Monday, September 19, 2011

Variational Analysis and Generalized Differentiation II by Boris S. Mordukhovich

- We can treat this firm stand by Euler [411] (“. . . nihil omnino in mundo contingint, in quo non maximi minimive ratio quapiam eluceat”) as the most fundamental principle of Variational Analysis. This principle justifies a variety of striking implementations of optimization/variational approaches to solving numerous problems in mathematics and applied sciences that may not be of a variational nature. Remember that optimization has been a major motivation and driving force for developing differential and integral calculus. Indeed, the very concept of derivative introduced by Fermat via the tangent slope to the graph of a function was motivated by solving an optimization problem; it led to what is now called the Fermat stationary principle. Besides applications to optimization, the latter principle plays a crucial role in proving the most important calculus results including the mean value theorem, the implicit and inverse function theorems, etc. The same line of development can be seen in the infinite-dimensional setting, where the Brachistochrone was the first problem not only of the calculus of variations but of all functional analysis inspiring, in particular, a variety of concepts and techniques in infinite-dimensional differentiation and related areas.

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