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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