I have the following constraint: $$ \begin{bmatrix} t & f(x) \\ g(y) & 1 \end{bmatrix} \succ 0$$
where $f(x),g(y)$ are convex (real) positive functions of $x$ and $y$, and $t\in \mathbb{R}_+$. Is this type of constraint convex and solvable in an interface like YALMIP or CVX? Can it be transformed into an LMI?
Thanks!
If you mean that this matrix is positive definite, then this constrain is not convex unless $f(x)$ and $g(y)$ are linear function.
For a 2-dimensional matrix, that it is positive definite is equivalent to: $$ t>0,1>0,f(x)=g(y), f(x)g(y)-t<0 $$ i.e.: $$ f(x)=g(y), f(x)g(y)-t<0 $$ If you want these contraints to be convex, then $f(x), g(y)$ should be linear and $f(x)g(y)$ should be convex.