|
Nonlinear complementarity problem (NCP) and the second-order cone programming (SOCP) problem are two types of optimization problems. They appeared widely in the field of science and engineering technology, so the research of solving methods have certain theoretical value and practical significance. Complementarity problems and nonlinear programming, minimax, game theory, fixed point theory, variational inequalities, and other branches of mathematics in close contact, and are widely used in mechanical, economic, transportation and other fields, and therefore attracted widespread attention, and in its theory and algorithms have achieved fruitful results. Which, by constructing a smooth function, smoothing Newton method for solving the NCP is one of the recent years, the research focus. Chapter II of this paper to consider a class of P 0 - mapping NCP (F). First, the introduction of a smooth function of the NCP (F) is equivalently transformed into a smooth equations, and the establishment of a smoothing Newton method for solving it. Secondly, prove the infinite sequence generated by the algorithm, any accumulation point are the solution of the original problem, and when the iterative sequence bounded nonempty bounded solution set of NCP (F). Then, when the NCP (F) has a local unique solution to meet a non-singular conditions prove local superlinear convergence and quadratic convergence of the algorithm. Finally, with five examples of numerical experiments show that the algorithm is feasible and effective. Compared with existing methods, the proposed method does not need to assume that the search direction sector does not require strict complementary conditions, but also by the special design of Newton's equations and linear search step, the smoothing parameter can be controlled with the right speed convergence. The SOCP issues are an important class of convex optimization problem. It is not only widely used in the field of engineering, and many other optimization problems can be transformed into it, the solution has been the focus of attention. Currently, there are many ways to solve SOCP problems, but they are basically traditional iterative method. The calculation of the time dependence of the scale of the problem, the structure and algorithm, making it difficult to meet the real-time requirements. Compared with the traditional numerical methods, due to the inherent parallel distributed processing characteristics and potential of the circuit, the neural network has many computational advantages and real-time applications. Since the introduction of the Hopfield neural network, and successfully applied to optimization problems using neural networks to solve the optimization problem to get a fairly in-depth research, and has made many important achievements. Chapter III of this paper to consider a class of SOCP problem. Two smooth functions of the second-order cone constraints into smooth convex constraints, the SOCP problem approximate transformation for the two types of convex optimization problem, and in accordance with the the projective theory established two new neural network for solving them. Then use the Lyapunov stability theory and LaSalle invariance principle to prove that the proposed neural network in the appropriate conditions is Lyapunov stable, and can converge to the solution of the original problem with arbitrary precision. Finally numerical experiments show that these networks is not only feasible, but also effective.
|