On the superlinear local convergence of a penalty-free method for nonlinear semidefinite programming

This paper is concerned with a sequentially semidefinite programming (SSDP) algorithm for solving nonlinear semidefinite programming problems (NLSDP), which does not use a penalty function or a filter. This method, inspired by the classic SQP method, calculates a trial step by a quadratic semidefini...

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 308; pp. 1 - 19
Main Authors: Zhao, Qi, Chen, Zhongwen
Format: Journal Article
Language:English
Published: Elsevier B.V 15.12.2016
Subjects:
Algorithms
Convergence
Global convergence
Iterative methods
Local convergence
Mathematical analysis
Mathematical models
Mathematical programming
Nonlinear semidefinite programming
Nonlinearity
Penalty-free method
Second order correction
Strategy
Nonlinear semidefinite programming
Global convergence
65K05
90C22
Second order correction
Local convergence
Penalty-free method
ISSN:0377-0427, 1879-1778
Online Access:Get full text
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Summary:This paper is concerned with a sequentially semidefinite programming (SSDP) algorithm for solving nonlinear semidefinite programming problems (NLSDP), which does not use a penalty function or a filter. This method, inspired by the classic SQP method, calculates a trial step by a quadratic semidefinite programming subproblem at each iteration. The trial step is determined such that either the value of the objective function or the measure of constraint violation is sufficiently reduced. In order to guarantee global convergence, the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit. We prove the global convergence properties of the algorithm under mild assumptions. We also analyze the local behaviour of the proposed method while using a second order correction strategy to avoid Maratos effect. We prove that, under the strict complementarity and the strong second order sufficient conditions with the sigma term, the rate of local convergence is superlinear. Finally, some numerical results with nonlinear semidefinite programming formulation of control design problem with the data contained in COMPleib are given.
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ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2016.05.007