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An optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications

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Abstract

We propose a modification of our MPGP algorithm for the solution of bound constrained quadratic programming problems so that it can be used for minimization of a strictly convex quadratic function subject to separable convex constraints. Our active set based algorithm explores the faces by conjugate gradients and changes the active sets and active variables by gradient projections, possibly with the superrelaxation steplength. The error estimate in terms of extreme eigenvalues guarantees that if a class of minimization problems has the spectrum of the Hessian matrix in a given positive interval, then the algorithm can find and recognize an approximate solution of any particular problem in a number of iterations that is uniformly bounded. We also show how to use the algorithm for the solution of separable and equality constraints. The power of our algorithm and its optimality are demonstrated on the solution of a problem of two cantilever beams in mutual contact with Tresca friction discretized by almost twelve millions nodal variables.

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Authors and Affiliations

  1. FEECS VŠB–Technical University of Ostrava, 17. listopadu 15, 70833, Ostrava, Czech Republic

    Zdeněk Dostál & Tomáš Kozubek

Authors
  1. Zdeněk Dostál
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  2. Tomáš Kozubek
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Correspondence to Tomáš Kozubek.

Additional information

This research has been supported by the grants GA CR 201/07/0294 and the Ministry of Education of the Czech Republic No. MSM6198910027.

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Dostál, Z., Kozubek, T. An optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications. Math. Program. 135, 195–220 (2012). https://doi.org/10.1007/s10107-011-0454-2

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