Semidefinite Representation of the k-Ellipse

Nie, Jiawang; Parrilo, Pablo A.; Sturmfels, Bernd

doi:10.1007/978-0-387-75155-9_7

Cited by 32 publications

(36 citation statements)

References 18 publications

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“…The family of focally generated 3D elements include: sphere, Cassini surface and m-ellipsoid, [3], [6]. This paper discussed well-known focally generated 3D elements and a new type, focally-directorially generated 3D elements.…”

Section: Resultsmentioning

confidence: 99%

Modeling of focal-directorial surfaces for application in architecture

Petrusevski¹,

Petrović²,

Devetaković³

et al. 2017

FME Transaction

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Section: Resultsmentioning

confidence: 99%

Modeling of focal-directorial surfaces for application in architecture

Petrusevski¹,

Petrović²,

Devetaković³

et al. 2017

FME Transaction

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“…However, in this case the size of the matrices is much bigger. Below we present a concrete statement; see [33] for a sharper result and an explicit construction of this representation.…”

Section: Spectraplexmentioning

confidence: 97%

“…In Figure 2.6 we illustrate these convex sets for the case of a 2 × 2 symmetric matrix given by k-ellipse: We consider a class of planar convex sets defined by the algebraic curves known as k-ellipses [33]. Recall that the standard ellipse in R 2 is defined as the locus of points with the sum of distances to two fixed points (the foci) a fixed constant.…”

Section: Spectraplexmentioning

confidence: 99%

Chapter 2: Semidefinite Optimization

Parrilo¹

2012

Semidefinite Optimization and Convex Algebraic Geometry

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In this chapter we introduce one of the core theoretical and computational techniques in convex algebraic geometry, namely, semidefinite optimization. We begin by reviewing linear programming and proceed to define and discuss semidefinite programs from the algebraic, geometric, and computational perspectives. We define spectrahedra as the feasible sets of semidefinite programs, study their properties, and discuss numerous examples. Despite the many parallels, the duality theory of semidefinite optimization is more complicated than in the case of linear programming, and we elaborate on the similarities and differences. We also showcase a number of applications of semidefinite optimization in several areas of applied mathematics and engineering and give a short discussion of algorithmic and software aspects. For the convenience of the reader, we present additional background material on convex geometry and optimization in Appendix A. From Linear to Semidefinite OptimizationSemidefinite optimization is a branch of convex optimization that is of great theoretical and practical interest. Informally, the main idea is to generalize linear programming and the associated feasible sets (polyhedra) to the case where the decision variables are symmetric matrices, and the inequalities are to be understood as matrices being positive semidefinite. Formal definitions and examples will be presented shortly in Subsection 2.1.2, preceded by a review of the familiar case of linear programming. A few selected standard references for linear programming and their applications are the books [5,12,29,42].3 Downloaded 03/21/15 to 169.230.243.252. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

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“…1. This is one connected component of the real algebraic curve of degree 8 given by {(x, y) ∈ R 2 : det E (x, y, 1) = 0} where E is defined in (12) (see Nie et al [16]). The region enclosed by this 3-ellipse is the z = 1 slice of the spectrahedral cone defined by E (x, y, z) …”

Section: Example 1 (Derivative Relaxations Of a 3-ellipse)mentioning

confidence: 99%

Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

Saunderson

Parrilo

2014

Math. Program.

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We give explicit polynomial-sized (in n and k) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k in n variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we construct explicit semidefinite representations (polynomial-sized in k, m, and n) of the hyperbolicity cones associated with kth directional derivatives of polynomials of the form p(x) = det( n i=1 A i x i ) where the A i are m × m symmetric matrices. These convex cones form an analogous family of outer approximations to any spectrahedral cone. Our 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.

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Semidefinite Representation of the k-Ellipse

Cited by 32 publications

References 18 publications

Modeling of focal-directorial surfaces for application in architecture

Modeling of focal-directorial surfaces for application in architecture

Chapter 2: Semidefinite Optimization

Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

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