Q:

Penalty method for $f(x,y,z) = x^2 + 2y^2 + z^2$

Exercise :

Given the constraints $x^2+y^2+z^2 \leq 9, \; \; x+ y = 1, \; \; x-y = 2$ find a sufficient penalty function. Then, construct a penalty function for the minimization optimization problem of the function $f(x,y,z) = x^2 + 2y^2 + z^2$ over the constraints given.

Attempt :
The penalty function $p(x)$ will be given as :
$$p(x) = \max\{0, x^2 + y^2 + z^2 - 9\}^2$$
How would one construct a penalty function for the minimization problem of the $f(x,y,z)$ given ?

A:

According to the barrier function technique (Fiacco-Mc Cormick) a good barrier formulation could be
$$
\phi(x,y,z,\lambda)=x^2-2 y^2+z^2+\lambda_1\max \left(0,x^2+y^2+z^2-9\right)^2+\lambda_2(x+y-1)^2+\lambda_3 (x-y-2)^2
$$
Here $\lambda_1,\lambda_2,\lambda_3$ are positive constants adapted to each barrier to enhance the restriction satisfaction. The present exercise with $\lambda_1=\lambda_2=\lambda_3=\lambda_0$ and calling
$$
f = x^2-2 y^2+z^2\\
g_1 = \max \left(0,x^2+y^2+z^2-9\right)^2\\
g_2 = (x+y-1)^2\\
g_3 = (x-y-2)^2
$$
gives after minimization a result which is $\lambda_0$ dependent.
$$
\begin{array}{cccccccc}
x & y & z & \lambda_0 & g_1 & g_2 & g_3 & f\\
 0.89668 & -2.89289 & 0 & 1 & 0.0298729 & 8.97726 & 3.20255 & -15.9336 \\
 1.42857 & -0.555556 & 0 & 10 & 0 & 0.016125 & 0.000251953 & 1.42353 \\
 1.49254 & -0.505051 & 0 & 100 & 0 & 0.00015658 & 0 & 1.71752 \\
 1.49925 & -0.500501 & 0 & 1000 & 0 & 0 & 0 & 1.74675 \\
\end{array}
$$

