Topological Aspects of Nonsmooth Optimization

ISBN-13: 9781461418962 / Angielski / Twarda / 2011 / 196 str.

Vladimir Shikhman

Opis

This book deals with nonsmooth structures arising within the optimization setting. It considers four optimization problems, namely, mathematical programs with complementarity constraints, general semi-infinite programming problems, mathematical programs with vanishing constraints and bilevel optimization. The author uses the topological approach and topological invariants of corresponding feasible sets are investigated. Moreover, the critical point theory in the sense of Morse is presented and parametric and stability issues are considered. The material progresses systematically and establishes a comprehensive theory for a rather broad class of optimization problems tailored to their particular type of nonsmoothness. Topological Aspects of Nonsmooth Optimization will benefit researchers and graduate students in applied mathematics, especially those working in optimization theory, nonsmooth analysis, algebraic topology and singularity theory. ​​

Recenzje

From the reviews:

"This monograph ... provides detailed topological investigations of nonsmooth optimization problems. ... the book highlights and motivates recent developments in a broad field of active research. The material is well presented, so that the book may be warmly recommended to graduate students and researchers in optimization." (Oliver Stein, Mathematical Reviews, October, 2013)

"The textbook (PhD thesis of the author) deals with mathematical programming problems which have smooth data but show, because of their structure, intrinsic non-smooth behavior ... . It can be used as a compendium of the critical-point theory also in the smooth case since nearly all important results are cited. New results are proved or it is referred to recently published results of the author. For researchers in the above described field of mathematical programming the book will become a very helpful and indispensable tool." (Armin Hoffmann, Zentralblatt MATH, Vol. 1234, 2012)

Spis treści

Preface. -Notation.- Introduction.-  Mathematical Programming Problems with Complementarity.- Constraints.-  General Semi-infinite Programming Problems.- Mathematical Programming Problems with Vanishing Constraints.- Bilevel Optimization.-  Impacts on Nonsmooth Analysis.- Appendix.- Bibliography.- References.- Index.