Consider the following problem:
Minimize $x(1-x) + y(1-y) + z(1-z)$ subject to: $$ a. 0\leq x, y, z \leq 1$$ $$ b. x+y+z = 1$$
If I plug in this problem in a standard quadratic optimizer, which minimizes $\displaystyle c^T \cdot \begin{pmatrix} x \\ y \\z \end{pmatrix} + \frac{1}{2}\cdot \begin{pmatrix} x&y&z \end{pmatrix} G \cdot \begin{pmatrix} x \\ y \\z \end{pmatrix} $, subject to constraints a) and b), I get the error message "the matrix G is not positive definite".
I'm aware that the minimum is obtained when exactly one of $x, y, z$ is equal to 1 ( and the others 0 ), but I need to translate this problem into a quadratic form with a positive definite matrix $G$ ( as part of a larger problem I am trying to solve, which contains more variables). Are there any suggestions on how to do so? Thanks a lot!
By the way, I am looking for ANY solution, not all solutions to such a problem.