Mean field annealing: a formalism for constructing GNC-like algorithms. 
Optimization problems are approached using mean field annealing (MFA), which is a deterministic approximation, using mean field theory and based on Peierls's inequality, to simulated annealing.  The MFA mathematics are applied to three different objective function examples.  In each case, MFA produces a minimization algorithm that is a type of graduated nonconvexity.  When applied to the ;weak-membrane' objective, MFA results in an algorithm qualitatively identical to the published GNC algorithm.  One of the examples, MFA applied to a piecewise-constant objective function, is then compared experimentally with the corresponding GNC weak-membrane algorithm.  The mathematics of MFA are shown to provide a powerful and general tool for deriving optimization algorithms.