Polynomial problem

I have simplified a complicated problem in to this form:
There exists a P=[p1 0;0 p2]>0
and alpha>0
Subject to
[p1+1 0 p1;
0 p2-1 p2;
p1 p2 -alpha]<=0
How can I solve it?

(also we can minimize alpha subject to the above constraint)


I am interpreting all your constraints as not being strict. if you need them to be strict, then accomplish that by making the RHS be a small positive number. (or subtract small number times identity matrix from LHS for senidefinite constraints, or lambda_min(LHS) >= small number).

Declare p1 and p2 and alpha to be CVX variables. Either declare them as nonnegative, or explicitly include p1 >= 0 and p2 >= 0. Or you can use [p1 0;0 p2] == semidefinite(2), which is equivalent.

For your last constraint, you want the negative of the LHS to be semidefinite, so negative of LHS == senidefinite(3). You can optionally add minimize(alpha).