Nonsmooth Vector Functions and Continuous Optimization (Springer Optimization and Its Applications (10))

A recent significant innovation in mathematical sciences has been the progressive use of nonsmooth calculus, an extension of the differential calculus, as a key tool of modern analysis in many areas of mathematics, operations research, and engineering. Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems and variational inequalities in finite dimensions. The treatment is motivated by a desire to expose an elementary approach to nonsmooth calculus by using a set of matrices to replace the nonexistent Jacobian matrix of a continuous vector function. Such a set of matrices forms a new generalized Jacobian, called pseudo-Jacobian. A direct extension of the classical derivative that follows simple rules of calculus, the pseudo-Jacobian provides an axiomatic approach to nonsmooth calculus, a flexible tool for handling nonsmooth continuous optimization problems. Illustrated by numerous examples of known generalized derivatives, the work may serve as a valuable reference for graduate students, researchers, and applied mathematicians who wish to use nonsmooth techniques and continuous optimization to model and solve problems in mathematical programming, operations research, and engineering. Readers require only a modest background in undergraduate mathematical analysis to follow the material with minimal effort.

a Recent Significant Innovation In Mathematical Sciences Has Been The Progressive Use Of Nonsmooth Calculus, An Extension Of The Differential Calculus, As A Key Tool Of Modern Analysis In Many Areas Of Mathematics, Operations Research, And Engineering. Focusing On The Study Of Nonsmooth Vector Functions, This Book Presents A Comprehensive Account Of The Calculus Of Generalized Jacobian Matrices And Their Applications To Continuous Nonsmooth Optimization Problems And Variational Inequalities In Finite Dimensions.

the Treatment Is Motivated By A Desire To Expose An Elementary Approach To Nonsmooth Calculus By Using A Set Of Matrices To Replace The Nonexistent Jacobian Matrix Of A Continuous Vector Function. Such A Set Of Matrices Forms A New Generalized Jacobian, Called Pseudo-jacobian. A Direct Extension Of The Classical Derivative That Follows Simple Rules Of Calculus, The Pseudo-jacobian Provides An Axiomatic Approach To Nonsmooth Calculus, A Flexible Tool For Handling Nonsmooth Continuous Optimization Problems.

illustrated By Numerous Examples Of Known Generalized Derivatives, The Work May Serve As A Valuable Reference For Graduate Students, Researchers, And Applied Mathematicians Who Wish To Use Nonsmooth Techniques And Continuous Optimization To Model And Solve Problems In Mathematical Programming, Operations Research, And Engineering. Readers Require Only A Modest Background In Undergraduate Mathematical Analysis To Follow The Material With Minimal Effort.

"Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems and variational inequalities in finite dimensions. Illustrated by numerous examples of known generalized derivatives, the work may serve as a valuable reference for graduate students, researchers, and applied mathematicians who wish to use nonsmooth techniques and continuous optimization to model and solve problems in mathematical programming, operations research, and engineering. Readers require only a modest background in undergraduate mathematical analysis to follow the material with minimal effort."--Jacket Preface......Page 8 Contents......Page 6 Pseudo-Jacobian Matrices......Page 10 Calculus Rules forPseudo-Jacobian......Page 66 Openness of ContinuousVector Functions......Page 107 Nonsmooth MathematicalProgramming Problems......Page 151 Monotone Operators andNonsmooth VariationalInequalities......Page 215 Bibliographical Notes......Page 263 References......Page 267 Notations......Page 273 Index......Page 274 Pseudo-Jacobian matrices Calculus rules for Pseudo-Jacobians Openness of continuous vector functions Nonsmooth mathematical programming problems Monotone operators and nonsmooth variational inequalities.