Title: Non-smooth singular Newton's method for positive semidefinite solution of nonlinear matrix equations
Authors: Boubakeur Benahmed; Khadra Nachi; Adnan Yassine
Addresses: Department of Mathematics and Informatics, National Polytechnic School of Oran, BP 1523 El m'naouer, Algeria ' Laboratory of Mathematic and its Applications, University of Oran, BP 1524 El m'naouer, Algeria ' Laboratory of Applied Mathematics, University of Le Havre, BP 540, 76058 Le Havre Cedex, France
Abstract: In this paper, we propose a new method for solving conic constrained nonlinear matrix equations. With the use of the orthogonal projection onto the positive semidefinite cone of matrices, the conic constrained equation is transformed to a non-smooth unconstrained equation which is solved by the non-smooth Newton's method. Here, we use an explicit expression of the Clarke generalised Jacobian of the projection onto the cone of positive semidefinite matrices as developed by several authors. We prove under natural assumptions that the method converges locally and superlinearly.
Keywords: conic equations; non-smooth equations; Newton's method; nonlinear matrix equations; generalised equations; superlinear convergence; positive semidefinite matrices; orthogonal projection.
International Journal of Operational Research, 2016 Vol.27 No.1/2, pp.303 - 315
Received: 02 Nov 2013
Accepted: 08 May 2014
Published online: 02 Aug 2016 *