Vtyjkyuk

Given

beginequation
X^TAX =
left(beginarraycc x_1 & x_2 endarrayright)
left(beginarraycc 1.5 & 1\ 1 & 2 endarrayright)
left(beginarrayc x_1 \ x_2 endarrayright)
=
endequation

Find $x_1$ and $x_2$ ratios that will minimize $X^TAX$ subject to $x_1+x_2=1$

Guide:

The quesiton is equivalent to

$$min 1.5x_1^2 + 2x_1x_2 + 2x_2^2$$

subject to $x_1+x_2=1$.

Once we perform a substitution, reduce the problem to a one dimensional quadratic optimization problem.

$$min 1.5x_1^2 + 2x_1(1-x_1) + 2(1-x_1)^2$$

Another way is to write

$$x_1+x_2=1$$ as $$bf SX-1=0, text where bf S = [1,1]$$

This will be solved where $min_Xbf SX-1$

So you can add this term to the minimization:

$$bf X_o = min_Xbf SX-1$$

Edit Sloppily, I forgot to mention that we need parameter $epsilon$ to say how much more important the fit to equality is than to minimize the first term. $epsilon=10^3$ seems to work well enough here.

If you are still skeptical, we can derive the solution and type it into Octave:

 A = [1.5,1;1,2];
 S = [1,1];
 lhs = A + 1e3*S'*S;
 rhs = [1e3*S'*1];
 Xo = [lhsrhs]'

displays result

 0.66578 0.33289

pretty close to the real solution [2/3 1/3]. If you don't have Octave and you can't install it, there apparently exists something called Octave Online, where you can type it in.

This approach is useful in case you have more similar constraints or costs you want to enforce at the same time. You can systematically stuff the linear combinations into such $bf S$ matrices. Nice if you have like 1000 000 dimensions and 1000 constraints to keep track of.