An O(root r(cond(G))(1/4) log epsilon(-1)) iteration predictor-corrector interkr-point method with a new one norm neighbourhood for symmetric cone optimization

In this paper, we propose a predictor-corrector interiorpoint method for symmetric cone optimization. The proposed algorithm is based on a new one-norm neighbourhood, which is an even wider neighbourhood than a given negative infinity neighbourhood. The convergence is shown for a commutative class of search directions, which includes the Nesterov-Todd direction and the xs and sx directions. We show that the algorithm has O(root r(cond(G))(1/4) log epsilon(-1)) iteration complexity bound which is better than that of the usual wide neighbourhood algorithm O(r root cond(G)) log epsilon(-1)). To our knowledge, these are the best complexity results obtained so far for the solution of symmetric cone optimization. We prove that beside the predictor steps, each corrector step also reduces the duality gap by a rate of 1 - (1/O(root r). Finally, numerical experiments show that the proposed algorithm is efficient and reliable.