Extending and improving some recent results of Hantoute, López, and Zălinescu and others, we provide characterization conditions for subdifferential formulas to hold for the supremum function of a family of convex functions on a real locally convex space.

In this note we give a Brøndsted-Rockafellar Theorem for diagonal subdifferential operators in Banach spaces. To this end we apply an Ekeland-type variational principle for monotone bifunctions.

We relate the argmin sets of a given function, not necessarily convex or lower semicontinuous, and its lower semicontinuous convex hull by means of explicit characterizations involving an appropriate concept of asymptotic functions. This question is connected to the subdifferential calculus of the Legendre–Fenchel conjugate function. The final expressions, which also involve a useful extension ...

In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we estab...

and Applied Analysis 3 where ∂ denotes the subdifferential in the sense of convex analysis. We need the subdifferential inequality Φ( 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩) ≤ Φ (‖x‖) + ⟨y, j (x + y)⟩ ∀x, y ∈ X, j (x + y) ∈ Jφ (x + y) . (14) For a smoothX, we have Φ( 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩) ≤ Φ (‖x‖) + ⟨y, Jφ (x + y)⟩ ∀x, y ∈ X, (15) or considering the normalized duality mapping J, we have 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2 ⟨y, J (...

In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton–Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that i...

In this paper, we deal with the subdierential concept onHadamard spaces. Flat Hadamard spaces are characterized, and nec-essary and sucient conditions are presented to prove that the subdif-ferential set in Hadamard spaces is nonempty. Proximal subdierentialin Hadamard spaces is addressed and some basic properties are high-lighted. Finally, a density theorem for subdierential set is established.