Neural Network Algorithm For Nonconvex Optimization Problems And Its Applications

There are many optimization problems in the science and engineering applications.Neural network algorithms for the optimization problems have been obtained the wide attention since the algorithms are able to obtain the real-time solution to the optimization problems.This paper proposes a discrete-time and continuous-time neural network algorithms for the pseudoconvex and nonconvex quadratic programming problems,respectively.The networks are constructed through the projection theory,set-valued maps,and the properties of normal cone.Furthermore,we apply the algorithms in solving the support vector machine.This paper is divided into the following three parts.For the pseudoconvex optimization problem,we propose a discrete-time neural network model.First,the projection equations are constructed through the Karush-Kuhn-Tucker(KKT)conditions and projection theory such that there is a one-to-one correspondence between the solution of projection equations and the optimal solution of optimization problem.Moreover,a discrete-time neural network is represented via projection equations.Second,the obtained theoretical results indicate that the equilibrium point of the proposed neural network corresponds to the optimal solution of the optimization problem,and the proposed neural network is globally exponentially convergent.Compared with some continuous-time neural networks,the architecture of proposed neural network is simple,which decreases the computational complexity.Finally,numerical examples show the efficiency of the proposed neural network for solving the pseudoconvex optimization problems.Considering the nonconvex quadratic programming problems,we construct two different penalty functions based on equality constraints and inequalities constraints.Then,according to the properties of set-valued map and regular function,we formulate a neural network model with differential inclusion.Theoretical results show that the penalty functions force the trajectories to the feasible region in finite time,and the equilibrium point of the neural network coincides with the criticalpoint the optimal problem.Besides,the obtained theoretical results further analysis the relationship between the critical points and the minimum points of the optimal problem.Finally,the convergence of network is proved under appropriate assumptions.The support vector classification and support vector regression problems are analyzed briefly.Both the classification and regression problems can be transformed into a quadratic programming problem with a uniform form under appropriate loss function.Then,we use the proposed discrete-time neural network to solve classification and regression problems.Simulation results showed the good performance of the proposed neural network.