Neural Computation
A novel optimizing network architecture with applications
Neural Computation
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This paper deals with the problem of mapping hybrid (i.e., bothdiscrete and continuous) constrained optimization problems ontoanalog networks. The saddle-point paradigm of mean-field methods instatistical physics provides a systematic procedure for findingsuch a mapping via the notion of effective energy.Specifically, it is shown that within this paradigm, to each closedbounded constraint set is associated a smooth convexpotential function. Using the conjugate (or the Legendre-Fencheltransform) of the convex potential, the effective energy can betransformed to yield a cost function that is a naturalgeneralization of the analog Hopfield energy. Descent dynamics anddeterministic annealing can then be used to find the global minimumof the original minimization problem. When the conjugate is hard tocompute explicitly, it is shown that a minimax dynamics, similar tothat of Arrow and Hurwicz in Lagrangian optimization, can be usedto find the saddle points of the effective energy. As anillustration of its wide applicability, the effective energyframework is used to derive Hopfield-like energy functions anddescent dynamics for two classes of networks previously consideredin the literature, winner-take-all networks and rotor networks,even when the cost function of the original optimizationproblem is not quadratic.