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Tensors in Optimization

  • Tamás Rapcsák
Chapter
  • 419 Downloads
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 19)

Abstract

Optimization problems can be formulated by using tensors and obtain in this way tensor field optimization problems introduced by (Rapcsák 1990. In differential geometry, theoretical physics and several applications of mathematics, the concept of tensor proved to be instrumental. In optimization theory, a new class of methods, called tensor methods, was introduced for solving systems of nonlinear equations (Schnabel and Frank, 1984) and for unconstrained optimization using second derivatives (Schnabel and Chowe, 1991). Tensor methods are of general purpose-methods intended especially for problems where the Jacobian matrix at the solution is singular or ill-conditioned. The description of a linear optimization problem in the tensor notation is proposed in order to study the integrability of vector and multivector fields associated with interior point methods by (Iri 1991). The most important feature of tensors is that their values do not change when they cause regular nonlinear coordinate transformations, and thus, this notion seems to be useful for the characterization of structural properties not depending on regular nonlinear coordinate transformations. This motivated the idea of using this notion within the framework of nonlinear optimization.

Keywords

Covariant Derivative Positive Semidefinite Tensor Field Coordinate Representation Nondegenerate Critical Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Tamás Rapcsák
    • 1
  1. 1.Computer and Automation Institute of Hungarian Academy of SciencesBudapestHungary

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