In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see [9]). The class of problems under consideration includes linear programming, semidefinite programming and quadratically constrained quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.

CONTENTS 1 Introduction 1 2 The Basics of Predictor-Corrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 Piecewise-Linear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (1881--1886), Klein (1882-- 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS-9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the

A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y- ffx)). Under the assumption that the mapping f is a P,,-function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x, y) = t(a, b) and (x, y) 8 0 until the parameter t> 0 attains 0. Here (a, b) denotes a 2n-dimensional constant positive vector. We establish the existence of a trajectory which leads to a solution of the problem, and then present a numerical method for tracing the trajectory. We also discuss the global and local convergence of the method.

We consider a family of primal/primal-dual/dual search directions for the monotone LCP over the space of n \Theta n symmetric block-diagonal matrices. We consider two infeasible predictor-corrector path-following methods using these search directions, with the predictor and corrector steps used either in series (similar to the Mizuno-Todd-Ye method) or in parallel (similar to Mizuno et al./McShane's method). The methods attain global linear convergence with a convergence ratio which, depending on the quality of the starting iterate, ranges from 1 \Gamma O( p n) \Gamma1 to 1 \Gamma O(n) \Gamma1 . Our analysis is fairly simple and parallels that for the LP and LCP cases. 1 Introduction Since the original work of Nesterov and Nemirovskii [26], followed by that of Alizadeh [1] and Jarre [11], there has been very active research on interior-point methods for the semi-definite linear programming problem (SDLP) and the semidefinite linear complementarity problem (SDLCP). In particular,...

. This paper proposes two sets of rules, Rule G and Rule P, for controlling step lengths in a generic primal-dual interior point method for solving the linear programming problem in standard form and its dual. Theoretically, Rule G ensures the global convergence, while Rule P, which is a special case of Rule G, ensures the O(nL) iteration polynomial-time computational complexity. Both rules depend only on the lengths of the steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions. They rely neither on neighborhoods of the central trajectory nor on potential function. These rules allow large steps without performing any line search. Rule G is especially flexible enough for implementation in practically efficient primal-dual interior point algorithms. Key words: Primal-Dual Interior Point Algorithm, Linear Program, Large Step, Global Convergence, Polynomial-Time Convergence Abbreviated Title: Large-Step Primal-Dual...

This paper presents some new methods for solving a class of linear projection equations. The search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization. Based on the idea of a projection and contraction method [7] we get a simple step length rule and are able to obtain global linear convergence.

We investigate the relation between interior-point algorithms applied to two homogeneous self-dual approaches to linear programming, one of which was proposed by Ye, Todd, and Mizuno and the other by Nesterov, Todd, and Ye. We obtain only a partial equivalence of path-following methods (the centering parameter for the first approach needs to be equal to zero or larger than one half), whereas complete equivalence of potential-reduction methods can be shown. The results extend to self-scaled conic programming and to semidefinite programming using the usual search directions. Abbreviated title: On two homogeneous systems for LP 1 Introduction Ye, Todd, and Mizuno [23] presented a homogeneous and self-dual interior-point algorithm for solving linear programming (LP) problems. The algorithm can start from arbitrary (infeasible) interior points and achieves the best known complexity in term of the number of iterations without using a big initial constant. Recently, Nesterov, Todd, and Ye [...

Version 1.02 1 Introduction This paper contains examples of various features from AMS-LATEX. 2 Enumeration of Hamiltonian paths in a graph Let A = (aij) be the adjacency matrix of graph G. The corresponding Kirchhoff matrix K = (kij) is obtained from A by replacing in \Gamma A each diagonal entry by the degree of its corresponding vertex; i.e., the ith diagonal entry is identified with the degree of the ith vertex. It is well known that

this paper,

This paper contains examples of various features from AMS-L ATEX.