I am trying to check if the dual expression of slide 13 of http://ee364a.stanford.edu/lectures/duality.pdf
$- \lambda^TAP^{-1}A^T\lambda - b^T\lambda$
is either convex or concave but am having a hard time. Computing the hessian gives me $-AP^{-1}A^T$ but I am not sure if that is positive or negative semi definite.
I have been looking through these slides on convex sets and convex functions but i still dont know http://ee364a.stanford.edu/lectures/functions.pdf
http://ee364a.stanford.edu/lectures/sets.pdf
Someone please help
The matrix $P$ has positive eigenvalues $\lambda_i$. The eigenvalues of $P^{-1}$ are $1/\lambda_i$ and therefore $P^{-1}$ is positive definite. Writing $P^{-1} = UU^T$ for some matrix $U$, you get for any $x \in R^n$: $$x^T AP^{-1}A^T x = x^T AUU^TA^T x = (U^TA^T x)^T(U^TA^T x) \geq 0$$ Therefore, $AP^{-1}A^T$ is positive semidefinite. This means that the optimization problem is concave. Another way to show that the objective is concave, is by noticing that it is the infimum of functions that are linear in $\lambda$.