Study Of Methods For Solving Systems Of Nonlinear Inequalities

As a basic mathematical structure, systems of inequalities are widely used in many fields, such as numerical analysis, linear and nonlinear programming, engineering area and so on. Thus, it’s pretty significant for theoretical study and application to explore how to solve systems of equalities effectively. The study for solving systems of nonlinear inequalities, the existing results need improvement and there are still many problems to be solved. It’s one of focuses in the optimization community now.The non-interior continuation algorithm has been proposed for solving various optimization problems successfully except for systems of nonlinear inequalities. As a very competitive evolutionary computation techniques, differential evolution algorithm has been extensively studied. The main features of DE are simple in principle, easy to implement; little information about problems needed, good generality; fast convergence speed, strong global searching ability, suitable for solving complex optimization problems. This paper will use the above two algorithms to solve nonlinear inequalities. The work of this paper is as follows:Firstly, a non-interior continuation algorithm is proposed to solve a system of nonlinear inequalities in this paper. The system of nonlinear inequalities is converted into a system of non-smoothing equalities by using projection function in this paper. Based on the idea of smoothing reformulation and the application of a special smoothing function, the problem is approximated via a family of parameterized smoothing equations. Then we obtain a solution of the original problem by utilizing the Newton-type method at each iteration, with the smoothing parameter tending to zero as the iteration is going on. When conducting this algorithm, we only need to solve at most one system of linear equations at each iteration. Additionally, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. Some preliminary numerical experiments have been carried out, whose results demonstrate that the algorithm is effective.Secondly, a differential evolution algorithm is presented for solving a system of nonlinear inequalities. The nonlinear inequalities are transformed into parameterized smoothing equations by using projection function and smoothing function. In order to facilitate solving, the problem is converted into unconstrained optimization problem, and then a differential evolution algorithm is applied for solving it. Numerical experiments show that the algorithm is feasible.