Our approach to the Karush-Kuhn-Tucker theorem in [OSC] was entirely based on subdifferential calculus (essentially, it was an outgrowth of the two subdifferential calculus rules contained in the Fenchel-Moreau and Dubovitskii-Milyutin theorems, i.e., Theorems 2.9 and 2.17 of [OSC]). On the other hand, Proposition B.4(v) in [OSC] gives an intimate connection between the subdifferential of a fun...

Monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite (not necessarily symmetric) matrices. In 1970, Asplund studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an “irreducible” monotone operator. In 2007, Borwein and Wiersma [...

The classical Jensen inequality for concave function φ is adapted for the Sugeno integral using the notion of the subdifferential. Some examples in the framework of the Lebesgue measure to illustrate the results are presented.

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...