Let s denote a compact convex object in IR. The f-width of s is the perpendicular distance between two distinct parallel lines of support of s with direction f . A set of disjoint convex compact objects in IR is of equal f -width if there exists a direction f such that every pair of objects have equal f -width. A visibility matching, for a set of equal f -width objects is a matching using non-c...

We consider the problem min{f(x) : x E G, T(x) tI. int D}, where fis a lower semicontinuous function, G a compact, nonempty set in JRn, D a closed convex set in JR2 with nonempty interior, and T a continuous mapping from JRn to JR2. The constraint T( x) tI. int D is areverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that...

In this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we obtain some related results. In particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. Then we study fuzzy...

Convex programming involves a convex set F ⊆ R and a convex cost function c : F → R. The goal of convex programming is to find a point in F which minimizes c. In online convex programming, the convex set is known in advance, but in each step of some repeated optimization problem, one must select a point in F before seeing the cost function for that step. This can be used to model factory produc...

In 1956, Frank and Wolfe extended the fundamental existence theorem of linear programming by proving that an arbitrary quadratic function f attains its minimum over a nonempty convex polyhedral set X provided f is bounded from below over X . We show that a similar statement holds if f is a convex polynomial and X is the solution set of a system of convex polynomial inequalities. In fact, this r...

Lemma A.1 (a) A real-valued convex function is also 0-convex and hence k-convex for all k ≥ 0. A k1-convex function is also a k2-convex function for k1 ≤ k2. (b) If f1(y) and f2(y) are k1-convex and k2-convex respectively, then for α, β ≥ 0, αf1(y) + βf2(y) is (αk1 + βk2)-convex. (c) If f(y) is k-convex and w is a random variable, then E{f(y − w)} is also k-convex, provided E{|f(y − w)|} < ∞ fo...

We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or globally subanalytic sets. A monotone map is a multi-dimensional generalization of a usual univariate monotone continuous function on an open interval, while the closure of the graph of a monotone map is a generalization of a compact convex set. In a particular case of an identically con...

We describe various structures of algebraic nature on the space of continuous valuations on convex sets, their properties (like versions of Poincaré duality and hard Lefschetz theorem), and their relations and applications to integral geometry. 2000 Mathematics Subject Classification: 46, 47.

Let Z and X be Banach spaces, U ⊂ Z an open convex set and f : U → X a mapping. We say that f is a delta-convex mapping (d. c. mapping) if there exists a continuous convex function h on U such that y ◦ f + h is a continuous convex function for each y ∈ Y , ‖y∗‖ = 1. We say that f : U → X is locally d. c. if for each x ∈ U there exists an open convex U ′ such that x ∈ U ′ ⊂ U and f |U ′ is d. c....

Convex optimization is a special class of optimization problems, that includes many problems of interest such as least squares and linear programming problems. Convex optimization problems are considered especially important because several efficient algorithms exist for solving them; as a result, many machine learning problems have been modeled as convex optimization. In a typical convex optim...