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 nullspace 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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 Fall '08
 Moss
 Linear Programming, Optimization, Charles B. Moss, penalty parameter, Nonlinear Constraints

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