Semidefinite Representation of the k-Ellipse

  title={Semidefinite Representation of the k-Ellipse},
  author={Jiawang Nie and Pablo A. Parrilo and Bernd Sturmfels},
  journal={arXiv: Algebraic Geometry},
The k-ellipse is the plane algebraic curve consisting of all points whose sum of distances from k given points is a fixed number. The polynomial equation defining the k-ellipse has degree 2 k if k is odd and degree \( 2^k - \left( {\begin{array}{*{20}c} k \\ {k/2} \\ \end{array} } \right) \) if k is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted k-ellipses and k-ellipsoids in arbitrary dimensions… 
Semidefinite Representation for Convex Hulls of Real Algebraic Curves
We show that the closed convex hull of any one-dimensional semialgebraic subset of $\mathbb{R}^n$ is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of
Discriminants and nonnegative polynomials
Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones
These representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their (less well understood) dual cones.
Singularities and genus of the k-ellipse
Matrix Cubes Parameterized by Eigenvalues
A linear matrix inequality (LMI) representation is given for the convex set of all feasible instances, and its boundary is studied from the perspective of algebraic geometry.
Matrix Cubes Parametrized by Eigenvalues
An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix cube problems whose constraints are the minimum and maximum eigenvalue function
The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach
We give a discrete geometric (differential-free) proof of the theorem underlying the solution of the well known Fermat–Torricelli problem, referring to the unique point having minimal distance sum to
Classical curve theory in normed planes
Certified Approximation Algorithms for the Fermat Point and n-Ellipses
A certified subdivision algorithm for computing x̃, enhanced by Newton operator techniques is devised, and the classic Weiszfeld-Kuhn iteration scheme for x∗ is revisited, turning it into an ε-approximate Fermat point algorithm.
New Fixed Figure Results with the Notion of $k$-Ellipse
In this paper, as a geometric approach to the fixed-point theory, we prove new fixed-figure results using the notion of k-ellipse on a metric space. For this purpose, we are inspired by the Caristi


n-Ellipses and the Minimum Distance Sum Problem
at the level f(r) = k. We show that if k is sufficiently large (as explained in Theorem 6), then the n-ellipse is a piecewise smooth Jordan curve whose interior is convex; it is nonsmooth only where
The algebraic degree of semidefinite programming
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of
A quadratic bound for the determinant and permanent problem
The determinantal complexity of a polynomial f is defined here as the minimal size of a matrix M with affine entries such that f = det M. This function gives a minoration of the more traditional size
The algebraic degree of geometric optimization problems
  • C. Bajaj
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 1988
Galoois methods are applied to certain fundamental geometric optimization problems whose exact computational complexity has been an open problem for a long time and show that the classic Weber problem, along with the line-restricted Weber problem and itsthree-dimensional version are in general not solvable by radicals over the field of rationals.
Algebraic optimization: The Fermat-Weber location problem
This work exhibits an explicit solution to the strong separation problem associated with the Fermat-Weber model and shows that anε-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.
Introduction to Matrix Analysis
This book discusses Maximization, Minimization, and Motivation, which is concerned with the optimization of Symmetric Matrices, and its applications in Programming and Mathematical Economics.
Handbook of semidefinite programming : theory, algorithms, and applications
The aim of this monograph is to provide a Discussion of the Foundations of Semidefinite Programming and its Applications, as well as some Applications and Extensions, which were developed after the original book was written.
k-Elliptic Optimization for Locational Problems Under Constraints
A solution is presented to a problem of finding a location with a minimum sum of weighted distances from a given set of fixed points. This is obtained by controlling the distance parameter of a
Linear matrix inequality representation of sets
This article concerns the question, Which subsets of ℝm can be represented with linear matrix inequalities (LMIs)? This gives some perspective on the scope and limitations of one of the most powerful