On the symmetry of induced norm cones

Orlitzky, Michael

doi:10.1080/02331934.2020.1789132

Cited by 4 publications

(2 citation statements)

References 10 publications

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“…However, checking that L n+1 p is never self-dual with respect to any inner product requires more work, see [IL19, Theorem 11 and Corollary 14] or [LLP21, Section 5.2]. This subtlety is important because, for example, a cone may become symmetric under a change of inner product [Orl22], and symmetric cones enjoy many favourable theoretical properties [FK94,Fay08].…”

Section: Inner Products and Self-dualitymentioning

confidence: 99%

Self-dual polyhedral cones and their slack matrices

Gouveia¹,

Lourenço²

2022

Preprint

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We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the DNN matrices, which correspond to a family of extreme rays not previously described. This leads to a curious consequence for 5 × 5 DNN matrices: the extreme rays that are not rank 1 must come from slack matrices of self-dual cones over a pentagon. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.

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Section: Inner Products and Self-dualitymentioning

confidence: 99%

Self-dual polyhedral cones and their slack matrices

Gouveia¹,

Lourenço²

2022

Preprint

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“…For example, a common source of confusion is as follows: in order to disprove that a cone is symmetric, it is not enough to show that K * = K. The reason is that the selfduality requirement, in the Jordan algebra context, can be met by arbitrary inner products, and K * changes if •, • varies. An interesting discussion on symmetrizing a cone by changing the inner product can be seen in [28]. In fact, the existence of an inner product making a cone K self-dual is equivalent to the existence of a positive definite matrix Q such that QK = K * , where K * is the dual cone obtained under the usual Euclidean inner product.…”

Section: Self-duality and Homogeneity Of P-conesmentioning

confidence: 99%

Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond

Lindstrom¹,

Lourenço²,

Pong³

2021

Preprint

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We prove tight Hölderian error bounds for all p-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate p-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with pnorm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of p. Another application is a (relatively) simple proof that most p-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight.

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Self-Dual Polyhedral Cones and Their Slack Matrices

Gouveia

Lourenço

2023

SIAM J. Matrix Anal. Appl.

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On the symmetry of induced norm cones

Cited by 4 publications

References 10 publications

Self-dual polyhedral cones and their slack matrices

Self-dual polyhedral cones and their slack matrices

Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond

Self-Dual Polyhedral Cones and Their Slack Matrices

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