No Slide Title by HC121010011826

VIEWS: 0 PAGES: 49

									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:
Good guys or bad guys?




                         Page 3
    Constraints:
Good guys or bad guys?

            reduces the region in
            which we search for
                 optimum.



                                Page 4
        Constraints:
    Good guys or bad guys?

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




              Page 16
Page 17
Test whether the stationary point
   corresponds to a minimum




        positive definite

                                    Page 18
Page 19
Example 5.3




              Page 20
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




              Page 28
  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

								
To top