Type of Document Master's Thesis Author Maruhn, Jan Hendrik URN etd-05022001-131450 Title An augmented Lagrangian algorithm for optimization with equality constraints in Hilbert spaces Degree Master of Science Department Mathematics Advisory Committee
Advisor Name Title Sachs, Ekkehard W. Committee Chair King, Belinda B. Committee Member Lang, James R. Committee Member Rogers, Robert C. Committee Member Keywords
- equality constraints
- optimization in Hilbert spaces
- augmented Lagrangian methods
- nonlinear optimization
- discrete approximations
Date of Defense 2001-04-30 Availability unrestricted AbstractSince augmented Lagrangian methods were introduced by Powell and Hestenes, this class of methods has been investigated very intensively. While the finite dimensional case has been treated in a satisfactory manner, the infinite dimensional case is studied much less.
The general approach to solve an infinite dimensional optimization problem subject to equality constraints is as follows: First one proves convergence for a basic algorithm in the Hilbert space setting. Then one discretizes the given spaces and operators in order to make numerical computations possible. Finally, one constructs a discretized version of the infinite dimensional method and tries to transfer the convergence results to the finite dimensional version of the basic algorithm.
In this thesis we discuss a globally convergent augmented Lagrangian algorithm and discretize it in terms of functional analytic restriction operators. Given this setting, we prove global convergence of the discretized version of this algorithm to a stationary point of the infinite dimensional optimization problem. The proposed algorithm includes an explicit rule of how to update the discretization level and the penalty parameter from one iteration to the next one - questions that had been unanswered so far. In particular the latter update rule guarantees that the penalty parameters stay bounded away from zero which prevents the Hessian of the discretized augmented Lagrangian functional from becoming more and more ill conditioned.
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