Finite-Dimensional Variational Inequalities and Complementarity Problems
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 of Volume II
Interior and Smoothing Methods 989
Solution Analysis I
Global Methods for Nonsmooth Equations 723
Solution Analysis II
Methods for Monotone Problems 1107
The Euclidean Projector and Piecewise Functions
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affine affine function algorithm arbitrary assume belongs bounded open closed convex set coherently oriented complementarity problem continuous function continuously differentiable converging convex cone convex function convex set copositive Corollary CRCQ deduce defined definition denote equations equilibrium equivalent error bound Euclidean feasible finite follows function F global holds homeomorphism implies index set KKT system Lemma Let F linear linear program Lipschitz continuous locally Lipschitz Mathematical Programming matrix MFCQ minimize MnorK Moreover natural map NCP F nonempty nonlinear nonlinear program nonnegative nonsingular nonsmooth nonzero normal map open neighborhood open set optimization problem pair polyhedral polyhedron positive scalars positive semidefinite proof Proposition pseudo monotone result satisfying Schur complement semicopositive sequence set-valued map SOL(K solution set stability Suppose symmetric Theorem theory unique solution variational inequalities vector vectors q VI/CP xref zero