A product-form Cholesky factorization method for handling dense columns in interior point methods for linear programming

Cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods (IPMs) for linear programming (LP), its main advantages being numerical stability and efficiency for sparse systems. However in the presence of dense columns there is dramatic loss of efficiency. A typical approach to remedy this problem is to apply the Sherman-Morrison-Woodbury (SMW) update formula to the dense part. This approach while being very efficient, is not numerically stable. Here we propose using product-form Cholesky factorization to handle dense columns. The proposed approach is essentially as stable as the original Cholesky factorization and nearly as efficient as the SMW approach. We demonstrate these properties both theoretically and computationally. A key part of our theoretical analysis is the proof that the elements of the Cholesky factors of the matrices that arise in IPMs for LP are uniformly bounded as the duality gap converges to zero.The doctoral research of this author was supported in part by an IBM Cooperative FellowshipResearch supported in part by NSF Grants DMS 91-06195, DMS 94-14438, DMS 95-27124, DMS 01-04282 and CDA 97-26385 and DOE Grant DE-FG02-92ER25126     

Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods

We devise a hybrid approach for solving linear systems arising from interior point methods applied to linear programming problems. These systems are solved by preconditioned conjugate gradient method that works in two phases. During phase I it uses a kind of incomplete Cholesky preconditioner such that fill-in can be controlled in terms of available memory. As the optimal solution of the problem is approached, the linear systems becomes highly ill-conditioned and the method changes to phase II. In this phase a preconditioner based on the LU factorization is found to work better near a solution of the LP problem. The numerical experiments reveal that the iterative hybrid approach works better than Cholesky factorization on some classes of large-scale problems.     

Product-form Cholesky factorization in interior point methods for second-order cone programming

Second-order cone programming (SOCP) problems are typically solved by interior point methods. As in linear programming (LP), interior point methods can, in theory, solve SOCPs in polynomial time and can, in practice, exploit sparsity in the problem data. Specifically, when cones of large dimension are present, the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an LP. Here we propose a product-form Cholesky factorization (PFCF) approach, and show that it is more numerically stable than the alternative Sherman-Morrison-Woodbury approach. We derive several PFCF variants and compare their theoretical perfomance. Finally, we prove that the elements of L in the Cholesky factorizations LDL^T that arise in interior point methods for SOCP are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path.Mathematics Subject Classification (1991): 90C25, 90C51, 15A23Research supported in part by NSF Grants CDA 97-26385, DMS 01-04282, ONR Grant N000140310514 and DOE Grant GE-FG01-92ER-25126     

Matrix stretching for sparse least squares problems

For linear least squares problems min _x ‖Ax − b‖₂, where A is sparse except for a few dense rows, a straightforward application of Cholesky or QR factorization will lead to catastrophic fill in the factor R. We consider handling such problems by a matrix stretching technique, where the dense rows are split into several more sparse rows. We develop both a recursive binary splitting algorithm and a more general splitting method. We show that for both schemes the stretched problem has the same set of solutions as the original least squares problem. Further, the condition number of the stretched problem differs from that of the original by only a modest factor, and hence the approach is numerically stable. Experimental results from applying the recursive binary scheme to a set of modified matrices from the Harwell‐Boeing collection are given. We conclude that when A has a small number of dense rows relative to its dimension, there is a significant gain in sparsity of the factor R. A crude estimate of the optimal number of splits is obtained by analysing a simple model problem. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient method for calculating smoothing splines using orthogonal transformations

Summary A procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m–1 using an orthogonal factorization is presented. The procedure substantially overcomes the problem of ill-conditioning encountered by a recently developed method which employs a Cholesky factorization, but still requires only orderm ² n operations and ordermn storage.     

A factorization with update procedures for a KKT matrix arising in direct optimal control

Quadratic programs obtained for optimal control problems of dynamic or discrete-time processes usually involve highly block structured Hessian and constraints matrices, to be exploited by efficient numerical methods. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite factorization codes. For active set methods, however, conventional dense matrix techniques suffer from the need to update base matrices in every active set iteration, thereby loosing the sparsity structure after a few updates. This contribution presents a new factorization of a KKT matrix arising in active set methods for optimal control. It fully respects the block structure without any fill-in. For this factorization, matrix updates are derived for all cases of active set changes. This allows for the design of a highly efficient block structured active set method for optimal control and model predictive control problems with long horizons or many control parameters.     

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

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.     

Avoiding numerical cancellation in the interior point method for solving semidefinite programs

 The matrix variables in a primal-dual pair of semidefinite programs are getting increasingly ill-conditioned as they approach a complementary solution. Multiplying the primal matrix variable with a vector from the eigenspace of the non-basic part will therefore result in heavy numerical cancellation. This effect is amplified by the scaling operation in interior point methods. A complete example illustrates these numerical issues. In order to avoid numerical problems in interior point methods, we propose to maintain the matrix variables in a Cholesky form. We discuss how the factors of the v-space Cholesky form can be updated after a main iteration of the interior point method with Nesterov-Todd scaling. An analogue for second order cone programming is also developed. Numerical results demonstrate the success of this approach. Received: June 16, 2001 / Accepted: April 5, 2002 Published online: October 9, 2002 Key Words. semidefinite programming – second order cone programming Mathematics Subject Classification (2000): 90C22, 90C20     

The almost chebyshev property in linear semi-infinite programming

We consider a linear semi-infinite programming problem where the index set of the constraints is compact and the constraint functions are continuous on it. The set of all continuous functions on this index set as right hand sides are the parameter set. We investigate how large various unicity sets are.We state a condition on the objective function vector and the “matrix” of the problem which characterizes when the set of a parameters with a non-unique optimal point is a set of the first Baire category in the solvability set. This is the case if and only if the unicity set is a dense subset of the solvability set. Under the same assumptions it is even true that the interior of the strong unicity set is I also dense. If the index set of the constraints contains a dense subset with the property that each point1 is a G ₈-set, then the parameters of the strong unicity set, such that the optimal point satisfies the linear independence constraint qualification, are also dense.

We apply our results to a characterization of a unique continuous selection for the optimal set I mapping and to a one-sided L ₁-approximation problem     

Optimization of unconstrained functions with sparse hessian matrices-newton-type methods

Newton-type methods for unconstrained optimization problems have been very successful when coupled with a modified Cholesky factorization to take into account the possible lack of positive-definiteness in the Hessian matrix. In this paper we discuss the application of these method to large problems that have a sparse Hessian matrix whose sparsity is known a priori. Quite often it is difficult, if not impossible, to obtain an analytic representation of the Hessian matrix. Determining the Hessian matrix by the standard method of finite-differences is costly in terms of gradient evaluations for large problems. Automatic procedures that reduce the number of gradient evaluations by exploiting sparsity are examined and a new procedure is suggested. Once a sparse approximation to the Hessian matrix has been obtained, there still remains the problem of solving a sparse linear system of equations at each iteration. A modified Cholesky factorization can be used. However, many additional nonzeros (fill-in) may be created in the factors, and storage problems may arise. One way of approaching this problem is to ignore fill-in in a systematic manner. Such technique are calledpartial factorization schemes. Various existing partial factorization are analyzed and three new ones are developed. The above algorithms were tested on a set of problems. The overall conclusions were that these methods perfom well in practice.     

Correlative Sparsity in Primal-Dual Interior-Point Methods for LP,SDP, and SOCP

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation which is determined by the sparsity of an optimization problem (linear program, semidefinite program or second-order cone program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fill-in. S. Kim’s research was supported by Kosef R01-2005-000-10271-0 and KRF-2006-312-C00062.     

A fast algorithm for equality-constrained quadratic programming on the alliant FX/8

An efficient implementation of the null-space method for quadratic programming on the Alliant FX/8 computer is described. The most computationally significant operations in this method are the orthogonal factorization of the constraint matrix and corresponding similarity transformation of the Hessian, and the Cholesky factorization of the reduced Hessian matrix. It is shown how these can be implemented in such a way as to take full advantage of the Alliant's parallel/vector capabilities and memory hierarchy. Timing results are given on a set of test problems for which the data can be easily accommodated in core memory. Note: Research partially supported by the Air Force office of Scientific Research under grant AFOSR-ISSA-870092 and the National Science Foundation under grant DMS-8619903.     

Combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods

In this work, the optimal adjustment algorithm for p coordinates, which arose from a generalization of the optimal pair adjustment algorithm is used to accelerate the convergence of interior point methods using a hybrid iterative approach for solving the linear systems of the interior point method. Its main advantages are simplicity and fast initial convergence. At each interior point iteration, the preconditioned conjugate gradient method is used in order to solve the normal equation system. The controlled Cholesky factorization is adopted as the preconditioner in the first outer iterations and the splitting preconditioner is adopted in the final outer iterations. The optimal adjustment algorithm is applied in the preconditioner transition in order to improve both speed and robustness. Numerical experiments on a set of linear programming problems showed that this approach reduces the total number of interior point iterations and running time for some classes of problems. Furthermore, some problems were solved only when the optimal adjustment algorithm for p coordinates was used in the change of preconditioners.     

Computing Row and Column Counts for Sparse QR and LU Factorization

We present algorithms to determine the number of nonzeros in each row and column of the factors of a sparse matrix, for both the QR factorization and the LU factorization with partial pivoting. The algorithms use only the nonzero structure of the input matrix, and run in time nearly linear in the number of nonzeros in that matrix. They may be used to set up data structures or schedule parallel operations in advance of the numerical factorization.The row and column counts we compute are upper bounds on the actual counts. If the input matrix is strong Hall and there is no coincidental numerical cancellation, the counts are exact for QR factorization and are the tightest bounds possible for LU factorization.These algorithms are based on our earlier work on computing row and column counts for sparse Cholesky factorization, plus an efficient method to compute the column elimination tree of a sparse matrix without explicitly forming the product of the matrix and its transpose.This revised version was published online in October 2005 with corrections to the Cover Date.     

Indefinite QR Factorization

Let A be a Hermitian positive definite matrix given by its rectangular factor G such that A=G^*G. It is well known that the Cholesky factorization of A is equivalent to the QR factorization of G. In this paper, an analogue of the QR factorization for Hermitian indefinite matrices is constructed. This problem has been considered by many authors, but the problem of zero diagonal elements has not been solved so far. Here we show how to overcome this difficulty. AMS subject classification (2000) 65F25, 46C20, 65F15     

An interior point method for quadratic programs based on conjugate projected gradients

We propose an interior point method for large-scale convex quadratic programming where no assumptions are made about the sparsity structure of the quadratic coefficient matrixQ. The interior point method we describe is a doubly iterative algorithm that invokes aconjugate projected gradient procedure to obtain the search direction. The effect is thatQ appears in a conjugate direction routine rather than in a matrix factorization. By doing this, the matrices to be factored have the same nonzero structure as those in linear programming. Further, one variant of this method istheoretically convergent with onlyone matrix factorization throughout the procedure.     

An Inversion-Free Estimating Equations Approach for Gaussian Process Models

One of the scalability bottlenecks for the large-scale usage of Gaussian processes is the computation of the maximum likelihood estimates of the parameters of the covariance matrix. The classical approach requires a Cholesky factorization of the dense covariance matrix for each optimization iteration. In this work, we present an estimating equations approach for the parameters of zero-mean Gaussian processes. The distinguishing feature of this approach is that no linear system needs to be solved with the covariance matrix. Our approach requires solving an optimization problem for which the main computational expense for the calculation of its objective and gradient is the evaluation of traces of products of the covariance matrix with itself and with its derivatives. For many problems, this is an O(nlog?n) effort, and it is always no larger than O(n²). We prove that when the covariance matrix has a bounded condition number, our approach has the same convergence rate as does maximum likelihood in that the Godambe information matrix of the resulting estimator is at least as large as a fixed fraction of the Fisher information matrix. We demonstrate the effectiveness of the proposed approach on two synthetic examples, one of which involves more than 1 million data points.     

A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problems

We present a parallel preconditioned iterative solver for large sparse symmetric positive definite linear systems. The preconditioner is constructed as a proper combination of advanced preconditioning strategies. It can be formally seen as being of domain decomposition type with algebraically constructed overlap. Similar to the classical domain decomposition technique, inexact subdomain solvers are used, based on incomplete Cholesky factorization. The proper preconditioner is shown to be near optimal in minimizing the so‐called K‐condition number of the preconditioned matrix. The efficiency of both serial and parallel versions of the solution method is illustrated on a set of benchmark problems in linear elasticity. Copyright © 2002 John Wiley & Sons, Ltd.     

Wilkinson's inertia‐revealing factorization and its application to sparse matrices

We propose a new inertia‐revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof‐of‐concept implementation and present experimental results, studying the method's numerical stability and performance.     

Bound constrained quadratic programming via piecewise quadratic functions

1 , the smallest eigenvalue of a symmetric, positive definite matrix, and is solved by Newton iteration with line search. The paper describes the algorithm and its implementation including estimation of λ₁, how to get a good starting point for the iteration, and up- and downdating of Cholesky factorization. Results of extensive testing and comparison with other methods for constrained QP are given. Received May 1, 1997 / Revised version received March 17, 1998 Published online November 24, 1998