Well-Posedness by Perturbations of Variational Problems

In this paper, we consider the extension of the notion of well-posedness by perturbations, introduced by Zolezzi for optimization problems, to other related variational problems like inclusion problems and fixed-point problems. Then, we study the conditions under which there is equivalence of the well-posedness in the above sense between different problems. Relations with the so-called diagonal well-posedness are also given. Finally, an application to staircase iteration methods is presented.

[1]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

[2]  T. Zolezzi.  Well-Posedness and Stability Analysis in Optimization , 1986 .

[3]  B. Lemaire.  Regularization of Fixed-Point Problems and Staircase Iteration , 1999 .

[4]  B. Lemaire,et al.  Convergence of diagonally stationary sequences in convex optimization , 1994 .

[5]  T. Zolezzi.  Extended well-posedness of optimization problems , 1996 .

[6]  A. N. Tikhonov,et al.  On the stability of the functional optimization problem , 1966 .

[7]  J. Revalski,et al.  Well-posed constrained optimization problems in metric spaces , 1993 .

[8]  R. Lucchetti,et al.  Well-Posed Optimization Problems and a New Topology for the Closed Subsets of a Metric Space , 1993 .

[9]  T. Zolezzi,et al.  Well-posedness criteria in optimization with application to the calculus of variations , 1995 .

[10]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[11]  T. Zolezzi,et al.  On equiwellset minimum problems , 1977 .

[12]  A. Kaplan,et al.  Stable methods for ill-posed variational problems : prox-regularization of elliptic variational inequalities and semi-infinite problems , 1994 .

[13]  B. Lemaire.  Bounded Diagonally Stationary Sequences in Convex Optimization , 1994 .

[14]  Jean-Paul Penot,et al.  Metrically well-set minimization problems , 1992 .

[15]  Massimo Furi,et al.  About well-posed optimization problems for functionals in metric spaces , 1970 .

[16]  R. Rockafellar,et al.  On the subdifferentiability of convex functions , 1965 .

[17]  Roberto Cominetti,et al.  Convex Functions with Unbounded Level Sets and Applications to Duality Theory , 1993, SIAM J. Optim..

[18]  F. Browder.  Nonlinear maximal monotone operators in Banach space , 1968 .

[19]  J. Diestel.  Geometry of Banach Spaces: Selected Topics , 1975 .

[20]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .