Weak Sharp Minima for Interval-Valued Functions and its Primal-Dual Characterizations Using Generalized Hukuhara Subdifferentiability

This article introduces the concept of weak sharp minima (WSM) for convex interval-valued functions (IVFs). To identify a set of WSM of a convex IVF, we provide its primal and dual characterizations. The primal characterization is given in terms of gHdirectional derivatives. On the other hand, to derive dual characterizations, we propose the notions of the support function of a subset of I(R) n and gH-subdifferentiability for convex IVFs. Further, we develop the required gH-subdifferential calculus for convex IVFs. Thereafter, by using the proposed gH-subdifferential calculus, we provide dual characterizations for the set of WSM of convex IVFs.

A Study of Convex Convex-Composite Functions via Infimal Convolution with Applications

In this paper, we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone convexity, is straightforward. The results are established under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity assumptions on the functions in play. The versatility of our findings is illustrated by a series of applications in optimization and matrix analysis, including conic programming, matrix-fractional, variational Gram, and spectral functions.

The core and superdifferential of a fuzzy TU-cooperative game

In the paper, we consider conditions providing coincidence of the cores and superdifferentials of fuzzy cooperative games with side payments. It turned out that one of the most simple sufficient conditions consists of weak homogeneity. Moreover, by applying so-called S*-representation of a fuzzy game introduced by the author, we show that for any vwith nonempty core C(v) there exists some game u such that C(v) coincides with the superdifferential of u. By applying subdifferential calculus we describe a structure of the core forboth classic fuzzy extensions of the ordinary cooperative game (e.g., Aubin and Owen extensions) and for some new continuations, like Harsanyi extensions and generalized Airport game.

Fréchet vector subdifferential calculus

In this paper, we study Fréchet vector subdifferentials of vector-valued functions in normed spaces which reduceto the known ones of extended-real-valued functions. We establish relations between two kinds of Fréchet vectorsubdifferentials and between subdifferential and coderivative; some of them improve the existing relations forextended-real-valued functions. Finally, sum and chain rules among others for Fréchet subdifferentials of vector-valued functions are formulated and verified. Many examples are provided

Hyperbolic Maxwell variational inequalities of the second kind

We analyze a class of hyperbolic Maxwell variational inequalities of the second kind. By means of a local boundedness assumption on the subdifferential of the underlying nonlinearity, we prove a well-posedness result, where the main tools for the proof are the semigroup theory for Maxwell’s equations, the Yosida regularization and the subdifferential calculus. The second part of the paper focuses on a more general case omitting the local boundedness assumption. In this case, taking into account more regular initial data and test functions, we are able to prove a weaker existence result through the use of the minimal section operator associated with the Nemytskii operator of the governing subdifferential. Eventually, we transfer the developed well-posedness results to the case involving Faraday’s law, which in particular allows us to improve the regularity property of the electric field in the weak existence result.