Primal-Dual Interior-Point Methods for Semidefinite
Optimization Based on Self-Regular Proximities
This chapter considers primal-dual algorithms for solving SDO problems based on self-regular functions defined on the positive definite cone Sn×n++. We start with a brief survey of major developments in the area of SDO, after which the notion of self-regular functions is generalized to the case of positive definite cones. Then, self-regular proximities for SDO are introduced as a combination of self-regular functions in Sn×n++ joint with the NT scaling. Several fundamental properties of self-regular proximities for SDO are described. The first and second derivatives of a function involving matrix functions are also estimated. We then propose some new search directions for SDO based on self-regular proximities. These new primal-dual IPMs for SDO enjoy the same complexity as their LO counterparts.