In this paper, we consider the positive semi-definite space tensor cone constrained convex program, its structure and algorithms. We study defining functions, defining sequences and polyhedral outer approximations for this positive semidefinite space tensor cone, give an error bound for the polyhedral outer approximation approach, and thus establish convergence of three polyhedral outer approxi...

Abstract We present recursive formulas that compute the recently defined “higher symplectic capacities” for all convex toric domains. In special case of four-dimensional ellipsoids, we apply homological perturbation theory to associated filtered $\mathcal{L}_\infty $ algebras and prove resulting structure coefficients count punctured pseudoholomorphic curves in cobordisms between ellipsoids. As...

Let L be the space of line transversals to a nite family of pairwise disjoint compact convex sets in R 3. We prove that each connected component of L can itself be represented as the space of transversals to some nite family of pairwise disjoint compact convex sets.

We consider the problem of reporting convex hull points in an orthogonal range query in two dimensions. Formally, let P be a set of n points in R. A point lies on the convex hull of a point set S if it lies on the boundary of the minimum convex polygon formed by S. In this paper, we are interested in finding the points that lie on the boundary of the convex hull of the points in P that also fal...

An important class of planar straight-line drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of te...

The paper is a supplement to [2]. Let L be a lattice and U an o-symmetric convex body inR n. The Minkowski functional I[ I1~ of U, the polar body U ~ the dual lattice L*, the covering radius/z(L, U), and the successive minima ~.i(L, U), i = 1 . . . . . n, are defined in the usual way. Let 12, be the family of all lattices in R". Given a convex body U, we define mh(U) = sup max ~.i(L, U)~.n-i+l(...

We study the conditions under which one is able to efficiently apply variancereduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of Õ((n + L/μ) ln(1/ )) for L-smooth and μ-strongly convex individual functions one must also know which individual fu...

We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of Õ((n + L/μ) ln(1/ǫ)) for L-smooth and μ-strongly convex individual functions one must also know which individual f...

A function f is said to be cone superadditive if there exists a partition of R into a family of polyhedral convex cones such that f(z + x) + f(z + y) ≤ f(z) + f(z + x+ y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by loca...