  In our recent work \cite{StojnicHopBnds10} we looked at a class of random
optimization problems that arise in the forms typically known as Hopfield
models. We viewed two scenarios which we termed as the positive Hopfield form
and the negative Hopfield form. For both of these scenarios we defined the
binary optimization problems whose optimal values essentially emulate what
would typically be known as the ground state energy of these models. We then
presented a simple mechanisms that can be used to create a set of theoretical
rigorous bounds for these energies. In this paper we create a way more powerful
set of mechanisms that can substantially improve the simple bounds given in
\cite{StojnicHopBnds10}. In fact, the mechanisms we create in this paper are
the first set of results that show that convexity type of bounds can be
substantially improved in this type of combinatorial problems.
