Linear programming using neural networks

We propose and analyze two classes of neural network models for solving linear programming (LP) problems. Our first models use the penalty function method to find solutions to the LP problems. We introduce a family of penalty functions that transform linear programming problems into unconstrained optimization problems. Subsequently, using a method from variable structure systems theory, we derive bounds on the weight parameters of the penalty functions for which the given linear program and the associated unconstrained optimization problems have the same solution. In our second model, we combine the gradient projection and the penalty function methods. For this model, we also derive the bound on the weight parameter of the penalty function resulting in the exact solution. Both proposed neural network models for solving linear programming problems are interpreted from the variable structure systems viewpoint. Simulation examples are given to illustrate the results obtained. We derive and compare the complexity of our models with the complexity of existing models