Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
An approximation method which combines Tikhonov method with the proximal point algorithm, is presented. Conditions which guarantee the convergence to a particular element of the solution set, are provided. A particular attention is given to the convex and convex-concave optimization cases.
ISSN: 0233-1934

hal-00815513
https://hal.univ-antilles.fr/hal-00815513
DOI : 10.1080/02331939608844217

Éditeur(s) : HAL CCSD
Résumé : International audience
The paper deals with iterative schemes for finding a zero of a maximal monotone operator. A combination of the auxiliary principle with the Tikhonov method is considered. Under some assumptions the sequence generated by such an iterative scheme is shown to converge to a solution of the problem that is required to satisfy an associated variational inequality
Lecture notes in economics and mathematical systems

hal-00773198
https://hal.univ-antilles.fr/hal-00773198

Éditeur(s) : HAL CCSD
Résumé : International audience
Relying on an idea of T. Pennanen, we use a localized version of maximal hypomonotonicity, and prove that it is enough to guarantee the local convergence of the Tikhonov regularization. In the process, we revisit this method which together with the help of the analysis developed by T. Pennanen allows us to obtain the desired result.
Mathematical Inequalities and Apllications

hal-00779258
https://hal.univ-antilles.fr/hal-00779258

Éditeur(s) : HAL CCSD
Résumé : International audience
In the paper the following mixed equilibrium problem in a Hilbert space is considered: find x 2 C such that F(x, x)+hG(x), x−xi 0, 8x 2 C, where C is a nonempty convex closed subset of a real Hilbert space. For its solution a version of the proximal approach is studied. The general proximal method always includes a quadratic term of the kind |x−xn|2, which provides a process of transition from a point xn to a point xn+1. The authors replace this quadratic term by its linearization. Thus, they obtain a method of the kind F(xn+1, x)+hG(xn)+ −1(h 0(xn+1)−h 0(xk)), x−xn+1i 0, 8x 2 C, where h0 is the derivative of a given function on C. It is proved that if the function F(x, y) is monotone and convex in y for any x, F(x, x) = 0, and the operator G(x) is co-coercive, then the method weakly converges in a real Hilbert space to an equilibrium solution. Note that this result will also be true if the co-coercive condition is replaced by a monotonicity property. The Tikhonov regularization method (in the absence of disturbances and calculating auxiliary solutions approximately) is justified and its weak convergence in a Hilbert space is proved. It is known that the proximal method can always be interpreted as an implicit discret approximation of some differential equation or inclusion. Therefore the authors consider a differential inclusion for the extreme mapping. Permitting "-approximation for the function F(x, y) by means of a parametric family F(x, y, ") and assuming the strong monotonicity of the operator @F(x, y, ") (it is a very strong condition), the authors prove the convergence of the trajectory of the differential inclusion to an equilibrium solution.
Ill-posed variational problems and regularization techniques

hal-00773204
https://hal.univ-antilles.fr/hal-00773204