I am working on Constrained Optimization, using the Penalty Method, and then using Uzawa's algorithm. I am looking to minimize the objective function f, given by
f = @(x) ( (1/2)*x'*A*x - x'*b + (1/4)*sum(x.^4) );
with A generated by the MATLAB command relative to the discrete Poisson’s problem $-\Delta{u} = f$ in a 2-dimensional region $R \subseteq \{1 \le x, y \le 1\}\ u = 0$ on the boundary of $R$.
function [A,b,N] = generate(n, c)
G = numgrid('S',n);
A = delsq(G);
N = length(A);
b = c*ones(N,1);
The constraints are such that $x_i \le 0$ whenever the index $i$ corresponds to points in the closed half-unit square $R_1 = \{1/2 \le s, t \le 1/2\}$.
Now, what I understood is that the constraints on the minimizer x* are such that elements of x*, which are $x_i$, must be negative or equal to zero, if they happen to be inside the half unit square. Right?
Moreover, how can I use the MATLAB function find to get these indices and write (or programme) my constraints to be able to write my algorithm (say, Uzawa's algorithm).
My $m$ constraints should be summarized as vector phi: $R^m$ to $R$, such that $\phi(x) = C*x - d <= 0$.
If anyone could please help me make sense of all this. I am not a math student but I need to understand these for a thesis work in engineering. This is not an assignment.