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Optimal iterative QP and QPQC algorithms

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Abstract

We review our recent results in the development of optimal algorithms for the minimization of a strictly convex quadratic function subject to separable convex inequality constraints and/or linear equality constraints. A unique feature of our algorithms is the theoretically supported bound on the rate of convergence in terms of the bounds on the spectrum of the Hessian of the cost function, independent of representation of the constraints. When applied to the class of convex QP or QPQC problems with the spectrum in a given positive interval and a sparse Hessian matrix, the algorithms enjoy optimal complexity, i.e., they can find an approximate solution at the cost that is proportional to the number of unknowns. The algorithms do not assume representation of the linear equality constraints by full rank matrices. The efficiency of our algorithms is demonstrated by the evaluation of the projection of a point to the intersection of the unit cube and unit sphere with hyperplanes.

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Authors and Affiliations

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

    Zdeněk Dostál & Lukáš Pospíšil

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  1. Zdeněk Dostál
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  2. Lukáš Pospíšil
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Correspondence to Zdeněk Dostál.

Additional information

This paper has been supported by the IT4Innovations Center of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme ‘Research and Development for Innovations’ funded by the Structural Funds of the European Union and the budget of the Czech Republic and by the Ministry of Education of the Czech Republic under contract No. MSM6198910027.

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Dostál, Z., Pospíšil, L. Optimal iterative QP and QPQC algorithms. Ann Oper Res 243, 5–18 (2016). https://doi.org/10.1007/s10479-013-1479-0

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