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Sequential equality-constrained optimization for nonlinear programming

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Abstract

A novel idea is proposed for solving optimization problems with equality constraints and bounds on the variables. In the spirit of sequential quadratic programming and sequential linearly-constrained programming, the new proposed approach approximately solves, at each iteration, an equality-constrained optimization problem. The bound constraints are handled in outer iterations by means of an augmented Lagrangian scheme. Global convergence of the method follows from well-established nonlinear programming theories. Numerical experiments are presented.

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Notes

  1. The inertia is freely computed as an outcome of the matrix factorization performed by the Harwell subroutine MA57. Given an indefinite sparse symmetric matrix A, subroutine MA57 computes the factorization given by \(PAP^T = LDL^T\), where P is a permutation matrix, L is lower triangular, and D is block diagonal with \(1 \times 1\) or \(2 \times 2\) diagonal blocks. By the Sylvester law of inertia, matrices A and D have the same inertia and computing the inertia of D is trivial (it is necessary to check the sign or to compute the eigenvalues of its \(1 \times 1\) or \(2 \times 2\) diagonal blocks).

  2. When analyzing the output of Algencan, the reason that motivates using a strict criterion that considers that the primal active variables are those exactly equal to zero is that the subproblems solved by Algencan preserve non-negative of the primal variables all along the calculations. Moreover, the searches used in the Algencan subproblems’ solver make it almost impossible the existence of positive primal variables with very small values. On the other hand, in SECO, the bound constraints may be violated (hopefully slightly) or variables may be strictly positive with tiny values, so that tolerances are necessary to declare almost feasibility and activity.

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Acknowledgments

This work was supported by PRONEX-CNPq/FAPERJ E-26/111.449/2010-APQ1, FAPESP (Grants 2010/10133-0, 2013/03447-6, 2013/05475-7, 2013/07375-0, and 2015/02528-8), and CNPq (Grants 309517/2014-1 and 303750/2014-6).

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Authors and Affiliations

  1. Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil

    E. G. Birgin

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

    L. F. Bueno

  3. Department of Applied Mathematics, Institute of Mathematics, Statistics, and Scientific Computing, State University of Campinas, Campinas, SP, Brazil

    J. M. Martínez

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  1. E. G. Birgin
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  2. L. F. Bueno
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  3. J. M. Martínez
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Correspondence to E. G. Birgin.

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Birgin, E.G., Bueno, L.F. & Martínez, J.M. Sequential equality-constrained optimization for nonlinear programming. Comput Optim Appl 65, 699–721 (2016). https://doi.org/10.1007/s10589-016-9849-6

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