Interior Point Algorithms And A Smoothing Newton Method For The Second-order Cone Programming

The second-order cone programming (SOCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of a finite number of second-order cones. It is well known that the constraint of the second-order cone programming is nonlinear , but convex, so the second-order cone programming is a convex optimization problem. The second-order cone programming unifies several problems, such as linear programming and quadratically constrained convex quadratic programming, but it is a special case of semidefinite programming. Because of the quick development of its primal-dual interior-point algorithms and its wide applications, the second-order cone programming is an important research field in mathematical programming.In the paper, firstly the theory, algorithm, and recent research of the second-cone programming is summarized, then our some work in algorithms is introduced. For detail, we conclude them as follows:1. A primal-dual inexact infeasible interior-point algorithm for the second-order cone programming is presented in this paper. This algorithm allows the search direction that is calculated with only moderate accuracy, and does not require feasibility of the iteration points. Under a mild assumption on the inexactness, we show that the algorithm can find an e -approximate solution of the second-order cone programming.2. Based on smoothing the Fischer-Burmeister function, a new smoothing Newton method is presented in this paper. The system which is employed in this method is equivalent to the optimality conditions and not to the central path conditions. This algorithm does not have restrictions regarding its starting point and it is Q-quadratically convergent.