Graphics Programs Reference
In-Depth Information
10
Introduction to Optimization
Find
x
that minimizes
F
(
x
)subject to
g
(
x
)
=
0,
h
(
x
)
≥
0
10.1
Introduction
Optimization
is the termoftenused forminimizing ormaximizing a function. It issuf-
ficienttoconsider the problemofminimization only;maximization of
F
(
x
) is achieved
by simplyminimizing
−
F
(
x
). In engineering,optimizationis closelyrelated to design.
The function
F
(
x
), called the
merit function
or
objective function
, is the quantity that
we wish to keepassmall as possible, such as cost orweight. The components of
x
,
known as the
design variables
, are the quantities that we arefree to adjust. Physical
dimensions (lengths, areas, angles, etc.) arecommon examples of design variables.
Optimizationis a largetopicwithmany topics dedicated to it. The bestwecando in
limited space istointroduce a fewbasic methodsthat are good enoughforproblems
that are reasonablywell behaved and don't involve too many design variables.By
omitting the moresophisticatedmethods, we may actuallynot miss all that much.
All optimizationalgorithms are unreliable to adegree—any oneofthemmay workon
one problem and failonanother. As arule of thumb, by going up in sophisticationwe
gain computationalefficiency, but not necessarilyreliability.
The algorithmsfor minimizationare iterative procedures that requirestarting
values of the design variables
x
. If
F
(
x
)hasseveral local minima, the initial choice of
x
determines which of these will becomputed. There is no guaranteed way of finding
the globaloptimal point. Onesuggestedprocedure istomake severalcomputerruns
using differentstarting points and pick the best result.
More often than not, the design is also subjected to restrictions, or
constraints
,
which may have the form of equalities orinequalities.As an example, take the min-
imum weight design of aroof truss thathastocarry a certain loading. Assumethat
382
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