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）3q1-2q2+1/2（q1-q2）n1/2 logn/ε) complexity results for large- and small-update methods, respectively.