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

Daniel P. Robinson ()
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Daniel P. Robinson: Johns Hopkins University

Journal of Optimization Theory and Applications, 2015, vol. 166, issue 1, No 7, 137-171

Abstract: 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.

Keywords: Constrained optimization; Primal-dual; Augmented Lagrangian; Large scale; 49M37; 65K05; 65K10; 90C06; 90C26; 90C30; 90C55 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10957-015-0708-x

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