# Minimize function with constraint

I have a Markowitz problem :

Min $x^T*C*x$

$x : {x_1 , x_2 ... x_n}$ is a vector of size $N$

$C$ is a known matrix $[N \times N]$

1) $∑ x_i$ = 1

2) $x_1 < 0$

I can minimize the function with the first constraint with Excel Solver. I find the optimal $x$ vector.

It doesn't work with the second one (because of the strict inequality) .

I don't know what kind of problem is this, so if someone have a path of research ..

Thanks

• Try for "Quadratic programming". – Khue Jun 24 '14 at 9:50
• With the 2nd constraint it seems to be no longer a quadratic problem :s – user159854 Jun 24 '14 at 10:05
• The second constraint can be seen as $Ax < 0$ where $A=[1 \ 0 \ \cdots \ 0]$. You can relax it to $Ax \le 0$ and use any quadratic solver. – Khue Jun 24 '14 at 10:12
• Ok but the solver will give me x = 0 as solution. I want to keep a strict inequality . – user159854 Jun 24 '14 at 11:30
• I guess you forgot to include the first constraint, i.e. $\mathbf{1}^Tx =1$, in the problem to be solved. – Khue Jun 24 '14 at 12:02