Study On Smoothing Algorithms For Conic Programming

Linear programming problem is a kind of optimization problem for the study of variables on the intersection of affine sets and convex polyhedrons.As the promotion of linear programming,second-order cone programming is also a kind of convex optimization problem.It is a convex optimization problem in which a linear function is minimized over the intersection of an affine linear manifold with Cartesian product of second-order cones.Many mathematical problems can be formulated as the second-order cone programming.Linear programming and second-order cone programming have become one of impor-tant research direction in the mathematical programming field for their wide range of applications in many fields,such as engineering,control and design and so on.This paper aims to study a smoothing Newton method for linear program-ming and second-order cone programming.It consists four parts.In the first chapter,we introduce the research background and the recent situations of linear programming and second-order cone programming.In Chapter 2,we present a new function by smooth approximating Fischer-Burmeister function and analyze its continuous differentiability.Then,we ap-ply the smoothing Newton method to solve the linear programming based on the new function.Furthermore,the global convergence property is established.Un-der the condition that jacobian matrix is reversible at solving point,the quadrat-ic convergence property is established.Numerical experiments demonstrate the efficiency of the proposed algorithm.In Chapter 3,a new complementary function is given by smoothing the symmetric perturbed Fischer-Burmeister function.Based on this function,we reformulate the second-order cone programming which concerned as a fami-ly of parameterized smooth equations and then propose a smoothing Newton method.Besides,the global convergence property is established.Under the condition that jacobian matrix is reversible at solving point,the quadratic con-vergence property is established.Finally,we give some numerical experiments.The numerical results show that our algorithm is efficient.The last chapter is a summary of this paper.