The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.

In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two di#erent ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over di#erent classes of SDPs show that these methods can be very e#cient for some problems. Keywords: Semidefinite programming; Primal-dual interior-point method; Matrix completion problem; Clique tree; Numerical results. # Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8565 Japan (nakata@zzz.t.u-tokyo.ac.jp ). + Department of Architecture and Architectural Systems, Kyoto University, Kyoto 606-8501 Japan (fujisawa@is-mj.archi.kyoto-u.ac.jp). # Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 OhOkayama, Meguro-ku, Tokyo 152-8552 Japan (mituhiro@is.titech.ac.jp). The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 OhOkayama, Meguro-ku, Toky...

We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the norm of the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the bounds arising from the simplex method. We also present explicit bounds for SDP using the AHO, H..K..M, and NT directions.

Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y

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 iteration bound, where q ≥ 1 is the so-called barrier degree of the self-regular ε proximity measure underlying the algorithm. When q increases, this � bound approaches the √n n best known complexity bound for interior point methods, namely O log. Our unified �√n ε n analysis provides also the O log best known iteration bound of small-update IPMs. ε 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.

The contribution of this paper is to describe a general technique to solve some classes of large but sparse semidefinite problems via a robust primal-dual interior-point technique which uses an inexact Gauss-Newton approach with a matrix free preconditioned conjugate gradient method. This approach avoids the ill-conditioning pitfalls that result from symmetrization and from forming the so-called normal equations, while maintaining the primal-dual framework.

We propose a new class of primal-dual methods for linear optimization (LO). By using some new analysis tools, we prove that the large update method for LO based on the new search direction has a polynomial complexity O i n 4 4+ae log n " j iterations where ae 2 [0; 2] is a parameter used in the system defining the search direction. If ae = 0, our results reproduce the well known complexity of the standard primal dual Newton method for LO. At each iteration, our algorithm needs only to solve a linear equation system. An extension of the algorithms to semidefinite optimization is also presented. Keywords: Linear Optimization, Semidefinite Optimization, Interior Point Method, PrimalDual Newton Method, Polynomial Complexity. AMS Subject Classification: 90C05 1 Introduction Interior point methods (IPMs) are among the most effective methods for solving wide classes of optimization problems. Since the seminal work of Karmarkar [7], many researchers have proposed and analyzed various ...

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. Interior-point methodology has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and column-generation algorithms for large linear programs. We review some core developments in the area and discuss their impact on these other problem areas.

ctions, and that the Nesterov-Todd process generates iterates that converge at a quadratic rate. Biographical Sketch Raphael Hauser was born on January 8th 1967 in Lucerne, Switzerland. After earning a Matura Type C from Kantonsschule Alpenquai Luzern in 1986, he decided to discover the world and become a mathematician. A long period of Lehr- und Wanderjahre led him first to ' Ecole Polytechnique F'ed'erale de Lausanne, and then to Georgia Tech in Atlanta and Eidgenossische Technische Hochschule in Zurich. Raphael became a Dipl. Math. ETH in 1993. After teaching Mathematics as an assistant at ETH Zurich and as an instructor at Abentechnikum der Innerschweiz in Lucerne, he entered the Ph.D. program of Cornell University's School of Operations Research and Industrial Engineering in September of 1995. iii To my parents iv Acknowledgements I would like to give my warmest thanks to my thesis supervisor, Professor Michael J. Todd, who first intro

We study numerical stability for interior-point methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms.