AN INTERIOR POINT APPROACH FOR SEMIDEFINITE OPTIMIZATION USING NEW PROXIMITY FUNCTIONS

Kernel functions play an important role in interior point methods (IPMs) for solving linear optimization (LO) problems to define a new search direction. In this paper, we consider primal-dual algorithms for solving Semidefinite Optimization (SDO) problems based on a new class of kernel functions defined on the positive definite cone $\mathcal{S}_{++}^{n\times n}$. Using some appealing and mild conditions of the new class, we prove with simple analysis that the new class-based large-update primal-dual IPMs enjoy an $O(\sqrt{n}\, {\rm log}\, n\, {\rm log}\, \frac{n}{\varepsilon})$ iteration bound to solve SDO problems with special choice of the parameters of the new class.