This came up in our game theory course. While doing the Lemke's algorithm for solving LP, it was said that the process terminates when the matrix $M$ is copositive plus.
Now copositive plus has a weird definition. So I'd like to know how different(as in weak) that condition is from PSD. Give an example of a matrix which copositive plus but not PSD.
The matrix $A = \begin{bmatrix}1 & 2\\ 2 & 1\end{bmatrix}$ is one such example. We have $f(x) = x^TAx = x_1^2 + x_2^2 + 4x_1x_2$. Clearly, $f(x) \geq 0$ for all $x \geq 0$. However, if $x_1=1$ and $x_2=-1$, we have $f(x) = -2 < 0$, clearly indicating that the matrix is not positive definite.