Lifting/lowering Hopfield models ground state energies

In our recent work citeStojnicHopBnds10 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 citeStojnicHopBnds10. 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.