A primal-dual augmented Lagrangian

Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we consider the formulation of subproblems in which the objective function is a generalization of the Hestenes-Powell augmented Lagrangian...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 51; no. 1; pp. 1 - 25
Main Authors: Gill, Philip E., Robinson, Daniel P.
Format: Journal Article
Language:English
Published: Boston Springer US 01.01.2012
Springer Nature B.V
Subjects:
Algorithms
Availability
Constraints
Convex and Discrete Geometry
Estimates
Grants
Lagrange multiplier
Linear programming
Management Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Nonlinear programming
Operations Research
Operations Research/Decision Theory
Optimization
Quadratic programming
Software packages
Stability
Statistics
Studies
Bound constrained Lagrangian methods
Linearly constrained Lagrangian methods
Sequential quadratic programming
Augmented Lagrangian methods
Primal-dual methods
Nonlinear programming
Nonlinear constraints
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we consider the formulation of subproblems in which the objective function is a generalization of the Hestenes-Powell augmented Lagrangian function. The main feature of the generalized function is that it is minimized with respect to both the primal and the dual variables simultaneously. The benefits of this approach include: (i) the ability to control the quality of the dual variables during the solution of the subproblem; (ii) the availability of improved dual estimates on early termination of the subproblem; and (iii) the ability to regularize the subproblem by imposing explicit bounds on the dual variables. We propose two primal-dual variants of conventional primal methods: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual ℓ 1 linearly constrained Lagrangian (pd ℓ 1 LCL) method. Finally, a new sequential quadratic programming (pdSQP) method is proposed that uses the primal-dual augmented Lagrangian as a merit function.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-010-9339-1