# 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} }

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…

## 52 Citations

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

- Mathematics, Computer ScienceSIAM J. Appl. Algebra Geom.
- 2018

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

- Computer Science, MathematicsJ. Symb. Comput.
- 2012

For a semialgebraic set K in R^n, let P"d(K)={[email protected]?R[x]"@?"d:f(u)>[email protected][email protected]?K} be the cone of polynomials in [email protected]?R^n of degrees at most d that are…

Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

- Mathematics, Computer ScienceMath. Program.
- 2015

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

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2021

The paper determines the singularities and genus of its Zariski closure in the complex projective plane and resolves an open problem stated by Nie, Parrilo and Sturmfels in 2008.

Matrix Cubes Parameterized by Eigenvalues

- Computer Science, MathematicsSIAM J. Matrix Anal. Appl.
- 2009

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

- Mathematics
- 2008

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

- Mathematics, Computer ScienceJ. Optim. Theory Appl.
- 2013

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

- Mathematics, Computer ScienceComput. Aided Geom. Des.
- 2014

It is verified that classical curve theory in Minkowski planes can be nicely developed to become a very wide and interesting research subject in the spirit of different modern fields, like Differential Geometry, Functional Analysis, Computational Geometry and related directions.

Certified Approximation Algorithms for the Fermat Point and n-Ellipses

- Computer ScienceESA
- 2021

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

- Mathematics
- 2021

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…

## References

SHOWING 1-10 OF 24 REFERENCES

n-Ellipses and the Minimum Distance Sum Problem

- Mathematics
- 1999

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 ScienceMath. 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

- Mathematics, Computer ScienceDiscret. 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 ScienceMath. 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

- Mathematics, Computer Science
- 1960

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

- Mathematics
- 1977

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…