# Semidefinite Representation of the k-Ellipse

@article{Nie2008SemidefiniteRO,
title={Semidefinite Representation of the k-Ellipse},
author={Jiawang Nie and Pablo A. Parrilo and Bernd Sturmfels},
journal={arXiv: Algebraic Geometry},
year={2008},
pages={117-132}
}
• Published 31 January 2007
• Mathematics
• 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…

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## References

SHOWING 1-10 OF 24 REFERENCES
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
• Mathematics, Computer Science
Math. Program.
• 2010
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
• Mathematics
• 2004
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
• Mathematics, Computer Science
Math. Program.
• 1990
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.
Applications of second-order cone programming
• Mathematics
• 1998
In a second-Order cone program (SOCP) a linear function is minimized over the intersection of an affine set and the product of second-Order (quadratic) cones. SOCPs are nonlinear convex Problems that
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
• Mathematics
• 2000
Contributing Authors. List of Figures. List of Tables. Preface. 1. Introduction H. Wolkowicz, et al. Part I: Theory. 2. Convex Analysis on Symmetric Matrices F. Jarre. 3. The Geometry of Semidefinite
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
• Mathematics
• 2003
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