1. L8 Optimal Design concepts pt D
Homework
Review
Inequality constraints
General LaGrange Function
Necessary Conditions
for general Lagrange Multiplier Method
Example
Summary

2. MV Optimization- E. CONSTRAINED
For x* to be a local minimum:

3. LaGrange Function
If we let x* be the minimum f(x*) in the feasible region:
All x* satisfy the equality constraints (i.e. hj =0) 
Note: when x is not feasible, hj is not equal to 0 and
by minimizing hj we are pushing x towards feasibility!

4. Necessary Condition Necessary condition for a stationary point
Given f(x), one equality constraint, and n=2

5. Example

6. Example cont’d

7. Lagrange Multiplier Method
1. Both f(x) and all hj(x) are differentiable
2. x* must be a regular point:
2.1 x* is feasible
2.2 Gradient vectors of hj(x) are linearly independent
3. LaGrange multipliers can be +, - or 0.

8. MV Optimization Inequality Constrained

9. To Use LaGrange Approach Convert Inequalities to Equalties?
Given an inequality
Add a variable sj to take up the slack
No longer an inequality
Can now use Lagrange Multiplier Approach

10. MV Optimization Active or Inactive Inequalities?

11. KKT Necessary Conditions for Min
1. Lagrange Function (in standard form)
2. Gradient Conditions

12. KKT Conditions Cont’d 3. Feasibility Check for Inequalities
4. Switching Conditions, e.g.
5. Non-negative LaGrange Multipliers for inequalities
6. Regularity check
gradients of active inequality constraints are linearly independent

13. KKT Necessary Conditions for Min
Regularity check - gradients of active inequality constraints are linearly independent

14. Ex 4.32 pg 150

15. LaGrange Function

16. 4 equations and 4 unknowns
Non-linear system of equations

17. Use Switching Conditions to Simplify

18. Case 1 Non-linear system of equations Both inequalities are VIOLATED
Therefore, x is INFEASIBLE

19. Case 2
System of 3 linear equations in 3 unknowns
Rewrite

20. Gaussian Elimination

21. Gaussian Elimination cont’d

22. Ex 4.32 cont’d Check last eqn (for s1 feasiblity)
Nope! The point is not feasible.

23. Case 4
We can use Gaussian elimination again
Where are
cases 1-4?

24. Check if regular point Rank= order of largest non-singular matrix in A
since det (A) is non-singular,
the A matrix is full rank
We can also see that the vectors are not parallel in figure.

25. All Equations must be satisfied

26. Summary General LaGrange Function L(x,v,u,s)
Necessary Conditions for Min
Use switching conditions
Check results