The main purpose of the present paper is to impose onesided Lipschitz condition for differential inclusions on a closed and convex domain of a uniformly convex Banach space. Both differential inclusions with almost upper demicontinuous and almost lower semicontinuous right–hand sides are considered. The existence theorems are proved and it is shown that the set of solutions is connected. AMS Ma...

This paper deals with a nonlinear multiobjective semi-infinite programming problem involving generalized (C,α, ρ, d)-convex functions. We obtain sufficient optimality conditions and formulate the Mond-Weirtype dual model for the nonlinear multiobjective semi-infinite programming problem. We also establish weak, strong and strict converse duality theorems relating the problem and the dual problem.

We study the weak* lower semicontinuity properties of functionals of the form F (u) = ess sup x∈Ω f(x,Du(x)) where Ω is a bounded open set of R and u ∈ W 1,∞(Ω). Without a continuity assumption on f(·, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub levels). In particular if F is weakly* lower semic...

Given γ ≥ 0, let us consider the following differential inclusion (S) ẍ(t) + γ ẋ(t) + ∂Φ(x(t)) 3 0, t ∈ R+, where Φ : Rd → R ∪ {+∞} is a lower semicontinuous convex function such that int(domΦ) 6= ∅. The operator ∂Φ denotes the subdifferential of Φ. When Φ = f + δK with f : Rd → R a smooth convex function and K ⊂ Rd a closed convex set, inclusion (S) describes the motion of a discrete mechanica...

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the ...

The iterative shrinkage/thresholding algorithm (ISTA) and its faster version FISTA have been widely used in the literature. In this paper, we consider general versions of the ISTA and FISTA in the more general “strongly + semi” convex setting, i.e., minimizing the sum of a strongly convex function and a semiconvex function; and conduct convergence analysis for them. The consideration of a semic...

of thesis entitled: Learning with Unlabeled Data Submitted by XU, Zenglin for the degree of Doctor of Philosophy at The Chinese University of Hong Kong in January 2009 We consider the problem of learning from both labeled and unlabeled data through the analysis on the quality of the unlabeled data. Usually, learning from both labeled and unlabeled data is regarded as semi-supervised learning, w...

We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of three-tuples of pairwise incompatible strains. In addition, we construct a fivedimensional continuum of T3s and show that its intersection with the boundary o...

There is a fundamental asymmetry between algebras and their dual objects, coalgebras, namely that the dual of a coalgebra is an algebra, but the converse is only true in finite dimensions. We prove that there exists a differential graded coalgebra whose continuous dual is the differential graded algebra of differential forms. This coalgebra will be constructed as an explicit subspace of de Rham...