Research And Application Of Augmented Lagrange Method For Second-order Cone Programming

Second-order cone programming is a constrained optimization problem defined on the intersection of ray-like manifolds and Cartesian products of finite second-order cones.Many mathematical programming problems,such as linear programs and convex quadratic programs,can all be formulated as second-order cone programming problems under certain conditions.In addition,because of its special cone structure and computational convenience,second-order cone programming has been widely used in engineering control,signal processing and other practical problems.Therefore,it is of great significance and value to study efficient algorithms for solving second-order cone programming problems.Among many algorithms for solving second-order cone programming,the augmented Lagrange method is one of the most effective methods.In this paper,the augmented Lagrangian method for solving non-second-order cone programming problems is studied further.The paper is divided into four chapters.In Chapter 1,We introduce the basic knowledge of second-order cone programming and augmented Lagragian method,including the standard model of second-order cone programming and the research status of second-order cone programming problems at home and abroad,the related theory of augmented Lagrangian method,and also summarizes the shortcomings of existing algorithms to lead to the problems to be solved and the main work of this paper.In Chapter 2,for the nonlinear second-order cone programming problem,we study the properties of the augmented Lagrangian function by using the implicit function theorem of semi-smooth functions and the variational analysis of projection operators on second-order cones,and propose an augmented Lagrangian method for solving this problem and give a concrete description of the algorithm.In the existing literatures,it is proved that the method has local convergence under second-order sufficient conditions,constrained non-degenerate conditions and strict complementarity conditions,and the convergence rate is proportional to 1/?,However,the strict complementarity condition is not always easy to hold.Therefore,we analyze the local convergence of the method without strict complementarity and obtain the same convergence results.In Chapter 3,In order to achieve global convergence of the augmented Lagrangian method,We present a globally augmented Lagrangian method based on second order cone programming.For each iteration k,We use the global optimization method,such as?BB method,to obtain the global optimal solution of the augmented Lagrange function.When ?_k??,it is proved that the method converges globally to the global optimal solution of the second-order cone programming problem and obtained a weaker global convergence.In Chapter 4,we study the application of second-order cone programming in support vector machine classification model.We transform the model of support vector machine into a second-order cone programming problem,Although the transformed problem contains r second-order cone constraint,each second-order cone is three dimensional,which simplifies the computational complexity and speeds up the solution of the problem.We use improved support vector machine to classify iris dataset and heart disease dataset.The numerical experiments show that this transformation speeds up the solution of the problem,but does not affect the number of support vectors and the classification error rate.