Self-Regular Proximities and New Search Directions for Linear and Semidefinite Optimization

In this paper, we first introduce the notion of self-regular functions. Various appealing properties of self-regular functions are explored and we also discuss the relation between selfregular functions and the well-known self-concordant functions. Then we use such functions to define self-regular proximity measure for path-following interior point methods for solving linear optimization (LO) problems. Any self-regular proximity measure naturally defines a primal-dual search direction. In this way a new class of primal-dual search directions for solving LO problems is obtained. Using the appealing properties of self-regular functions, we prove that these new large-update path-following methods for LO enjoy a polynomial, O ( n q+1 2q log n ε ) iteration bound, where q ≥ 1 is the so-called barrier degree of the self-regular proximity measure underlying the algorithm. When q increases, this bound approaches the best known complexity bound for interior point methods, namely O (√ n log n ε ) . Our unified analysis provides also the O (√ n log n ε ) best known iteration bound of small-update IPMs. At each iteration, we need only to solve one linear system. As a byproduct of our results, we remove some limitations of the algorithms presented in [24] and improve their complexity as well. An extension of these results to semidefinite optimization (SDO) is also discussed.