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Towards an efficient augmented Lagrangian method for convex quadratic programming

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Abstract

Interior point methods have attracted most of the attention in the recent decades for solving large scale convex quadratic programming problems. In this paper we take a different route as we present an augmented Lagrangian method for convex quadratic programming based on recent developments for nonlinear programming. In our approach, box constraints are penalized while equality constraints are kept within the subproblems. The motivation for this approach is that Newton’s method can be efficient for minimizing a piecewise quadratic function. Moreover, since augmented Lagrangian methods do not rely on proximity to the central path, some of the inherent difficulties in interior point methods can be avoided. In addition, a good starting point can be easily exploited, which can be relevant for solving subproblems arising from sequential quadratic programming, in sensitivity analysis and in branch and bound techniques. We prove well-definedness and finite convergence of the method proposed. Numerical experiments on separable strictly convex quadratic problems formulated from the Netlib collection show that our method can be competitive with interior point methods, in particular when a good initial point is available and a second-order Lagrange multiplier update is used.

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Notes

  1. Freely available at: www.ime.usp.br/~egbirgin/tango.

  2. Problems forplan, gfrd-pnc and pilot.we were removed from the Netlib collection, since they were not available in the presolved library [23].

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Acknowledgements

We dedicate this paper, in honor of his 70th birthday, to Professor J. M. Martínez, who greatly influenced our careers. Moreover, we would like to thank him for important discussions we had in designing the original ideas of this work. In addition, we would like to express gratitude to Professors Roger Behling, Ernesto Birgin and Thadeu Senne for their discussions along different stages of the trajectory of the results presented here. We also thank the anonymous referees whose valuable suggestions improved this paper.

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Authors and Affiliations

  1. Institute of Science and Technology, Federal University of São Paulo, São José dos Campos, SP, Brazil

    Luís Felipe Bueno

  2. Department of Applied Mathematics, University of São Paulo, São Paulo, SP, Brazil

    Gabriel Haeser

  3. Department of Mathematics, Federal University of Santa Catarina, Blumenau, SC, Brazil

    Luiz-Rafael Santos

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  1. Luís Felipe Bueno
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  2. Gabriel Haeser
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  3. Luiz-Rafael Santos
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Correspondence to Luís Felipe Bueno.

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This work was supported by Brazilian Agencies Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) (Grants 2015/02528-8, 2017/18308-2 and 2018/24293-0), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

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Bueno, L.F., Haeser, G. & Santos, LR. Towards an efficient augmented Lagrangian method for convex quadratic programming. Comput Optim Appl 76, 767–800 (2020). https://doi.org/10.1007/s10589-019-00161-2

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