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Lecture13-2003

# Lecture13-2003 - Lecture XIII Algorithms for Nonlinear...

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Lecture XIII Algorithms for Nonlinear Constraints I. General Objective A. The general objective is to maximize or minimize a nonlinear objective function subject to nonlinear constraints. Gill, Murray, and Wright lay out the basic problems: NEP f x st c x i t x i min ( ) ( ) ,.. = = 0 1 and NIP f x st c x i t x i min ( ) ( ) ,... = 0 1 B. These optimization problems are much more difficult than the equality or inequality constraints posed in the preceding sections due to the difficulties involved in maintaining feasiblity. 1. By the formulation of the null-space in the linear equality scenario, it was always possible to guarantee that x k +1 was feasible given that x k was feasible. 2. Similarly, expansion to linear inequality constraints only added caveats to the step length algorithm and checks on whether a constraint could be deleted. 3. However, the solution of nonlinear constraints may be difficult, if not impossible, without the incorporation of a nonlinear objective function. C. I want to discuss three different methodologies for optimization under nonlinear constraints. The first two are the use of penalty and barrier functions. The third procedure is the projected augmented Lagrangian method.

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Lecture13-2003 - Lecture XIII Algorithms for Nonlinear...

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