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

MV Optimization- E. 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)  Note: when x is not feasible, hj is not equal to 0 and by minimizing hj we are pushing x towards feasibility!

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

Example

Example cont’d

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.

MV Optimization Inequality Constrained

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

MV Optimization Active or Inactive Inequalities?

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

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

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

Ex 4.32 pg 150

LaGrange Function

4 equations and 4 unknowns Non-linear system of equations

Use Switching Conditions to Simplify

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

Case 2 System of 3 linear equations in 3 unknowns Rewrite

Gaussian Elimination

Gaussian Elimination cont’d

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

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

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.

All Equations must be satisfied

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