tailieunhanh - Báo cáo hóa học: " A smoothing-type algorithm for solving inequalities under the order induced by a symmetric cone"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A smoothing-type algorithm for solving inequalities under the order induced by a symmetric cone | Lu and Zhang Journal of Inequalities and Applications 2011 2011 4 http content 2011 1 4 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access A smoothing-type algorithm for solving inequalities under the order induced by a symmetric cone Nan Lu1 and Ying Zhang2 Correspondence yingzhang@tju. department of Mathematics School of Science Tianjin University Tianjin 300072 PR China Full list of author information is available at the end of the article SpringerOpen0 Abstract In this article we consider the numerical method for solving the system of inequalities under the order induced by a symmetric cone with the function involved being monotone. Based on a perturbed smoothing function the underlying system of inequalities is reformulated as a system of smooth equations and a smoothing-type method is proposed to solve it iteratively so that a solution of the system of inequalities is found. By means of the theory of Euclidean Jordan algebras the algorithm is proved to be well defined and to be globally convergent under weak assumptions and locally quadratically convergent under suitable assumptions. Preliminary numerical results indicate that the algorithm is effective. AMS subject classifications 90C33 65K10. Keywords Symmetric cone Euclidean Jordan algebra smoothing-type algorithm global convergence local quadratic convergence 1 Introduction Let V be a finite dimensional vector space over 4 with an inner product . If there exists a bilinear transformation from V X V to V denoted by o such that for any x y z e V x o y y o x x o x2 o y x2 o x o y x o y z x y o z where x2 x o x then V o is called a Euclidean Jordan algebra. Let K x2 x e V then K is a symmetric cone 1 . Thus K could induce a partial order for any x e V x 0 means xe K. Similarly x 0 means x e intK where intK denotes the interior of K and x 0 means -x 0. Let nk x denote the orthogonal projection of x onto K. By Moreau .