Finite-Dimensional Variational Inequalities and Complementarity ProblemsThe ?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). |
Contents
Contents of Volume II | 92 |
Interior and Smoothing Methods 989 | 124 |
Solution Analysis I | 125 |
Global Methods for Nonsmooth Equations 723 | 236 |
Solution Analysis II | 243 |
Methods for Monotone Problems 1107 | 266 |
The Euclidean Projector and Piecewise Functions | 339 |
Sensitivity and Stability | 419 |
Algorithms for VIs 891 | 528 |
Theory of Error Bounds | 531 |
Bibliography for Volume I | I-2 |
I-51 | |
I-56 | |
Common terms and phrases
affine analysis Applications arbitrary assume assumption belongs Clearly closed convex complementarity computational condition cone Consequently Consider constant constraints containing contradiction converging convex set copositive Corollary deduce defined definition denote derivative differentiable directional discussion equal equation equilibrium equivalent error bound establish example Exercise exists extended feasible finite follows function given global Hence holds homeomorphism implies important inequality introduced latter Lemma limit linear Lipschitz continuous locally Mathematical matrix methods minimize monotone Moreover multiplier natural neighborhood nonempty nonlinear nonnegative nonzero normal obtain open set optimization pair polyhedral positive present problem proof Proposition prove provides result satisfying scalar sequence SOL(K,F solution solution set solving stability statements strictly strong strongly subset sufficient sufficiently Suppose Theorem theory tion unique solution vector zero