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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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
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Abstract	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.
AbstractList	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. 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 ^sub 1^ linearly constrained Lagrangian (pd ^sub 1^LCL) method. Finally, a new sequential quadratic programming (pdSQP) method is proposed that uses the primal-dual augmented Lagrangian as a merit function.[PUBLICATION ABSTRACT] 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 sub(1) linearly constrained Lagrangian (pd sub(1)LCL) method. Finally, a new sequential quadratic programming (pdSQP) method is proposed that uses the primal-dual augmented Lagrangian as a merit function.
Author	Gill, Philip E. Robinson, Daniel P.
Author_xml	– sequence: 1 givenname: Philip E. surname: Gill fullname: Gill, Philip E. organization: Department of Mathematics, University of California – sequence: 2 givenname: Daniel P. surname: Robinson fullname: Robinson, Daniel P. email: robinson@maths.ox.ac.uk organization: Mathematical Institute, University of Oxford
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Keywords	Bound constrained Lagrangian methods Linearly constrained Lagrangian methods Sequential quadratic programming Augmented Lagrangian methods Primal-dual methods Nonlinear programming Nonlinear constraints
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Snippet	Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this...
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SubjectTerms	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
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Title	A primal-dual augmented Lagrangian
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