How can I minimize a quadratic function which some of the quadratic terms have a coefficient of zero?
e.g. $\min x_1^2 + x_1 + x_2$
(subject to some linear constraints on $x_i$)
As a quadratic programming problem, this is
$\min q^\intercal x + \frac{1}{2}x^\intercal Q x$
with
$Q = \begin{pmatrix}2 & 0 \\0 & 0\end{pmatrix}, q = \begin{pmatrix}1 \\ 1 \end{pmatrix}$
However $Q$ is not positive-definite and therefore a standard QP solver cannot be used. What other methods can I use?
Matrix $Q$ has zero eigenvector that isn't orthogonal to $q$. That means that without constraints the function has no global minimum.
So the minimum of the problem is achieved on a boundary. We have 4 linear boundaries and 3 points (vertices). We solve boundary problems with Lagrange multipliers.
For boundaries $x_1=\pm c$ there is no minimum. For boundary $x_1-x_2=0$, there is minimum $f=-1$ at $(-1,-1)$, which lies outside of condition $x_1+x_2\ge0$. For boundary $x_1+x_2=0$ there is minimum $f=0$ at $(0,0)$ which coincides with one vertex. For other two vertices $(-c,c)$ and $(c,c)$, $f=c^2>0$ and $f=c^2+2c>0$.
Thus, we conclude that the minimum $f=0$ is achieved at vertex $(0,0)$.