Exterior penalty function method

Question

I am trying to implement an exterior penalty function method for minimizing the below problem.

min f(x)=100*(x(2)-x(1)^2)^2+(1-x(1))^2

s.t, x(1)+2x(2)<=1

2x(1)+x(2)=1

First of all, I have found the minimum using fmincon, which the answer is x: array([ 0.4149, 0.1701]) and f(x)=0.34.

Then I am trying to find the minimum using my implementation of exterior penalty function method. I am using this penalty function:

F(x,a)=f(x)+a*(x(1)+2*x(2)-1)^2+a*(2*x(1)+x(2)-1)^2

with start point x_0=[1,1], a=10 (in each iteration a= a^2) , which gives me x: array([ 0.3333, 0.3333]) and f(x)=5.3.

Where is the mistake in my implementation? Thanks.

Answer 1

Your algorithm solves the following equality constrained problem!

minimize f(x)=100*(x(2)-x(1)^2)^2+(1-x(1))^2

subject to x(1)+2x(2)=1 2x(1)+x(2)=1

and fmincon (correctly) solved the original problem (with one inequality and one equality constraint).