On a global implicit function theorem for locally Lipschitz maps via nonsmooth critical point theory

We prove a non-smooth generalization of the global implicit function theorem. More precisely we use the non-smooth local implicit function theorem and the non-smooth critical point theory in order to prove a non-smooth global implicit function theorem for locally Lipschitz functions. A comparison between several global inversion theorems is discussed

BANACH FAMILIES AND THE IMPLICIT FUNCTION THEOREM

We generalise the classical implicit function theorem (IFT) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms.Banach spaces, Implicit Function Theorem

The Implicit Function Theorem for continuous functions

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily C^1) maps.Comment: 9 pages, no figure

A comprehensive view on optimization: reasonable descent

Reasonable descent is a novel, transparent approach to a well-established field: the deep methods and applications of the complete analysis of continuous optimization problems. Standard reasonable descents give a unified approach to all standard necessary conditions, including the Lagrange multiplier rule, the Karush-Kuhn-Tucker conditions and the second order conditions. Nonstandard reasonable descents lead to new necessary conditions. These can be used to give surprising proofs of deep central results outside optimization: the fundamental theorem of algebra, the maximum and the minimum principle of complex function theory, the separation theorems for convex sets, the orthogonal diagonalization of symmetric matrices and the implicit function theorem. These optimization proofs compare favorably with the usual proofs and are all based on the same strategy. This paper is addressed to all practitioners of optimization methods from many fields who are interested in fully understanding the foundations of these methods and of the central results above.optimization;fundamental theorem of algebra;Lagrange multiplier;Karush-Kuhn-Tucker;descent;implicit function theorem;necessary conditions;orthogonal diagonalization