On nonconvex version of the inequality of Clarke and Ledyaev
Nonlinear Analysis: Theory, Methods & Applications
Variational pairs and applications to stability in nonsmooth analysis
Nonlinear Analysis: Theory, Methods & Applications
SIAM Journal on Optimization
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We establish a general Fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a Banach space. Our basic tool is a constrained variational principle derived from the "smooth" variational principle of Borwein and Preiss. Specializing the Fermat rule to the case when the convex set is a "drop," we obtain a multidirectional Rolle-type inequality from which, in turn, we deduce a multidirectional mean value inequality, in the line of Clarke and Ledyaev. We follow the abstract approach of our previous paper [Trans. Amer. Math. Soc., 347 (1995), pp. 4147--4161], thus covering all standard situations met in applications, while stressing the links between the results and the few key properties that are needed.