1. L10 Optimal Design L.Multipliers
Homework
Review
Meaning & of Lagrange Multiplier
Summary

2. Homework 4.44 Now is a “minimize”
We have only used “necessary conditions”
We cannot yet conclude that the pt is a MIN!

3. 4.44 cont’d
H(x) is negative definite, therefore the candidate pt is not a local min. (therefore Pt A is NOT a max of the original F(x)). Unbounded?

4. Prob 4.54

5. Gaussian Elimination Case 2
x R1 by -1
+ to R2
x R3 by -13
+ to R2

6. Prob 4.57

7. Gaussian Elmination 4.57 Case 1 u=0
+R1 to R2
x R2 by -1/2
+ to R3
Check feasibility
Backsub
Using R2

8. Gaussian Elmination 4.57 Case 2 s=0
+R3 to R4
Check feasibility
Backsub
Using R3

9. Prob 4.57

10. Prob 4.59

11. Prob 4.59

12. Prob 4.59

13. Gaussian Elmination 4.59 Case 4 s1,2=0
+R3 to R4
Backsub
Using R3
Check feasibility
Both s1 and s2 =0

14. Prob 4.59
Where is:
Case 1
Case 2
Case 3
Case 4

15. MV Optimization Inequality & Equality Constrained

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

17. Relax both constraints (Prob 4.59)

18. Constraint Variation Sensitivity Theorem
The instantaneous rate of change in the objective function with respect to relaxing a constraint IS the LaGrange multiplier!

19. Practical Use of Multipliers in 4.59
The first-order approximation on f(x), of relaxing a constraint is obtained from a Taylor Series expansion:
f(actual)=1
versus
f(approx)=0

20. Summary Min =-Max, i.e. f(x)=-F(x) Necessary Conditions for Min
KKT point is a CANDIDATE min!
(need sufficient conditions for proof)
Use switching conditions, Gaussian Elimination to find KKT pts
LaGrange multipliers are the instantaneous rate of change in f(x) w.r.t. change in constraint relaxation.