A PRIMAL-DUAL IPMS FOR SDO PROBLEM BASED ON A NEW KERNEL FUNCTION WITH A LOGARITHMIC BARRIER TERM

Abstract

In this paper, we consider primal-dual Interior Point Method $(IPMs)$ for semidefinite optimization problem $(SDO)$ problems, based on a new kernel function with a logarithmic barrier term, which play an important role for generating a new design of primaldual (IPM) algorithms. New search directions and proximity functions are proposed, based on this kernel function.We proved that our algorithm has $\mathbf{O}\left( qsn^{\frac{sq+1}{2sq}}\log \left( \frac{n}{\epsilon }\right) \right) $ iteration bound for large-update methods and $\mathbf{O}\left(q^{2}s^{2}\sqrt{n}\log \left( \frac{n}{\epsilon }\right) \right) $ iteration bound for small-update methods. Finally, for its numerical tests, some strategies are used and indicate that the algorithm is efficient.