This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: sup-, sup⋆- and symmetric subdifferentials. They satisfy most fundamental properties convexity, weak∗-closedness and, if function is bounded, weak∗-compactness. Moreover, they are nonempty for Rainwater function, possess certain calculus rules, provide necessary sufficient optimality conditions not ne...

In this short note, we derive an upper estimate of Clarke's subdifferential marginal functions in Banach spaces. The structure the is very similar to other results already obtained literature. novelty lies on fact that our assertions general spaces, and avoid use Asplund assumption.

The idea behind the Clarke Tax Algorithm is that a tax will be assessed to each individual. The amount of this tax depends on how the individual’s utilities affect the rest of the group. Since people do not want to pay taxes, a big assessment will decrease an individual’s overall utility. To get a better understanding of this statement, it is helpful to restrict the problem a bit. Formally, the...

and Applied Analysis 3 Furthermore, if h is a convex function, then ∂ c h(?̂?) is in accordance with the subdifferential ∂h(?̂?) in convex analysis; that is,