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Interior Epigraph Directions method for nonsmooth and nonconvex optimization via generalized augmented Lagrangian duality

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  • Regina Burachik
  • Wilhelm Freire
  • C. Kaya

Abstract

We propose and study a new method, called the Interior Epigraph Directions (IED) method, for solving constrained nonsmooth and nonconvex optimization. The IED method considers the dual problem induced by a generalized augmented Lagrangian duality scheme, and obtains the primal solution by generating a sequence of iterates in the interior of the dual epigraph. First, a deflected subgradient (DSG) direction is used to generate a linear approximation to the dual problem. Second, this linear approximation is solved using a Newton-like step. This Newton-like step is inspired by the Nonsmooth Feasible Directions Algorithm (NFDA), recently proposed by Freire and co-workers for solving unconstrained, nonsmooth convex problems. We have modified the NFDA so that it takes advantage of the special structure of the epigraph of the dual function. We prove that all the accumulation points of the primal sequence generated by the IED method are solutions of the original problem. We carry out numerical experiments by using test problems from the literature. In particular, we study several instances of the Kissing Number Problem, previously solved by various approaches such as an augmented penalty method, the DSG method, as well as several popular differentiable solvers. Our experiments show that the quality of the solutions obtained by the IED method is comparable with (and sometimes favourable over) those obtained by the differentiable solvers. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Regina Burachik & Wilhelm Freire & C. Kaya, 2014. "Interior Epigraph Directions method for nonsmooth and nonconvex optimization via generalized augmented Lagrangian duality," Journal of Global Optimization, Springer, vol. 60(3), pages 501-529, November.
    • Handle: RePEc:spr:jglopt:v:60:y:2014:i:3:p:501-529
    DOI: 10.1007/s10898-013-0108-4
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    References listed on IDEAS

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    1. Regina S. Burachik & C. Yalçın Kaya, 2010. "A Deflected Subgradient Method Using a General Augmented Lagrangian Duality with Implications on Penalty Methods," Springer Optimization and Its Applications, in: Regina S. Burachik & Jen-Chih Yao (ed.), Variational Analysis and Generalized Differentiation in Optimization and Control, pages 109-132, Springer.
    2. Regina Burachik & Alfredo Iusem & Jefferson Melo, 2010. "A primal dual modified subgradient algorithm with sharp Lagrangian," Journal of Global Optimization, Springer, vol. 46(3), pages 347-361, March.
    3. R. S. Burachik & A. N. Iusem & J. G. Melo, 2010. "Duality and Exact Penalization for General Augmented Lagrangians," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 125-140, October.
    4. Regina Burachik & C. Kaya & Musa Mammadov, 2010. "An inexact modified subgradient algorithm for nonconvex optimization," Computational Optimization and Applications, Springer, vol. 45(1), pages 1-24, January.
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