ESI 4313 Operations Research 2

1
ESI 4313Operations Research 2

• Nonlinear Programming
• KKT conditions
• Lecture 10
• (February 13, 2007)

2
example 1

• consider the following problem

3
Constrained optimizationsufficient conditions

• If
• f is a concave function
• g1,,gm are convex functions
• then any solution x satisfying the KKT
  conditions is an optimal solution to the NLP
• A similar result can be formulated for
  minimization problems

4
Constrained optimizationKKT conditions

• We can interpret the KKT conditions using the
  interpretation of the multipliers ?i as dual
  prices
• If we increase the value of by some small
  amount ?, then the value of the objective
  function changes to

5
example 1

• solve

6
example 2

• solve

7
example 2

• Let be the dual multiplier for the first
  constraint be the dual multiplier of the
  second constraint.
• We need to consider 4 cases

8
example 2 cont.

• we find all KKT points and select the best one.