A Note on Convergence Analysis of an SQP-Type Method for Nonlinear Semidefinite Programming

A sequentially semidefinite programming method is proposed for solving nonlinear semidefinite programming problem (NLSDP). Inspired by the sequentially quadratic programming (SQP) method, the algorithm generates a search direction by solving a quadratic semidefinite programming subproblem at each iteration. The [Formula: see text] exact penalty function and a line search strategy are used to determine whether the trial step can be accepted or not. Under mild assumptions, the proposed algorithm is globally convergent. In order to avoid the Maratos effect, we present a modified SQP-type algorithm with the second-order correction step and prove that the fast local superlinear convergence can be obtained under the strict complementarity and the second-order sufficient condition with the sigma term. Finally, some numerical experiments are given to show the effectiveness of the algorithm.

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Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽

10.3390/math7090804 ◽

2019 ◽

Vol 7 (9) ◽

pp. 804

Author(s):

Ioannis K. Argyros ◽

Neha Gupta ◽

J. P. Jaiswal

Keyword(s):

Approximate Solution ◽

Convergence Analysis ◽

Banach Spaces ◽

Recurrence Relation ◽

Convergence Theorem ◽

Local Convergence ◽

Existence And Uniqueness ◽

Numerical Illustration ◽

Type Method ◽

Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

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A Penalty-Free Method with Trust Region for Nonlinear Semidefinite Programming

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595915400060 ◽

2015 ◽

Vol 32 (01) ◽

pp. 1540006 ◽

Cited By ~ 2

Author(s):

Zhongwen Chen ◽

Shicai Miao

Keyword(s):

Semidefinite Programming ◽

Objective Function ◽

Penalty Function ◽

Trust Region ◽

Numerical Results ◽

New Method ◽

Penalty Parameter ◽

Nonlinear Semidefinite Programming ◽

Globally Convergent ◽

Nonlinear Objective Function

In this paper, we propose a class of new penalty-free method, which does not use any penalty function or a filter, to solve nonlinear semidefinite programming (NSDP). So the choice of the penalty parameter and the storage of filter set are avoided. The new method adopts trust region framework to compute a trial step. The trial step is then either accepted or rejected based on the some acceptable criteria which depends on reductions attained in the nonlinear objective function and in the measure of constraint infeasibility. Under the suitable assumptions, we prove that the algorithm is well defined and globally convergent. Finally, the preliminary numerical results are reported.

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