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

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

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

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

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

2D Example

Example cont’d

Example cont’d

Geometric meaning?

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

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.

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