Convex optimization using constraint projection matrices

Question

I have a convex optimization of the form $$ \min_x \frac{1}{2} x^TAx-x^Tb \\ \text{s.t.}\ (I-P)x=0 $$ where $A$ is a $n$ by $n$ positive definite matrix, and $P$ is a $n$ by $n$ projection matrix (it has $p$ eigenvalues equal to zero, and $n-p$ eigenvalues equal to one)

Intuitively, it seems like I can separate the subspaces by rewriting it to: $$ \min_x \frac{1}{2} x^TP^TAPx+\frac{\gamma}{2} x^T(I-P)^T(I-P)x-x^TP^Tb $$ And since $P$ is a projection matrix, it is symmetric and idempotent, it can be simplified to: $$ \min_x \frac{1}{2} x^T(PAP+\gamma(I-P))x-x^TPb $$ This results in an unconstrained optimization, and the result is obtained by solving: $$(PAP+\gamma(I-P))x=Pb$$ This formulation is particularly handy because the matrix $(PAP+\gamma(I-P))$ is still positive definite for any positive $\gamma$ value (which is in fact the eigenvalues of the constraint subspace), so it can be solved numerically using the conjugate gradient method.
In practice, it works very well, but to be honest, I don't know if there is a flaw in my intuition, and I haven't found any resources on the subject.
Is there a known method that I can use to demonstrate my intuition is correct?

Answer 1

First of all,

$$\frac{1}{2}x^{T}Ax-x^{T}b=\frac{1}{2}x^{T}(P+I-P)^{T}A(P+I-P)x-x^{T}(P+I-P)^{T}b. $$

Notice that

$$\frac{1}{2}x^{T}(P+I-P)^{T}A(P+I-P)x=\frac{1}{2}x^{T}P^{T}APx+\frac{1}{2}x^{T}(I-P)^{T}A(I-P)x+\frac{1}{2}x^{T}P^{T}A(I-P)x+\frac{1}{2}x^{T}(I-P)^{T}APx, $$

and

$$x^{T}(P+I-P)^{t}b=x^{t}P^{t}b+x^{t}(I-P)^{t}b. $$

Now with the condition $(I-P)x=0$ we have that,

$$\frac{1}{2}x^{T}Ax-x^{T}b=\frac{1}{2}x^{T}P^{T}APx-x^{T}P^{t}b. $$

The problem is that P is semi-defined positive so $P^{T}AP$ isn't guranteed to be defined positive and thus it isn't guranteed that the minimuns are global and unique.

However, since $f(x)=\frac{1}{2}x^{t}Ax-x^{t}b$ is a convex function, and the restriction set has the form $h(x)=0$ with $h$ affine, the problems belongs to the category of convex optimization problems, that are easily solved by softwares like AMPL.

That problems could be also solved by hand using KKT conditions (becasuse convex optimization problems satisfies the hypothesis of that method).

When the restriction set has complicated functions ($h(x)=cos(x)=0$, for example) a penalization method is usually used. You can search information about exterior penalty function methods or augmented lagrangian method.