Related papers: Dualities in Convex Algebraic Geometry
In this paper, we introduce new properties of the relative interior calculus for nearly convex sets, functions, and set-valued mappings. These properties are important for the development of duality theory in optimization. Then we…
This article describes a method to compute successive convex approximations of the convex hull of a set of points in R^n that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of…
The notion of geometrical duality is discussed in the context of both Brans-Dicke theory and general relativity. It is shown that, in some particular solutions, the spacetime singularities that arise in usual Riemannian general relativity…
The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest…
The robust truss topology optimization against the uncertain static external load can be formulated as mixed-integer semidefinite programming. Although a global optimal solution can be computed with a branch-and-bound method, it is very…
Some properties of generalized convexity for sets and for functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is…
In this paper, we focus on the following general shape optimization problem: $$ \min\{J(\Om), \Om convex, \Om\in\mathcal S_{ad}\}, $$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\to\R$ is a…
We discuss a weak constraint qualification for conic linear programs and its applications for a few classes of cones. This constraint qualification is used to give a solution to a problem proposed by Shapiro and Z\v{a}linescu and show that…
We study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one…
Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify…
Error bounds are central objects in optimization theory and its applications. They were for a long time restricted only to the theory before becoming over the course of time a field of itself. This paper is devoted to the study of error…
An effort has been made to show mathematicians some new ideas applied to image analysis. Gray images are presented as tilings. Based on topological properties of the tiling, a number of gray convex hulls: maximal, minimal, and oriented ones…
The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact…
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…
The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying…
The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral…
We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…
We define a class of L-convex-concave subsets of $\Bbb{R}P^n$, where L is a projective subspace of dimension l in $\Bbb{R}P^n$. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on…
In the field of algebraic geometric codes (AG codes), the characterization of dual codes has long been a challenging problem which relies on differentials. In this paper, we provide some descriptions for certain differentials utilizing…
Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a…