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Nonlinear coordinate representations of smooth optimization problems

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Abstract

Nonlinear coordinate representations of smooth optimization problems are investigated from the point of view of variable metric algorithms. In other words, nonlinear coordinate systems, in the sense of differential geometry, are studied by taking into consideration the structure of smooth optimization problems and variable metric methods.

Both the unconstrained and constrained cases are discussed. The present approach is based on the fact that the nonlinear coordinate transformation of an optimization problem can be replaced by a suitable Riemannian metric belonging to the Euclidean metric class. In the case of equality and inequality constraints, these questions are related closely to the right inverses of full-rank matrices; therefore, their characterization is a starting point of the present analysis. The main results concern a new subclass of nonlinear transformations in connection with the common supply of coordinates to two Riemannian manifolds, one immersed in the other one. This situation corresponds to the differentiable manifold structure of nonlinear optimization problems and improves the insight into the theoretical background of variable metric algorithms. For a wide class of variable metric methods, a convergence theorem in invariant form (not depending on coordinate representations) is proved. Finally, a problem of convexification by nonlinear coordinate transformations and image representations is studied.

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Authors and Affiliations

  1. Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary

    T. Rapcsák (Senior Researcher) & T. T. Thang (Researcher)

Authors
  1. T. Rapcsák
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  2. T. T. Thang
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Additional information

Communicated by F. Giannessi

This research was supported by the Hungarian National Research Foundation, Grant Nos. OTKA-2568 and OTKA-2116, and by the Project “Trasporti” of the Italian National Research Council (CNR).

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Rapcsák, T., Thang, T.T. Nonlinear coordinate representations of smooth optimization problems. J Optim Theory Appl 86, 459–489 (1995). https://doi.org/10.1007/BF02192090

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