Stable methods for ill-posed variational problems: prox-regularization of elliptic variational inequalities and semi-infinite problems.

*(English)*Zbl 0804.49011
Mathematical Topics. 3. Berlin: Akademie Verlag. 437 S. (1994).

This book presents iterative prox-regularization methods for solving ill- posed convex variational problems in Hilbert spaces. Chapters I and II present several stability concepts in optimization and other fields of mathematics. Classical approaches to the solution of ill-posed problems are considered and the regularizing properties of different optimization methods are investigated. In Chapter III the iterative prox- regularization principle for finite-dimensional convex optimization problems is presented. Some stable variants of penalty methods and augmented Lagrangian methods are discussed. Chapters IV and V are devoted to the general framework for the investigation of iterative prox- regularization for infinite-dimensional convex variational problems, including a new class of multi-step regularization methods. In Chapters VI and VII the general framework is applied to the treatment of some specific types of variational problems: convex semi-infinite and convex parametric semi-infinite problems, ill-posed elliptic variational inequalities, elasticity theory problems, obstacle problems. Chapter VIII contains several remarks about the numerical realization of the suggested methods and some computational results.

Reviewer: S.Anita (Iaşi)

##### MSC:

49J40 | Variational inequalities |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

90C25 | Convex programming |

90C05 | Linear programming |

90C34 | Semi-infinite programming |