A neural network theory for constrained optimization

A variety of real-world problems can be formulated into continuous optimiza tion problems with constraint equalities. The real-world problem here can i nclude, for example, the traveling salesman problem, the Dido's isoperimetr ic problem, the Hitchcock's transportation problem, the network flow proble m and the associative memory problem. In spite of the significance, there h as not yet been developed any robust solving method that works efficiently across a broad spectrum of optimization problems. The recent Hopfield's neu ral network method to solve the traveling salesman problem is a potentially promising candidate because of the efficiency due to its parallel processi ng. His method, however, has certain drawbacks that must be removed away be fore it can be qualified for an efficient, robust solving method. That is: (a) locally minimum solutions instead of globally minimum; (b) possible inf easible solutions; (c) heuristic choice of network parameters and an initia l state; (d) quadratic objective functions instead of arbitrary nonlinear o bjective functions with arbitrary nonlinear equality constraints; and (e) u norganized mathematical formulation of the network: for extension. This pap er develops from the Hopfield method an efficient, robust network solving m ethod of the continuous optimization problem with constraint equalities tha t resolves all the drawbacks except for (a) that has already been resolved by others. The development is mathematically rigorous and thus constitutes a solid foundation of a neural network theory for constrained optimization. (C) 1999 Elsevier Science B.V. All rights reserved.