Chapter 4 Basic Concepts of Optimization MEF

Chapter 4

Chapter 4

Basic Concepts of Optimization

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Chapter 4

Objective Function
issues:
(1) continuity of f (discrete cases, e.g., insulation,
pipe sizes)
(2) convexity, concavity
(3) stationary points (necessary condition) f  0
X i
(4) quadratic vs. non-quadratic functions
(5) scalar vs. vector case
f
df

0
0
X i
dX
(6) sufficiency condition (min)
d2 f
0
H positive definite
2
dX
all eigenvalue s of H  0
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Chapter 4

Continuity of Functions
• In analytical or numerical optimization,
preferable and more convenient to work
with continuous functions of one or more
variables than with functions containing

discontinuities.
• Functions having continuous derivatives
are preferred.
• The property of continuity 
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A discontinuity in a function may or may not cause difficulty in optimization.

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FIGURE 4.9
Convex and nonconvex sets.
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f ( X )  X  2 X 2 X  X  X  2 X1  5
4
1

2
1

2
2

2
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Chapter 4

f
 0  4 X 13  4 X 2 X 1  2 X 1  2
X 1
f
2
 0  2 X 1  2 X 2
X 2
min at (1,1)
12 X 12  4 X 2  2  4 X 1 
H

 4 X1
2 

 10  4
at (1,1), H  
0

 4 2 

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