| Convexity and Well-Posed Problems 2006 Edition Contributor(s): Lucchetti, Roberto (Author) |
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ISBN: 0387287191 ISBN-13: 9780387287195 Publisher: Springer OUR PRICE: $52.24 Product Type: Hardcover - Other Formats Published: November 2005 Annotation: This book deals with the study of convex functions and of their behavior from the point of view of stability with respect to perturbations. Convex functions are considered from the modern point of view that underlines the geometrical aspect: thus a function is defined as convex whenever its graph is a convex set. A primary goal of this book is to study the problems of stability and well-posedness, in the convex case. Stability means that the basic parameters of a minimum problem do not vary much if we slightly change the initial data. On the other hand, well-posedness means that points with values close to the value of the problem must be close to actual solutions. In studying this, one is naturally led to consider perturbations of functions and of sets. This approach fits perfectly with the idea of regarding functions as sets. Thus the second part of the book starts with a short, yet rather complete, overview of the so-called hypertopologies, i.e. topologies in the closed subsets of a metric space. While there exist numerous classic texts on the issue of stability, there only exists one book on hypertopologies ÝBeer 1993¨. The current book differs from Beers in that it contains a much more condensed explication of hypertopologies and is intended to help those not familiar with hypertopologies learn how to use them in the context of optimization problems. |
| Additional Information |
| BISAC Categories: - Mathematics | Linear & Nonlinear Programming - Mathematics | Mathematical Analysis - Mathematics | Calculus |
| Dewey: 515.8 |
| LCCN: 2005932085 |
| Series: CMS Books in Mathematics |
| Physical Information: 0.77" H x 6.32" W x 9.54" (1.22 lbs) 305 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can be seen also as properties of a certain convex set related to it. Moreover, we shall consider extended real valued functions, i. e., functions taking possibly the values and +?. The reason for considering the value +? is the powerful device of including the constraint set of a constrained minimum problem into the objective function itself (by rede?ning it as +? outside the constraint set). Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set. And the value is allowed because useful operations, such as the inf-convolution, can give rise to functions valued even when the primitive objects are real valued. Observe that de?ning the objective function to be +? outside the closed constraint set preserves lower semicontinuity, which is the pivotal and mi- mal continuity assumption one needs when dealing with minimum problems. Variational calculus is usually based on derivatives. |