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Primal-Dual Active-Set Methods for Large-Scale Optimization

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  • Daniel P. Robinson

    (Johns Hopkins University)

Abstract

In this paper, we introduce two primal-dual active-set methods for solving large-scale constrained optimization problems. The first method minimizes a sequence of primal-dual augmented Lagrangian functions subject to bounds on the primal variables and artificial bounds on the dual variables. The basic structure is similar to the well-known optimization package Lancelot (Conn, et al. in SIAM J Numer Anal 28:545–572, 1991), which uses the traditional primal augmented Lagrangian function. Like Lancelot, our algorithm may use gradient projection-based methods enhanced by subspace acceleration techniques to solve each subproblem and therefore may be implemented matrix-free. The artificial bounds on the dual variables are a unique feature of our method and serve as a form of dual regularization. Our second algorithm is a two-phase method. The first phase computes iterates using our primal-dual augmented Lagrangian algorithm, which benefits from using cheap gradient projections and matrix-free linear CG calculations. The final iterate produced during this phase is then used as input for phase two, which is a stabilized sequential quadratic programming method (Gill and Robinson in SIAM J Opt 1–45, 2013). Obtaining superlinear local convergence under weak assumptions is an important benefit of the transition to a stabilized sequential quadratic programming algorithm. Interestingly, the bound-constrained subproblem used in phase one is equivalent to the stabilized subproblem used in phase two under certain assumptions. This fact makes our choice of algorithms a natural one.

Suggested Citation

  • Daniel P. Robinson, 2015. "Primal-Dual Active-Set Methods for Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 137-171, July.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:1:d:10.1007_s10957-015-0708-x
    DOI: 10.1007/s10957-015-0708-x
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    References listed on IDEAS

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    1. C. V. Rao & S. J. Wright & J. B. Rawlings, 1998. "Application of Interior-Point Methods to Model Predictive Control," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 723-757, December.
    2. Philip Gill & Daniel Robinson, 2012. "A primal-dual augmented Lagrangian," Computational Optimization and Applications, Springer, vol. 51(1), pages 1-25, January.
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