If you like, you might look at cones of sums of squares of polynomials (cones of PSD matrices are the same thing as cones of sums of squares of linear polynomials). This is the starting point of the modern technique of solving optimisation problems on semi-algebraic sets, due to J.Lasserre and others.

More generally, you might look at cones of nonnegative polynomials, and this opens up the whole Hilbert 17th problem business.