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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.

Nonsmooth optimization. --- Vector valued functions. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Functions, Vector --- Functions, Vector valued --- Functional analysis --- Functions of real variables --- Nonsmooth analysis --- Optimization, Nonsmooth --- Mathematical optimization --- Global analysis (Mathematics). --- Operations research. --- Engineering mathematics. --- Calculus of Variations and Optimal Control; Optimization. --- Analysis. --- Optimization. --- Operations Research, Management Science. --- Operations Research/Decision Theory. --- Mathematical and Computational Engineering. --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Engineering --- Engineering analysis --- Mathematics --- Calculus of variations. --- Mathematical analysis. --- Analysis (Mathematics). --- Management science. --- Decision making. --- Applied mathematics. --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving --- Isoperimetrical problems --- Variations, Calculus of --- Quantitative business analysis --- Statistical decision --- 517.1 Mathematical analysis --- Decision making