A variety of real-world problems can be formulated as continuous optimization problems with variable constraint. It is, however, well known that it is quite hard to develop a unified method for obtaining their feasible solutions. We have recognized that the recent work of solving the traveling salesman problem by the Hopfield model explores an innovative approach to them as well as combinatorial optimization problems. The Hopfield model is generalized into the Cohen-Grossberg model to which a specific Liapunov function has been found. This paper thus extends the Hopfield method onto the Cohen-Grossberg model in order to develop a unified solving method of continuous optimization problems with variable constraint. Specifically, we consider a certain class of continuous optimization problems with a constraint equation including the Hopfield version of the traveling salesman problem as a particular member. Then we theoretically develop a method that, from any given problem of that class, derives a network of an extended Cohen-Grossberg model to provide feasible solutions to it. The main idea for constructing that extended Cohen-Grossberg model lies in adding to the Cohen-Grossberg model a synapse dynamical system concurrently operating with its current unit dynamical system so that the constraint equation can be enforced to satisfaction at final states. This construction is also motivated by previous neuron models in biophysics and learning algorithms in neural networks.