C. Roos

On implementing a primal-dual interior-point method for conic quadratic optimization

Abstract. Based on the work of the Nesterov and Todd on self-scaled cones an implementation of a primal-dual interior-point method for solving large-scale sparse conic quadratic optimization problems is presented. The main features of the implementation are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type predictor-corrector extension and sparse linear algebra to improve the computational efficiency. Finally, the implementation exploits fixed variables which naturally occurs in many conic quadratic optimization problems. This is a novel feature for our implementation. Computational results are also presented to document that the implementation can solve very large problems robustly and efficiently.

A Comparative Study of Kernel Functions for Primal-Dual Interior-Point Algorithms in Linear Optimization

Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on these kernel functions. In both the algorithm and its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of kernel function in the new class. We apply this scheme to seven specific kernel functions. Some of these functions are self-regular, and others are not. One of the functions differs from the others, and from all self-regular functions, in the sense that its growth term is linear. Iteration bounds for both large- and small-update methods are derived. It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely,

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

O(\sqrt{n}\log\frac{n}{\e})

Self-regularity : a new paradigm for primal-dual interior-point algorithms

. For large-update methods the best obtained bound is

On maximization of quadratic form over intersection of ellipsoids with common center

O(\sqrt{n}(\log n)\log\frac{n}{\e})

On Copositive Programming and Standard Quadratic Optimization Problems

, which until now has been the best known bound for such methods.

A Full-Newton Step O ( n ) Infeasible Interior-Point Algorithm for Linear Optimization

Abstract.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 ?(n

A New Efficient Large-Update Primal-Dual Interior-Point Method Based on a Finite Barrier

\frac{q+1}{2q}

SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING: JUST BE CAREFUL

log

A polynomial method of approximate centers for linear programming

\frac{n}{\varepsilon}