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```									ENGINEERING OPTIMIZATION
Methods and Applications
A. Ravindran, K. M. Ragsdell, G. V. Reklaitis

Book Review

Page 1
Chapter 5: Constrained Optimality
Criteria
Part 1: Ferhat Dikbiyik
Part 2:Yi Zhang

Review Session
July 2, 2010

Page 2
Constraints:

Page 3
Constraints:

reduces the region in
which we search for
optimum.

Page 4
Constraints:

makes optimization
process very
complicated

Page 5
( x  2)   2

x4
25

20

15

10

5

0
-2        -1        0   1   2   3   4   5   6

Page 6
Outline of Part 1
• Equality-Constrained Problems

• Lagrange Multipliers

• Economic Interpretation of Lagrange Multipliers

• Kuhn-Tucker Conditions

• Kuhn-Tucker Theorem

Page 7
Outline of Part 1
• Equality-Constrained Problems

• Lagrange Multipliers

• Economic Interpretation of Lagrange Multipliers

• Kuhn-Tucker Conditions

• Kuhn-Tucker Theorem

Page 8
Equality-Constrained Problems

GOAL
solving the problem as an unconstrained
problem by explicitly eliminating K
independent variables using the equality
constraints

Page 9
Example 5.1

Page 10
What if?

Page 11
Outline of Part 1
• Equality-Constrained Problems

• Lagrange Multipliers

• Economic Interpretation of Lagrange Multipliers

• Kuhn-Tucker Conditions

• Kuhn-Tucker Theorem

Page 12
Lagrange Multipliers

Converting constrained problem to an
unconstrained problem with help of
certain unspecified parameters known
as   Lagrange Multipliers

Page 13
Lagrange Multipliers

Lagrange
function

Page 14
Lagrange Multipliers

Lagrange
multiplier

Page 15
Example 5.2

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Test whether the stationary point
corresponds to a minimum

positive definite

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Page 19
Example 5.3

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Page 21
Page 22
max

positive
definite

negative
definite

Page 23
Outline of Part 1
• Equality-Constrained Problems

• Lagrange Multipliers

• Economic Interpretation of Lagrange Multipliers

• Kuhn-Tucker Conditions

• Kuhn-Tucker Theorem

Page 24
Economic Interpretation of Lagrange Multipliers

The Lagrange multipliers have an
important economic interpretation as
shadow prices of the constraints, and
their optimal values are very useful in
sensitivity analysis.

Page 25
Outline of Part 1
• Equality-Constrained Problems

• Lagrange Multipliers

• Economic Interpretation of Lagrange Multipliers

• Kuhn-Tucker Conditions

• Kuhn-Tucker Theorem

Page 26
Kuhn-Tucker Conditions

Page 27
NLP problem

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Kuhn-Tucker conditions
(aka Kuhn-Tucker Problem)

Page 29
Example 5.4

Page 30
Example 5.4

Page 31
Example 5.4

Page 32
Outline of Part 1
• Equality-Constrained Problems

• Lagrange Multipliers

• Economic Interpretation of Lagrange Multipliers

• Kuhn-Tucker Conditions

• Kuhn-Tucker Theorem

Page 33
Kuhn-Tucker Theorems

1. Kuhn – Tucker Necessity Theorem

2. Kuhn – Tucker Sufficient Theorem

Page 34
Kuhn-Tucker Necessity Theorem

Let
• f, g, and h be differentiable functions
• x* be a feasible solution to the NLP problem.
•
•                 and          for k=1,….,K are
linearly independent

Page 35
Kuhn-Tucker Necessity Theorem
Let
h be differentiable functions
f, g, andConstraint qualification x* be a
feasible solution to the NLP problem.

•               and          for k=1,….,K are
linearly independent at the optimum

! is an to verify, since it requires that
If x*Hardoptimal solution to the NLP problem,
the exists a (u*, v*) such that (x*,u*,
then there optimum solution be known v*)
solves the KTP given by KTC.!
beforehand
Page 36
Kuhn-Tucker Necessity Theorem

For certain special NLP problems, the
constraint qualification is satisfied:
1. When all the inequality and equality
constraints are linear
2. When all the inequality constraints are
concave functions and equality
constraints are linear
! When the constraint qualification is
not met at the optimum, there may not
exist a solution to the KTP
Page 37
Example 5.5

x* = (1, 0)

and       for k=1,….,K are
linearly independent at the optimum
Page 38
Example 5.5

x* = (1, 0)

No Kuhn-Tucker
point at the
optimum

Page 39
Kuhn-Tucker Necessity Theorem

Given a feasible point                           not
that satisfies the
constraint qualification
optimal

optimal             If it does not satisfy the
KTCs

If it does satisfy the
KTCs

Page 40
Example 5.6

Page 41
Kuhn-Tucker Sufficiency Theorem

Let
• f(x) be convex
• the inequality constraints gj(x) for j=1,…,J be
all concave function
•the equality constraints hk(x) for k=1,…,K be
linear
If there exists a solution (x*,u*,v*) that
satisfies KTCs, then x* is an optimal solution
Page 42
Example 5.4

• f(x) be convex
• the inequality constraints gj(x) for
j=1,…,J be all concave function
•the equality constraints hk(x) for
k=1,…,K be linear

Page 43
Example 5.4

• f(x) be convex

semi-definite

Page 44
Example 5.4

• f(x) be convex v
• the inequality constraints gj(x) for
j=1,…,J be all concave function

g1(x) linear, hence both convex and
concave
negative definite

Page 45
Example 5.4

• f(x) be convex v
• the inequality constraints gj(x) for
j=1,…,J be all concave function
• the equality constraints hk(x) for
k=1,…,K be linear

Page 46
Remarks

For practical problems, the constraint qualification
will generally hold. If the functions are
differentiable, a Kuhn–Tucker point is a possible
candidate for the optimum. Hence, many of the
NLP methods attempt to converge to a Kuhn–
Tucker point.

Page 47
Remarks

When the sufficiency conditions of Theorem 5.2
hold, a Kuhn–Tucker point automatically becomes
the global minimum. Unfortunately, the
sufficiency conditions are difficult to verify, and
often practical problems may not possess these
nice properties. Note that the presence of one
nonlinear equality constraint is enough to violate
the assumptions of Theorem 5.2

Page 48
Remarks

The sufficiency conditions of Theorem 5.2 have
been generalized further to nonconvex inequality
constraints, nonconvex objectives, and nonlinear
equality constraints. These use generalizations of
convex functions such as quasi-convex and
pseudoconvex functions

Page 49

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