Self-regular functions and new search directions for linear and semidefinite optimization

In this paper, we introduce the notion of a self-regular function. Such a function is strongly convex and smooth coercive on its domain, the positive real axis. We show that any such function induces a so-called self-regular proximity function and a corresponding search direction for primal–dual path-following interior-point methods (IPMs) for solving linear optimization (LO) problems. It is proved that the new large-update IPMs enjoy a polynomial O(n^((q+1)/(2q)) log(n/ϵ)) iteration bound, where q ≥ 1 is the so-called barrier degree of the kernel function underlying the algorithm. The constant hidden in the O-symbol depends on q and the growth degree p ≥ 1 of the kernel function. When choosing the kernel function appropriately the new large-update IPMs have a polynomial O(√(n) log n log (n/ϵ)) iteration bound, thus improving the currently best known bound for large-update methods by almost a factor √(n). Our unified analysis provides also the O(√(n) log (n/ϵ)) best known iteration bound of small-update IPMs. At each iteration, we need to solve only one linear system. An extension of the above results to semidefinite optimization (SDO) is also presented.