Convergence to a second-order critical point by a primal-dual interior point trust-region method for nonlinear semidefinite programming. (2nd November 2022)

• Record Type: Journal Article
• Authors: Yamashita, Hiroshi
• Is Part Of: Optimization methods and software. Volume 37:Number 6(2022)
• Page Start: 2190
• Page End: 2224
• Publication Date: 2022-11-02
• Subjects: Nonlinear semidefinite programming -- primal-dual interior point method -- trust-region method -- negative-curvature direction -- primal-dual merit function -- global convergence -- second-order critical point; 90C22 -- 90C26 -- 90C51; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2022.2060973
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems, in which the iterates converge to a point that satisfies the first-order and second-order optimality conditions. The method consists of the outer iteration (SDPIP-revised) that finds a Karush-Kuhn-Tucker (KKT) point which satisfies the second-order optimality condition, and the inner iteration (SDPTR-revised) that calculates an approximate barrier KKT point. Algorithm SDPTR-revised uses a commutative class of Newton-like directions within the framework of the trust-region method in the primal-dual space. In addition, we also use a direction of negative curvature when it exists. The proposed algorithm employs a new method that generates negative-curvature directions in the existence of l 1 -type penalty term for equality constraints. It is proved that there exists a limit point of the generated sequence which satisfies the second-order optimality condition along with the barrier KKT conditions.