Subdifferential estimate of the directional derivative and optimality criterion for lower semicontinuous functions

Abstract

We provide an inequality relating the radial directional derivative and the subdifferential
of proper lower semicontinuous functions, which extends the known formula for convex functions.
We show that this property is equivalent to other subdifferential properties, such as
controlled dense subdifferentiability, controlled compact separation and weak star controlled Lipschitz separation. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces. Maximal monotonicity of the subdifferential of convex lower semicontinuous functions follows.

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