A Primal-dual Interior Point Method for Nonlinear Programming

In this paper, we propose a primal-dual interior point method for solving general constrained nonlinear programming problems. To avoid the situation that the algorithm we use may converge to a saddle point or a local maximum, we utilize a merit function to guide the iterates toward a local minimum. Especially, we add the parameter ε to the Newton system when calculating the decrease directions. The global convergence is achieved by the decrease of a merit function. Furthermore, the numerical results confirm that the algorithm can solve this kind of problems in an efficient way.

Path-following interior point algorithms for the Cartesian P_*(κ)-LCP over symmetric cones

In this paper, we establish a theoretical framework of path-following interior point al- gorithms for the linear complementarity problems over symmetric cones (SCLCP) with the Cartesian P*(κ)-property, a weaker condition than the monotonicity. Based on the Nesterov-Todd, xy and yx directions employed as commutative search directions for semidefinite programming, we extend the variants of the short-, semilong-, and long-step path-following algorithms for symmetric conic linear programming proposed by Schmieta and Alizadeh to the Cartesian P*(κ)-SCLCP, and particularly show the global convergence and the iteration complexities of the proposed algorithms.

Complexity Analysis of an Interior Point Algorithm for the Semidefinite Optimization Based on a Kernel Function with a Double Barrier Term

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization（SDO） problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 〉 q2 〉 1, the algorithm has O（（q1 ＋ 1） nq1＋1/2（q1-q2）logn/ε）and O（（q1 ＋ 1）2（q1-q2）^3q1-2q2＋1√n logn/c） complexity results for large- and small-update methods, respectively.

A ROBUST INTERIOR POINT METHOD FOR COMPUTING THE ANALYTIC CENTER OF AN ILL-CONDITIONED POLYTOPE WITH ERRORS

In this paper we propose an efficient and robust method for computing the analytic center of the polyhedral set P={x€R^n|Ax=b,x>0},where the matrix A€ Rm×n is ill-conditioned,and there are errors in A and b.Besides overcoming the difficulties caused by ill-cond计ioning of the matrix A and errors in A and b,our method can also detect the infeasibility and the unboundedness of the polyhedral set P automatically during the compu tation.Det ailed mat hematical analyses for our method are presen ted and the worst case complexity of the algorithm is also given.Finally some numerical results are presented to show the robustness and effectiveness of the new method.

AN EXTENSION OF PREDICTOR-CORRECTOR ALGORITHM TO A CLASS OF CONVEX SEPARABLE PROGRAM

redictor-corrector algorithm for linear programming, proposed by Mizuno et al. [1], becomes the best-known in the interior point methods. In this paper it is modified and then extended to solving a class of convex separable programming problems.