1. L8 Optimal Design concepts pt D
Homework
Review
Equality Constrained MVO
LaGrange Function
Necessary Condition EC-MVO
Example
Summary

2. MV Optimization- UNCONSTRAINED
For x* to be a local minimum:
1rst order
Necessary
Condition
2nd order
Sufficient
Condition
i.e. H(x*) must be positive definite

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

4. 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)
Let’s create the LaGrange Function by augmenting the objective function with “0’s”
Using parameters, known as LaGrange multipliers, and the equality constraints

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

6. 2D Example

7. Example cont’d

8. Example cont’d

9. Geometric meaning?

10. Stationary Points
Points that satisfy the necessary condition of the LaGrange Function are stationary points
Also called “Karush-Kuhn-Tucker” or KKT points

11. Lagrange Multiplier Method
1. Both f(x) and all hj(x) are differentiable
2. x* must be a regular point:
x* is feasible (i.e. satisfies all hj(x)
Gradient vectors of hj(x) are linearly independent (not parallel, otherwise no unique solution)
3. LaGrange multipliers can be +, - or 0.
Can multiply h(x) by -1, feasible region is the same.

12. Summary LaGrange Function L(x,u) Necessary Conditions for EC-MVO
Example
Geometric meaning of multiplier