For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rn , let K(Y ) be the family of all fuzzy sets of Rn , which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space K(Y ) with the topology of endograph metric is homeomorphic to the Hilbert cube Q = [−1, 1] iff Y is compact; and the space K(Y ) is homeomorphic to {(...

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi-continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided.

and Applied Analysis 3 Example 2.3. Let X be a subset in Rn defined as follows: X : { x1, x2 | 0 < x2 < x2 1, 0 < x1 < 2 } ∪ { 0, 0 }. 2.2 Consider the point u 0, 0 . Since the tangent line of the curve x2 x2 1 at point u is the line x2 0. Then, for any x ∈ X \ {u}, there exists 0 < λ0 < 1 such that u λη x, u / ∈ X, ∀λ ∈ 0, λ0 . 2.3 Therefore, there exists no vector-valued function η x, u / 0 s...

Several characterizations of solution sets of a class of convex infinite programs are given using Lagrange multiplier conditions. The results are then applied to some classes of optimization problems: cone-constrained convex programs and fractional programs. A class of semi-convex problems with convex constraints are also examined. Optimality conditions are obtained and several characterization...

The Reformulation-Linearization Technique (RLT) provides a hierarchy of relaxations spanning the spectrum from the continuous relaxation to the convex hull representation for linear 0-1 mixed-integer and general mixed-discrete programs. We show in this paper that this result holds identically for semi-infinite programs of this type. As a consequence, we extend the RLT methodology to describe a ...

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for two problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. Using these results, we solve the followin...

A convex body K in R has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi–infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We then investigate the automorphism groups o...

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a ba...

The Kantorovich function (xT Ax)(xT A−1x), where A is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we prove that the 2dimensional Kantorovich function is convex if and only if the condition number of its matrix is less than or equal to 3 + 2 √ 2. Thus ...

The following result of convex analysis is well–known [2]: If the function f : X → [−∞, +∞] is convex and some x0 ∈ core (dom f) satisfies f(x0) > −∞, then f never takes the value −∞. From a corresponding theorem for convex functions with values in semi–linear spaces a variety of results is deduced, among them the mentioned theorem, a theorem of Deutsch and Singer on the single–valuedness of co...