Section 4.1: Minimization of Functions of One Variable
Unconstrained Optimization
4
In this chapter we study mathematical programming techniques that are commonly
used to extremize nonlinear functions of single and multiple (
n
) design variables
subject to no constraints. Although most structural optimization problems involve
constraints that bound the design space, study of the methods of unconstrained op
timization is important for several reasons.
First of all, if the design is at a stage
where no constraints are active then the process of determining a search direction and
travel distance for minimizing the objective function involves an unconstrained func
tion minimization algorithm. Of course in such a case one has constantly to watch
for constraint violations during the move in design space.
Secondly, a constrained
optimization problem can be cast as an unconstrained minimization problem even if
the constraints are active. The penalty function and multiplier methods discussed in
Chapter 5 are examples of such indirect methods that transform the constrained min
imization problem into an equivalent unconstrained problem. Finally, unconstrained
minimization strategies are becoming increasingly popular as techniques suitable for
linear and nonlinear structural analysis problems (see Kamat and Hayduk[1]) which
involve solution of a system of linear or nonlinear equations. The solution of such
systems may be posed as finding the minimum of the potential energy of the system
or the minimum of the residuals of the equations in a least squared sense.
4.1 Minimization of Functions of One Variable
In most structural design problems the objective is to minimize a function with
many design variables, but the study of minimization of functions of a single de
sign variable is important for several reasons.
First, some of the theoretical and
numerical aspects of minimization of functions of
n
variables can be best illustrated,
especially graphically, in a one dimensional space. Secondly, most methods for un
constrained minimization of functions
f
(
x
) of
n
variables rely on sequential one
dimensional minimization of the function along a set of prescribed directions,
s
k
, in
the multidimensional design space
R
n
. That is, for a given design point
x
0
and a
specified search direction at that point
s
0
, all points located along that direction can
be expressed in terms of a single variable
α
by
x
=
x
0
+
α
s
0
,
(4
.
1
.
1)
115
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Chapter 4: Unconstrained Optimization
where
α
is usually referred to as the step length. The function
f
(
x
) to be minimized
can, therefore, be expressed as
f
(
x
) =
f
(
x
0
+
α
s
0
) =
f
(
α
)
.
(4
.
1
.
2)
Thus, the minimization problem reduces to finding the value
α
*
that minimizes the
function,
f
(
α
).
In fact, one of the simplest methods used in minimizing functions
of
n
variables is to seek the minimum of the objective function by changing only
one variable at a time, while keeping all other variables fixed, and performing a one
dimensional minimization along each of the coordinate directions of an
n
dimensional
design space. This procedure is called the
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 PETERIFJU
 Optimization, The Land

Click to edit the document details