scholarly journals Optimization Methods on Riemannian Manifolds and Their Application to Shape Space

2012 ◽  
Vol 22 (2) ◽  
pp. 596-627 ◽  
Author(s):  
Wolfgang Ring ◽  
Benedikt Wirth
Keyword(s):  
Riemannian Manifolds ◽  
Optimization Methods ◽  
Shape Space

scholarly journals Optimization Methods on Riemannian Manifolds via Extremum Seeking Algorithms

2018 ◽  
Vol 56 (5) ◽  
pp. 3867-3892
Author(s):  
Farzin Taringoo ◽  
Peter M. Dower ◽  
Dragan Nešić ◽  
Ying Tan
Keyword(s):  
Riemannian Manifolds ◽  
Optimization Methods ◽  
Extremum Seeking

scholarly journals Riemannian optimization of isometric tensor networks

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Markus Hauru ◽  
Maarten Van Damme ◽  
Jutho Haegeman
Keyword(s):  
Riemannian Manifolds ◽  
State Of The Art ◽  
Optimization Methods ◽  
Quantum Circuits ◽  
Matrix Product ◽  
Riemannian Optimization ◽  
Tensor Networks ◽  
Gradient Based ◽  
Nonlinear Conjugate Gradient ◽  
Quasi Newton

Several tensor networks are built of isometric tensors, i.e. tensors satisfying W\dagger W = \mathbb{1}W†W=1. Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.

Optimization Methods in Finance

2018 ◽  
Author(s):  
Gérard Cornuéjols ◽  
Javier Peña ◽  
Reha Tütüncü
Keyword(s):  
Optimization Methods

Optimization Methods in Finance

2006 ◽  
Author(s):  
Gerard Cornuejols ◽  
Reha Tutuncu
Keyword(s):  
Optimization Methods

Economics of coated paper production made from deinked pulp

TAPPI Journal ◽  
2013 ◽  
Vol 12 (4) ◽  
pp. 19-27
Author(s):  
PATRICK HUBER ◽  
LAURENT LYANNAZ ◽  
BRUNO CARRÉ
Keyword(s):  
Mechanical Properties ◽  
Optimization Methods ◽  
Coating Process ◽  
Coated Paper ◽  
Paper Production ◽  
Base Paper

The fraction of deinked pulp for coated paper production is continually increasing, with some mills using 100% deinked pulp for the base paper. The brightness of the coated paper made from deinked pulp may be reached through a combination of more or less extensive deinking, compensated by appropriate coating, to optimize costs overall. The authors proposed general optimization methods combined with Kubelka-Munk multilayer calculations to find the most economical combination of deinking and coating process that would produce a coated paper made from DIP, at a given target brightness, while maintaining mechanical properties.

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