  The purpose of this article is to obtain a better understanding of the
extended variational principle (EVP). The EVP is a formula for the
thermodynamic pressure of a statistical mechanical system as a limit of a
sequence of minimization problems. It was developed for disordered mean-field
spin systems, spin systems where the underlying Hamiltonian is itself random,
and whose distribution is permutation invariant. We present the EVP in the
simpler setting of classical mean-field spin systems, where the Hamiltonian is
non-random and symmetric. The EVP essentially solves these models. We compare
the EVP with another method for mean-field spin systems: the self-consistent
mean-field equations. The two approaches lead to dual convex optimization
problems. This is a new connection, and it permits a generalization of the EVP.
