We give a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope’s vertices. Our result provides a constructive proof for the Approximate Carathéodory Problem [2], which states that any point inside a polytope contained in the lp ball of radius D can be approximated to within ǫ in lp norm by a convex combi...

In 1906 Jensen founded the theory of convex functions. This enabled him to prove a considerable extension of the AM-GM inequality. Recall that a subset D of a real vector space is called convex if every convex linear combination of a pair of points of D is in D. Equivalently, if x, y ∈ D, then tx+ (1− t)y ∈ D for every t ∈ [0, 1]. Given a convex set D, a function f : D → R is called convex if f...

in this paper we introduce the concept of topological number for locally convex topological spaces and prove some of its properties. it gives some criterions to study locally convex topological spaces in a discrete approach.

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic prog...

Unsupervised learning algorithms based on convex and conic encoders are proposed. The encoders find the closest convex or conic combination of basis vectors to the input. The learning algorithms produce basis vectors that minimize the reconstruction error of the encoders. The convex algorithm develops locally linear models of the input, while the conic algorithm discovers features. Both algorit...

We consider a different L-Minkowski combination of compact sets in R than the one introduced by Firey and we prove an L-BrunnMinkowski inequality, p ∈ [0, 1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function t 7→ μ(t 1 pA), p ∈ (0, 1], for unconditional con...

“Discrete Convex Analysis” is aimed at establishing a novel theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. The theoretical framework of convex analysis is adapted to discrete settings and the mathematical results in matroid/submodular function theory are generalized. Viewed from ...

In this paper, firstly, we obtain some new results about bornological convergence in locally convex cones (which was studied in [1]) and then we introduce the concept of bornological completion for locally convex cones. Also, we prove that the completion of a bornological locally convex cone is bornological. We illustrate the main result by an example.

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result a...