Well-posed optimization problems / Asen Dontchev, Tullio Zolezzi

Preface: This book aims to present, in a unified way, some basic aspects of the mathematical theory of well-posedness in scalar optimization. The first fundamental concept in this area is inspired by the classical idea of J. Hadamard, which goes back to the beginning of this century. It requires existence and uniqueness of the optimal solution together with continuous dependence on the problem’s data.

In the early sixties A. Tykhonov introduced another concept of well- posedness imposing convergence of every minimizing sequence to the unique minimum point. Its relevance to (and motivation from) the approximate (numerical) solution of optimization problems is clear.

In the book, the authors study both the Tykhonov and the Hadamard concepts of well-posedness, the links between them and also some extensions (e.g. relaxing the uniqueness). Both the pure and the applied sides of our topic are presented. The first four chapters are devoted to abstract optimization problems. Applications to optimal control, calculus of variations and mathematical programming are the subject matter of the remaining five chapters.