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Le Thi H.A.,CNRS Theoretical and Applied Informatics | Le H.M.,CNRS Theoretical and Applied Informatics | Pham Dinh T.,National Institute for Applied Sciences, Strasbourg | Van Huynh N.,University of Quynhon
Journal of Global Optimization | Year: 2013

In this paper, we consider a binary supervised classification problem, called spherical separation, that consists of finding, in the input space or in the feature space, a minimal volume sphere separating the set A from the set B (i.e. a sphere enclosing all points of A and no points of B. The problem can be cast into the DC (Difference of Convex functions) programming framework and solved by DCA (DC Algorithm) as shown in the works of Astorino et al. (J Glob Optim 48(4):657-669, 2010). The aim of this paper is to investigate more attractive DCA based algorithms for this problem. We consider a new optimization model and propose two interesting DCA schemes. In the first scheme we have to solve a quadratic program at each iteration, while in the second one all calculations are explicit. Numerical simulations show the efficiency of our customized DCA with respect to the methods developed in Astorino et al. © 2012 Springer Science+Business Media, LLC. Source


Kruger A.,University of Ballarat | Van Ngai H.,University of Quynhon | Thera M.,University of Limoges
SIAM Journal on Optimization | Year: 2010

This paper studies stability of error bounds for convex constraint systems in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered. © 2010 Society for Industrial and Applied Mathematics. Source


Van Ngai H.,University of Quynhon | Kruger A.,University of Ballarat | Thera M.,University of Limoges
SIAM Journal on Optimization | Year: 2010

In this paper, we are concerned with the stability of the error bounds for semi-infinite convex constraint systems. Roughly speaking, the error bound of a system of inequalities is said to be stable if all its "small" perturbations admit a (local or global) error bound. We first establish subdifferential characterizations of the stability of error bounds for semi-infinite systems of convex inequalities. By applying these characterizations, we extend some results established by Azé and Corvellec [SIAM J. Optim., 12 (2002), pp. 913-927] on the sensitivity analysis of Hoffman constants to semi-infinite linear constraint systems. Copyright © 2010, Society for Industrial and Applied Mathematics. Source


Le Thi H.A.,CNRS Theoretical and Applied Informatics | Dinh T.P.,Bp08 Avenue Of Luniversite | Van Ngai H.,University of Quynhon
Journal of Global Optimization | Year: 2012

In the present paper, we are concerned with conditions ensuring the exact penalty for nonconvex programming. Firstly, we consider problems with concave objective and constraints. Secondly, we establish various results on error bounds for systems of DC inequalities and exact penalty, with/without error bounds, in DC programming. They permit to recast several class of difficult nonconvex programs into suitable DC programs to be tackled by the efficient DCA. © Springer Science+Business Media, LLC. 2011. Source


Adly S.,University of Limoges | Cibulka R.,University of Limoges | Cibulka R.,University of West Bohemia | Van Ngai H.,University of Quynhon
SIAM Journal on Optimization | Year: 2015

Results on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton-type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton's method, the nonsmooth Newton's method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton's method is discussed. © 2015 Society for Industrial and Applied Mathematics. Source

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