Self-Regular Proximities and New Search Directions for Linear and Semidefinite Optimization Jiming Peng † ‡∗ Cornelis Roos † Tam´asTerlaky ‡ January 4, 2001 modified: October 6, 2004 † Faculty of Information Technology and Systems, Delft University of Technology P.O.Box 5031, 2600 GA Delft, The Netherlands. Email: C.Roos@its.tudelft.nl. ‡ Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada, L8S 4L7. Email: terlaky@cas.mcmaster.ca Abstract 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 self- regular 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. Keywords: Linear Optimization, Semidefinite Optimization, Interior Point Method, Primal- Dual Newton Method, Self-Regularity, Self-Concordance, Polynomial Complexity. AMS Subject Classification: 90C05 * This work was finished when the first author visited the Department of Computing and Software, McMaster University, Canada. Email: pengj@cas.mcmaster.ca, J.Peng@its.tudelft.nl. 1