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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).
List of contents
Local Methods for Nonsmooth Equations.- Global Methods for Nonsmooth Equations.- Equation-Based Algorithms for CPs.- Algorithms for VIs.- Interior and Smoothing Methods.- Methods for Monotone Problems.
About the author
Jong-Shi Pang was awarded the 2003 Dantzig Prize, the worlds top prize in the area of Mathematical Programming.
Summary
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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Aus den Rezensionen:
"… Die behandelten Aufgabenstellungen lassen sich rasch präzisieren. … Vorbildlich ist die Organisation des Stoffes: … jedes Kapitel mit einer Einleitung beginnend, die klar die abgehandelten Themen benennt … sowie zu jedem Kapitel abschließend ein Abschnitt ‘Notes and Comments‘, der den Stoff einordnet und die Quellen zu den wichtigsten Ergebnissen nennt … So spricht dieses Werk nicht nur den Fachmann an. Es ist gleichermaßen bestens für fortgeschrittene Studenten, einschließlich Doktoranden … geeignet … Es ist ein großartiges Werk, das sich über längere Zeit hinweg zur Standardreferenz etablieren wird."
(J. Gwinner, in: Jahresbericht der Deutschen Mathematiker-Vereinigung, 2006, Vol. 108, Issue 3, S. 16 ff.)
Report
Aus den Rezensionen:
"... Die behandelten Aufgabenstellungen lassen sich rasch präzisieren. ... Vorbildlich ist die Organisation des Stoffes: ... jedes Kapitel mit einer Einleitung beginnend, die klar die abgehandelten Themen benennt ... sowie zu jedem Kapitel abschließend ein Abschnitt 'Notes and Comments', der den Stoff einordnet und die Quellen zu den wichtigsten Ergebnissen nennt ... So spricht dieses Werk nicht nur den Fachmann an. Es ist gleichermaßen bestens für fortgeschrittene Studenten, einschließlich Doktoranden ... geeignet ... Es ist ein großartiges Werk, das sich über längere Zeit hinweg zur Standardreferenz etablieren wird."
(J. Gwinner, in: Jahresbericht der Deutschen Mathematiker-Vereinigung, 2006, Vol. 108, Issue 3, S. 16 ff.)
