In 1965, Njastad [2] introduced the notion of α-sets in topological space. In 1983, Mashhour et al. [1] introduced, with the help of α-sets, a weak form of continuity which they termed as α-continuity. Noiri [3] introduced the same concept, but under the name strong semicontinuity. Noiri [4] defined with the aid of α-sets a new weakened form of continuous mapping called weakly α-continuous mapp...

In this paper we introduce the concepts of upper and lower weakly B*continuous multifunction, and obtain some characterizations and several properties concerning upper and lower B*-continuous as well as upper and lower weakly B*continuous multifunction. The relationship between these multifunctions and their graphs are investigated.

We find conditions under which direct and inverse limits of arbitrary indexed systems of topological Abelian groups are related via the duality defined by the continuous convergence structure. This generalizes known results by Kaplan about duality of direct and inverse sequences of locally compact Abelian groups.

Using a nonsymmetric duality for abstract continuous convex control problems opti-mality conditions are derived for calculating the primal and dual solutions in the case of linear on state depending dual operators. Functional and pointwise conditions are considered. Subject: 49K22, 49K27, 49N15, 90C42. Keywords: abstract optimal control , nonsymmetric duality, suucient conditions of optimality,...

Duality is an important notion for constrained optimization which provides a theoretical foundation for a number of constraint decomposition schemes such as separable programming and for deriving lower bounds in space decomposition algorithms such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex optimization prob...

Inspired by the Linear Programming based algorithms for discrete MRFs, we show how a corresponding infinite-dimensional dual for continuous-state MRFs can be approximated by a hierarchy of tractable relaxations. This hierarchy of dual programs includes as a special case the methods of Peng et al. [17] and Zach & Kohli [33]. We give approximation bounds for the tightness of our construction, stu...

In this note we prove that the weak topology of a Banach space has the Continuous Representability Property which means that every (weakly) continuous total preorder defined on a Banach space can be represented by a weakly continuous utility function. This shows that weak topologies are perfect to look for continuous utility functions that represent preferences on infinite-dimensional commodity...

A few Karush-Kuhn-Tucker type of sufficient optimality conditions are given in this paper for nonsmooth continuous-time nonlinear multi-objective optimization problems in the Banach space L∞ [0, T ] of all n-dimensional vector-valued Lebesgue measurable functions which are essentially bounded, using Clarke regularity and generalized convexity. Further, we establish duality theorems for Wolfe an...