Up to orthogonal transformation, a solid closed convex cone K in the Euclidean space Rn+1 is the epigraph of a nonnegative sublinear function f : Rn → R. This work explores the link between the geometric properties of K and the analytic properties of f . Mathematics Subject Classification: 46B10, 46B20, 52A41.

We introduce and study two operations on functions that may be useful on one hand for computing the distance to an epigraph and, on the other hand, for dealing with constraints in mathematical programming. Both of them involve the concept of monotone norms. A special attention is payed to convex functions.

a L∗∗(y(t), y′(t)) dt : y ∈ AC ([a, b],RN) , y(a) = A, y(b) = B } · (P ∗∗) It is well known that inf F = inf F ∗∗ if L is super-linear and continuous. Recently Cellina in [5] proved that the same conclusion holds true assuming, instead of superlinearity, a weaker growth condition that we will call (GA). Roughly, a convex function L(x, ξ) satisfies (GA) if the intersection of the supporting hype...

For a class of infinite-dimensional minimization problems with nonlinear equality constraints, an iterative algorithm for finding global solutions is suggested. A key assumption is the convexity of the “epigraph”, a set in the product of the image spaces of the constraint and objective functions. A convexification method involving randomization is used. The algorithm is based on the extremal sh...

As a generalization of geodesic function, this paper introduces the notion φE-convex function. Some properties function and are established. The concepts set φE-epigraph also given. characterization functions in terms their φE-epigraphs, obtained.

The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows ...

This paper presents a novel approach to modeling the propagation of the flu virus throughout a realistic interconnection network based on actual individual interactions which we extract from social networks. We allow the individual interconnections to change during the propagation by making them time-dependent. We have implemented a scalable, fully distributed simulator and we validated the epi...

We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection’s epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP, and Jameson’s (G)-property) are established. With suitable continuity assumption we show how this prop...