An efficient simplified neural network for solving linear and quadratic programming problems

Abstract

We present a high-performance and efficiently simplified new neural network which improves the existing neural networks for solving general linear and quadratic programming problems. The network, having no need for parameter setting, results in a simple hardware requiring no analog multipliers, is shown to be stable and converges globally to the exact solution. Moreover, using this network we can solve both linear and quadratic programming problems and their duals simultaneously. High accuracy of the obtained solutions and low cost of implementation are among the features of this network. We prove the global convergence of the network analytically and verify the results numerically.

Introduction

Solving linear and quadratic programming problems of large size is considered to be one of the basic problems encountered in operations research. In many applications a real time solution of linear or quadratic programming problem is desired. Also, most optimization problems with nonlinear objective functions are usually approximated by second order models and solved numerically by a quadratic programming technique [4], [5]. Traditional algorithms such as simplex algorithm or Karmarkar’s method for solving linear programming problems are computationally too expensive. One possible alternative approach is to employ neural networks on the basis of analog circuits [1], [2], [3]. The most important advantages of the neural networks are their massively parallel processing capacity and fast convergence properties.

In 1986, Tank and Hopfield [3] proposed a neural network for solving linear programming problems which was mapped onto a closed-loop circuit. Although the equilibrium point of Tank and Hopfield network may not be a solution of the original problem, this seminal work has inspired many researchers to investigate other neural networks for solving linear and nonlinear programming problems (see [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]). Kennedy and Chua [11] extended the Tank and Hopfield network by developing a neural network for solving nonlinear programming problems, by satisfaction of the Karush–Kuhn–Tucker optimality conditions [12]. The network proposed by Kennedy and Chua contains a penalty parameter. Thus, it generates approximate solutions only and implementation problems arise when the penalty parameter is large. To avoid the use of penalty parameters, significant work has been carried out in recent years [7], [9], [10]. For example, Rodriguez-Vazquez et al. [13] proposed a switched-capacitor neural network for solving a class of nonlinear convex programming problems. This network is suitable only for cases in which the optimal solutions lie within the feasible region. Otherwise, the network may have no equilibrium point [14]. Although the model proposed in [15] overcomes the aforementioned drawbacks and is robust for both continuous and discrete-time implementations, but still, the main disadvantage of the network is the requirement to use plenty of rather expensive analog multipliers for variables. Thus, not only the cost of the hardware implementation is very expensive, but also accuracy of solutions is greatly affected. The network of Xia [16], [17] is an improvement over the proposal in [15] in terms of accuracy and implementation cost. The network we discuss here will be both more efficient and less costly than Xia et al.’s [15], [16], [17].

The paper is organized as follows. In Section 2, we introduce the basic problem and the model for the new neural network. Section 3, discusses some theoretical aspects of the model and analyzes its global convergence. The circuit implementation of the new model and a comparative analysis are given in Section 4. Simulation results are shown in Section 5. Conclusions are given in Section 6.