A Second-Order Corrector Infeasible Interior-Point Method for Semidefinite Optimization Based on a Wide Neighborhood

This paper proposes a second-order corrector infeasible interior-point method for semidefinite optimization in a large neighborhood of the central path. Our algorithm uses the Nesterov–Todd search directions as a Newton direction. We make use of the scaled Newton directions for symmetric search directions. Based on Ai and Zhang idea, we decompose the Newton directions into two orthogonal directions corresponding to positive and negative parts of the right-hand side of the Newton equation. In such a way, we use different step lengths for each of them and the corrector step is multiplied by the square of the step length of the infeasible directions in the expression of the new iterate. Starting with a point \((X^0, y^0, S^0)\) in the neighborhood in terms of Frobenius norm, we show that the algorithm terminates after at most \(\mathcal {O}(n^{\frac{5}{4}}\log \varepsilon ^{-1})\) iterations, improving by a factor \(n^{\frac{1}{4}}\) results on these kinds of algorithms. We believe that this is the first infeasible interior-point algorithm in a large neighborhood, which uses a Newton direction decomposition and a second-order corrector step to improve the iteration complexity. The main difference of the proposed method comparing with the existing methods in literature is the strategy for generating corrector directions. Some preliminary numerical results are given to verify the efficiency of the algorithm.