Optimization Methods for the Sphere Packing Problem on Grassmannians

Lageman, Christian; Helmke, Uwe

doi:10.1002/9780470612217.ch22

Cited by 3 publications

(3 citation statements)

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“…In case the Clarke generalized directional derivative of f is not given explicitly (as is the case for instance in [21, Sections 3.2 and 3.3] and [31]), in Section 4 we also propose a numerical approximation which only solves (3.1). The choice of matrices {B k } is quite general; in our results they simply need to be uniformly bounded but even this property can be weakened; see for instance [42].…”

Section: A Nonsmooth Trust Region Methods On Riemannian Manifoldsmentioning

confidence: 99%

Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds

Grohs

Hosseini

2015

IMA J Numer Anal

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This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function Φ : T M → R on the tangent bundle T M , and at k-th iteration, using the restricted function Φ| Tx k M where Tx k M is the tangent space at x k , a local model function Q k that carries both first and second order information for the locally Lipschitz objective function f : M → R on a Riemannian manifold M , is defined and minimized over a trust region. We establish the global convergence of the proposed algorithm. Moreover, using the Riemannian εsubdifferential, a suitable model function is defined. Numerical experiments illustrate our results.over a restricted region centered at the current iterate. It is worth pointing out that in this model function, B k is adequately selected and the model function preserves the first and second order information of the objective function f . The

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Section: A Nonsmooth Trust Region Methods On Riemannian Manifoldsmentioning

confidence: 99%

Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds

Grohs

Hosseini

2015

IMA J Numer Anal

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“…In our experience, this ad hoc modification of the Armijo rule produces better results for the sphere packing problem. For alternative implementations of the algorithm we refer to [19].…”

Section: Sphere Packings On Grassmanniansmentioning

confidence: 99%

“…We present here a nonsmooth descent approach to this sphere packing problem. For a thorough discussion we refer the reader to [18,19]. The real Grassmann manifold is the smooth manifold of k-dimensional linear subspaces of R n .…”

Section: Sphere Packings On Grassmanniansmentioning

confidence: 99%

Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping

Dirr

Helmke

Lageman

Lagrangian and Hamiltonian Methods for Nonlinear Control 2006

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Summary. This paper presents a survey on Riemannian geometry methods for smooth and nonsmooth constrained optimization. Gradient and subgradient descent algorithms on a Riemannian manifold are discussed. We illustrate the methods by applications from robotics and multi antenna communication. Gradient descent algorithms for dextrous hand grasping and for sphere packing problems on Grassmann manifolds are presented respectively.

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A Collection of Nonsmooth Riemannian Optimization Problems

Absil

Hosseini

2019

Nonsmooth Optimization and Its Applications

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Optimization Methods for the Sphere Packing Problem on Grassmannians

Cited by 3 publications

References 11 publications

Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds

Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds

Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping

A Collection of Nonsmooth Riemannian Optimization Problems

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