Finite-Dimensional Variational Inequalities and Complementarity Problems
Springer Science & Business Media, Jun 4, 2007 - Business & Economics - 704 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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accumulation point Algorithm Assume assumption bounded boundedness C-function consider constant constrained continuously diﬀerentiable convergence rate converges to zero convex function convex set Corollary D-gap deduce deﬁned diﬀerent exists F(xk FB regular Fi(x ﬁnite ﬁrst follows function F G(xk gap function given global gradient Hence holds homeomorphism implies index sets inequality inexact IP methods iteration Jacobian JF(x Lemma Let F Let xk limit point limsup line search linear Newton approximation Lipschitz continuous matrix maximal monotone merit function methods for solving MiCP minimization NCP F neighborhood Newton approximation scheme Newton method nonempty nonnegative nonsingular open set positive deﬁnite positive scalars Proof Proposition proximal point reformulation result satisﬁes satisfying scalars Schur complement Section sequence xk smooth SOL(K,F Speciﬁcally stationary point strongly monotone subset superlinear superlinearly Suppose Theorem Tikhonov regularization trust region vector
Page 1271 - Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan...
Page 630 - A complex-valued function on [a, b] is in fiip^fa, &]) [see the definition in (17.31)] if and only if for every e > 0 there is a <5 > 0 such that for all sequences ([ah, &*])"=,! of subintervals of [a, b] for which Z (b, -<*»)<* k=i holds, the inequality holds. That is, / is "absolutely continuous with overlap permitted".