In this paper, we describe our implementation of a primal-dual infeasible-interior-point algorithm for large-scale linear programming under the MATLAB 1 environment. The resulting software is called LIPSOL -- Linear-programming Interior-Point SOLvers. LIPSOL is designed to take the advantages of MATLAB's sparse-matrix functions and external interface facilities, and of existing Fortran sparse Cholesky codes. Under the MATLAB environment, LIPSOL inherits a high degree of simplicity and versatility in comparison to its counterparts in Fortran or C language. More importantly, our extensive computational results demonstrate that LIPSOL also attains an impressive performance comparable with that of efficient Fortran or C codes in solving large-scale problems. In addition, we discuss in detail a technique for overcoming numerical instability in Cholesky factorization at the end-stage of iterations in interior-point algorithms. Keywords: Linear programming, Primal-Dual infeasible-interior-p...

Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling. We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra. The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as

Abstract. Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today’s codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used. Keywords: interior-point methods, iterative solvers, preconditioners 1.

A practical warm-start procedure is described for the infeasible primal-dual interior-point method employed to solve the restricted master problem within the cutting-plane method. In contrast to the theoretical developments in this field, the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow. Moreover, it treats systematically the case when a large number of cuts are added at one time. The technique proposed in this paper has been implemented in the context of HOPDM, the state of the art, yet public domain, interior-point code. Numerical results confirm a high degree of efficiency of this approach: regardless of the number of cuts added at one time (can be thousands in the largest examples) and regardless of the depth of the new cuts, reoptimizations are usually done with a few additional iterations. Key words. Warm start, primal-dual algorithm, cutting-plane methods. Supported by the Fonds National de la Recherche Scientifique Su...

An interior-point method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search method that computes steps by factoring the primal-dual equations and a trust region method that uses a conjugate gradient iteration. Steps computed by direct factorization are always tried first, but if they are deemed ineffective, a trust region iteration that guarantees progress toward stationarity is invoked. To demonstrate its effectiveness, the algorithm is implemented in the Knitro [6, 28] software package and is extensively tested on a wide selection of test problems. 1

Most of the current techniques for the direct solution of linear equations are based on supernodal or multifrontal approaches. An important feature of these methods is that arithmetic is performed on dense submatrices and Level 2 and Level 3 BLAS (matrixvector and matrix-matrix kernels) can be used. Both sparse LU and QR factorizations can be implemented within this framework. Partitioning and ordering techniques have seen major activity in recent years. We discuss bisection and multisection techniques, extensions to orderings to block triangular form, and recent improvements and modifications to standard orderings such as minimum degree. We also study advances in the solution of indefinite systems and sparse least-squares problems. The desire to exploit parallelism has been responsible for many of the developments in direct methods for sparse matrices over the last ten years. We examine this aspect in some detail, illustrating how current techniques have been developed or ...

A new concept is proposed for linking algebraic modeling language and the structure exploiting solver. SPI (Structure Passing Interface) is a program that enables retrieving structure from the anonymous mathematical program built by the algebraic modeling language. SPI passes the special structure of the problem to a SES (Structure Exploiting Solver). An integration of SPI and SES leads to SET (Structure Exploiting Tool) and can be used with any algebraic modeling language. Key words. Algebraic modeling language, large scale optimization, structure exploiting solver. 1 Introduction Practitioners who use mathematical programming are confronted with a dilemma. On the one hand, their problems are usually so large and so complex that they cannot be modeled without the aid of an algebraic modeling language. On the other hand, large models often necessitate the use of a specialized structure exploiting solver. Unfortunately, algebraic modeling languages only access general purpose This r...

A new primal-dual algorithm is proposed for the minimization of non-convex objective functions subject to simple bounds and linear equality constraints. The method alternates between a classical primal-dual step and a Newton-like step in order to ensure descent on a suitable merit function. Convergence of a well-defined subsequence of iterates is proved from arbitrary starting points. Algorithmic variants are discussed and preliminary numerical results presented. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA Email : arconn@watson.ibm.com 2 Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU Email : nimg@letterbox.rl.ac.uk 3 Current reports available by anonymous ftp from joyous-gard.cc.rl.ac.uk (internet 130.246.9.91) in the directory "pub/reports". 4 Department of Mathematics, Facult'es Universitaires ND de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium, EU Email : pht@ma...

We consider solving a sequence of weighted linear least squares problems where the changes from one problem to the next are the weights and the right hand side (or data). This is the case for primaldual interior-point methods. We derive a class of preconditioners based on a low rank correction to a Cholesky factorization of a weighted normal equation coefficient matrix with the previous weight. Key Words. Weighted linear least squares, Preconditioners, Preconditioned conjugate gradient for least squares, Linear programming, Primaldual infeasible-interior-point algorithms. 1 Introduction In this paper, we present a class of preconditioners based on low rank corrections to the Cholesky factorization of a weighted normal equation coefficient matrix. This class of preconditioners leads to good performance for interiorpoint methods for linear programming. Particularly, we have implemented primal-dual Newton method to test this class of preconditioners. The numerical results on large scale...

this paper we develop an adaptive algorithm that changes strategy over the course of the interior point algorithm. It determines dynamically whether the preconditioner should be held constant, updated, or recomputed, and it switches to a direct method when it predicts that an iterative method will be too expensive. In our experiments, we use a preconditioned conjugate gradient iteration on the linear system involving the matrix ADA