Multivalued analysis and nonlinear programming problems with perturbations
The aim of this book is to study, in finite dimensional spaces, multivalued mappings and the associated marginal functions and to apply the results to optimization problems with quite general constraints. More precisely, the following class of programming problems is the final object of investigation: \[ (\overline P_x)\qquad f(x,y)\to\inf_y,\;y\in F(x). \] Here \(f:\mathbb{R}^n\times \mathbb{R}^m\to\mathbb{R}\) and \(F\) is a multivalued mapping of \(\mathbb{R}^n\) to \(\mathbb{R}^m\). To a great extent, the analysis of such problems is sensitivity analysis, i.e., it consists in studying the marginal function (or optimal value function) \[ \varphi (x)=\inf\bigl\{f(x,y),\mid y\in F(x)\bigr\} \] and the set of optimal solutions \[ \omega(x)=\bigl\{y\in F(x)\mid\varphi(x)=f(x,y)\bigr\}. \] These functions are in general not differentiable, and this is the starting point of the book. The text is divided into six chapters. After a list with the notation used, Chapter 2 contains elements of convex analysis and of quasidifferentiable calculus in the sense of Demyanov and Rubinov. Chapter 3 deals with continuity and differentiability properties of multivalued mappings. Several notions of semicontinuity and (pseudo-)Lipschitz continuity are introduced and their relationship to each other and to the associated marginal function is studied. Moreover, various kinds of directional derivatives of multivalued mapppings are defined via tangent cones and are investigated in terms of the distance function. Chapter 4 is devoted to subdifferentials of marginal functions. In particular, estimates (in the sense of set inclusion) are established for the Clarke subdifferential and for subdifferentials of Mordukhovich. Chapter 5 presents sufficient conditions for the existence of the directional derivative of marginal functions. In this connection, weakly uniformly differentiable as well as strongly differentiable multifunctions are considered. The concluding chapter deals with the sensitivity analysis of perturbed programming problems such as \((\overline P_x)\). In this connection, various regularity concepts are introduced and studied. The results are then applied to programming problems with a finite number of inequalities and equations as constraints and in particular to programming problems with quasidifferentiable data. Finally, sufficient conditions are established for the second-order directional derivative of the marginal function to exist. The material is presented in a clear, rigorous manner. Beside the bibliographical comments at the end of the book, various references to the literature are given within the text. As the authors point out, the selection of the material was strongly influenced by the former Soviet school of optimization, in particular by Demyanov and Rubinov. Consequently, a significant part of the results have not been published in English before. Within the setting of finite dimensional spaces the unified approach to the directional differentiability of multifunctions and their associated marginal functions is a remarkable feature of the book and can be the starting point for further research. Unfortunately, the book contains a number of missprints. In particular, when the authors refer to Chapter \(k\) in the Preface and in the Bibliographical Comments, they mean Chapter \(k+1\). Also the page numbers in the Index are not correct. Despite these shortcomings the book is a useful contribution to nonsmooth analysis and optimization.
scientific article; zbMATH DE number 1855753