Editing Hopfield network (section)

== Hopfield network in optimization ==

[[John Hopfield|Hopfield]] and [[David Tank|Tank]] presented the Hopfield network application in solving the classical traveling-salesman problem in 1985.<ref name="hopfieldtank85">{{cite journal |first1=J.J. |last1=Hopfield |first2=D.W. |last2=Tank |title=Neural computation of decisions in optimization problems |journal=Biological Cybernetics |volume=52 |pages=141–6 |year=1985 |issue=3 |doi=10.1007/BF00339943 |pmid=4027280 |s2cid=36483354 |url=}}</ref> Since then, the Hopfield network has been widely used for optimization. The idea of using the Hopfield network in optimization problems is straightforward: If a constrained/unconstrained cost function can be written in the form of the Hopfield energy function E, then there exists a Hopfield network whose equilibrium points represent solutions to the constrained/unconstrained optimization problem.  Minimizing the Hopfield energy function both minimizes the objective function and satisfies the constraints also as the constraints are "embedded" into the synaptic weights of the network. Although including the optimization constraints into the synaptic weights in the best possible way is a challenging task, many difficult optimization problems with constraints in different disciplines have been converted to the Hopfield energy function: Associative memory systems, Analog-to-Digital conversion, job-shop scheduling problem, quadratic assignment and other related NP-complete problems, channel allocation problem in wireless networks, mobile ad-hoc network routing problem, image restoration, system identification, combinatorial optimization, etc, just to name a few. However, while it is possible to convert hard optimization problems to Hopfield energy functions, it does not guarantee convergence to a solution (even in exponential time).<ref>{{Cite journal |last1=Bruck |first1=Jehoshua |last2=Goodman |first2=Joseph W |date=1990-06-01 |title=On the power of neural networks for solving hard problems |url=https://linkinghub.elsevier.com/retrieve/pii/0885064X9090001T |journal=Journal of Complexity |volume=6 |issue=2 |pages=129–135 |doi=10.1016/0885-064X(90)90001-T |issn=0885-064X}}</ref>