L8 Optimal Design concepts pt D

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