order cone programs

Katō

Fukushima

2006

Optimization Letters

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We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm.

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“…Nowadays, there are several methods for NSDP problems. We mention the extension of a primal predictor-corrector interior point method of Jarre [21], the sequentially linear SDP method of Correa and Ramirez [13], which is a generalization of the method presented by Fares, Noll and Apkarian [14] and the augmented Lagrangian approach of Kočvara and Stingl [33]. Also Kanzow, Nagel, Kato and Fukushima [22] presented a successive linearization method with a trust region-type globalization.…”

Section: Introductionmentioning

confidence: 99%

A feasible direction interior point algorithm for nonlinear semidefinite programming

Aroztegui

Herskovits

Roche

et al. 2014

Struct Multidisc Optim

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We present a new algorithm for nonlinear semidefinite programming. It is based on the iterative solution, in the primal and dual variables, of KarushKuhn-Tucker first order optimality conditions. This method generates a decreasing feasible sequence. At each iteration, two linear systems with the same coefficient matrix are solved and an inexact line search is then performed. A proof of global convergence is given in the convex case. Some numerical tests involving nonlinear programming problems as well linear and nonlinear matrix inequalities are described. We also solve structural topology optimization problems employing a mathematical model based on semidefinite programming. The results suggest efficiency and high robustness of the proposed method.

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A Global Algorithm for Nonlinear Semidefinite Programming

Cited by 87 publications

References 21 publications

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

An SQP-type algorithm for nonlinear second-order cone programs

A feasible direction interior point algorithm for nonlinear semidefinite programming

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