Finite-Dimensional Variational Inequalities and Complementarity Problems

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Springer Science & Business Media, Jun 14, 2007 - Mathematics - 693 pages
The ?nite-dimensional nonlinear complementarity problem (NCP) is a s- tem of ?nitely many nonlinear inequalities in ?nitely many nonnegative variables along with a special equation that expresses the complementary relationship between the variables and corresponding inequalities. This complementarity condition is the key feature distinguishing the NCP from a general inequality system, lies at the heart of all constrained optimi- tion problems in ?nite dimensions, provides a powerful framework for the modeling of equilibria of many kinds, and exhibits a natural link between smooth and nonsmooth mathematics. The ?nite-dimensional variational inequality (VI), which is a generalization of the NCP, provides a broad unifying setting for the study of optimization and equilibrium problems and serves as the main computational framework for the practical solution of a host of continuum problems in the mathematical sciences. The systematic study of the ?nite-dimensional NCP and VI began in the mid-1960s; in a span of four decades, the subject has developed into a very fruitful discipline in the ?eld of mathematical programming. The - velopments include a rich mathematical theory, a host of e?ective solution algorithms, a multitude of interesting connections to numerous disciplines, and a wide range of important applications in engineering and economics. As a result of their broad associations, the literature of the VI/CP has bene?ted from contributions made by mathematicians (pure, applied, and computational), computer scientists, engineers of many kinds (civil, ch- ical, electrical, mechanical, and systems), and economists of diverse exp- tise (agricultural, computational, energy, ?nancial, and spatial).
 

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Contents

Contents of Volume II
92
Interior and Smoothing Methods 989
94
Solution Analysis I
125
Global Methods for Nonsmooth Equations 723
236
Solution Analysis II
243
Methods for Monotone Problems 1107
335
The Euclidean Projector and Piecewise Functions
339
Sensitivity and Stability
419
Theory of Error Bounds
531
Bibliography for Volume I
I-2
Index of Definitions and Results I51
I-51
Index of Definitions Results and Algorithms II39
I-56
Algorithms for VIs 891
I-59
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