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We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification…
We study a general smallest intersecting ball problem and its soft-margin variant in high-dimensional Euclidean spaces for input objects that are compact and convex. These two problems link and unify a series of fundamental problems in…
We introduce Patchwork, a new general-purpose shape representation capable of modeling 2D and 3D geometry with a small number of parameters. Patchwork is grounded in a rigorous mathematical framework, providing provable complexity bounds…
Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a…
Consider a rational family of planar rational curves in a certain region of interest. We are interested in finding an approximation to the implicit representation of the envelope. Since exact implicitization methods tend to be very costly,…
To simulate open boundaries within finite computation domain, real-function coordinate transformation in the framework of generally covariant formulation of Maxwell equations is proposed. The mapping--realized with arctangent function…
We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and…
A collection of algorithms is described for numerically computing with smooth functions defined on the unit disk. Low rank approximations to functions in polar geometries are formed by synthesizing the disk analogue of the double Fourier…
Computing the volume of a polytope in high dimensions is computationally challenging but has wide applications. Current state-of-the-art algorithms to compute such volumes rely on efficient sampling of a Gaussian distribution restricted to…
A polyhedral approximation of a convex body can be calculated by solving approximately an associated multiobjective convex program (MOCP). An MOCP can be solved approximately by Benson type algorithms, which compute outer and inner…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…
In this paper we analyze several new methods for solving nonconvex optimization problems with the objective function formed as a sum of two terms: one is nonconvex and smooth, and another is convex but simple and its structure is known.…
A new algorithm for the efficient numerical approximation of weakly singular integrals over convex polytopes is introduced. Such integrals appear in the Galerkin discretizations of integral equations and nonlocal partial differential…
We expand the basic geometric elements of the simplex method to linear programs in locally convex topological vector spaces and provide conditions under which the method converges in value to optimality. This setting generalizes many…
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian…
We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates…
We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most…