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A semidefinite framework for trust region subproblems with applications to large scale minimization

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Abstract

Primal-dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem (TRS). This latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and, therefore, it is a generalization of the minimum eigenvalue problem. The importance of (TRS) is due to the fact that it provides the step in trust region minimization algorithms. The semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for (TRS). In particular, a dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem. This method uses the Lanczos algorithm for the smallest eigenvalue as a black box. Therefore, the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited. A primal simplex type method provides steps for the so-called hard case. Extensive numerical tests for large sparse problems are discussed. These tests show that the cost of the algorithm is 1 +α(n) times the cost of finding a minimum eigenvalue using the Lanczos algorithm, where 0<α(n)<1 is a fraction which decreases as the dimension increases.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Graz, Kopernikusgasse 24, A-8010, Graz, Austria

    Franz Rendl

  2. Department of Combinatorics and Optimization, University of Waterloo, N2L 3GI, Waterloo, Ontario, Canada

    Henry Wolkowicz

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  1. Franz Rendl
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  2. Henry Wolkowicz
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Correspondence to Franz Rendl.

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Research supported by the National Science and Engineering Research Council Canada.

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Rendl, F., Wolkowicz, H. A semidefinite framework for trust region subproblems with applications to large scale minimization. Mathematical Programming 77, 273–299 (1997). https://doi.org/10.1007/BF02614438

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  • DOI: https://doi.org/10.1007/BF02614438

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