Journal ArticleDOI
On concepts of directional differentiability
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In this paper, the authors compared various definitions of directional derivatives in topological vector spaces and pointed out that in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent.Abstract:
Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of Gâteaux, Frechet, and Hadamard are singled out from the general framework of σ-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent. The chain rule for directional derivatives of a composite mapping is discussed.read more
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Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications
Patrick T. Harker,Jong-Shi Pang +1 more
TL;DR: The field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory as mentioned in this paper.
Journal ArticleDOI
A nonsmooth version of Newton's method
TL;DR: It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth, and the extended Newton's method can be used in the augmentedlagrangian method for solving nonlinear programs.
Journal ArticleDOI
Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations
TL;DR: Convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions and a hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is presented.
Journal ArticleDOI
Newton's method for B -differentiable equations
TL;DR: The classical Newton method for solving continuously differentiable systems of nonlinear equations to B -differentiable systems to provide a unified framework for the nonlinear complementarity, variational inequality and nonlinear programming problems is extended.
Journal ArticleDOI
Nonsmooth Equations: Motivation and Algorithms
Jong-Shi Pang,Liqun Qi +1 more
TL;DR: This paper reports on some recent developments in the area of solving of nonsmooth equations by generalized Newton methods, with the emphasis on three topics: motivation, characterization of superlinings, and motivation and characterization of motivation.
References
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Book
Optimization and nonsmooth analysis
TL;DR: The Calculus of Variations as discussed by the authors is a generalization of the calculus of variations, which is used in many aspects of analysis, such as generalized gradient descent and optimal control.