Research On The Smoothing Algorithm For Solving A System Of Inequalities Associated With Two Kinds Of Cones

The system of inequalities has a very wide application background,so far,the research of the traditional system of inequalities in the finite-dimensional Euclidean s-pace IR~nhas been achieved many good conclusions.In recent years,the research has spread from the traditional system of inequalities to the system of inequalities under the order induced by symmetric cones(including non-negative quadrant cones,second-order cones and semi-definite cones),which have a wider significance with a unified framework.And we have got some good theoretical and algorithmic results.But the system of inequalities under the order induced by non-symmetric cones(for example circular cones)is still in a blank stage.Moreover,compared with the classical inequal-ity systems,the system of inequalities under the order of the cone has a wider range of applications and stronger application value.Hence,the study on the cone linear complementarity problem is significant.In this paper,we consider a system of inequalities under the order induced by second-order cones and circular cones,respectively.Using the special structure of the second-order cone and circular cone,we can construct a new smoothing function,and thus the problem is reformulated as a system of parameterized smooth equations.Ac-cordingly,we propose a smoothing-type Newton algorithm to solve the reformulation,and show that the proposed algorithm is globally convergent and locally quadratically convergent under suitable assumptions.Preliminary numerical results demonstrate that the approach is effective.