EGM6365 Homework #2
1. Consider the following design optimization problem:
22
12
1
11
3
32
1
Minimize
( )
4
4
Subject to
( )
0
()
0
(
1
) 0
fx
x
x
gx
x
=+− +
=− ≤
=−
≤
=−− ≤
x
x
x
x
(i)
Find the optimum point graphically
(ii)
Show that the optimum point does not satisfy KT condition. Explain
2. An engineering design problem is formulated as:
Minimize
121
2
2
5
2
1
0
x
x
x
=+ − − +
x
Subject to the constraints
2
112
23
0
326
0
hx x
gxx
=+ −=
=+
−
≤
In all of the following questions, justify your answers.
(i)
Write KT necessary conditions
(ii)
How many cases are there to be considered? Identify those cases.
(iii)
Find the solution for the case where g
1
is active. Is this acceptable case?
(iv)
Regardless of the solution you obtained in (iii), suppose the Lagrange
multiplier for the constraint
h
1
= 0 is
λ
1
= 2 and the Lagrange multiplier for
the constraint
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 Spring '08
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 Optimization, 0 g, optimization problem, KT necessary conditions

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