Irvin J. Lustig

Computational experience with a primal-dual interior point method for linear programming

Abstract A new comprehensive implementation of a primal-dual algorithm for linear programming is described. It allows for easy handling of simple bounds on the primal variables and incorporates free variables, which have not previously been included in a primal-dual implementation. We discuss in detail a variety of computational issues concerning the primal-dual implementation and barrier methods for linear programming in general. We show that, in a certain way, Lustigs method for obtaining feasibility is equivalent to Newtons method. This demonstrates that the method is in some sense the natural way to reduce infeasibility. The role of the barrier parameter in computational practice is studied in detail. Numerical results are given for the entire expanded NETLIB test set for the basic algorithm and its variants, as well as version 5.3 of MINOS .

ON IMPLEMENTING MEHROTRA'S PREDICTOR-CORRECTOR INTERIOR-POINT METHOD FOR LINEAR PROGRAMMING*

Mehrotra [Tech. Report 90-03, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1990] recently described a predictor–corrector variant of the primal–dual interior-point algorithm for linear programming. This paper describes a full implementation of this algorithm, with extensions for solving problems with free variables and problems with bounds on primal variables. Computational results on the NETLIB test set are given to show that this new method almost always improves the performance of the primal–dual algorithm and that the improvement increases dramatically as the size and complexity of the problem increases. A numerical instability in using Schur complements to remove dense columns is identified, and a numerical remedy is given.

Very large-scale linear programming: a case study in combining interior point and simplex methods

Experience with solving a 12.753.313 variable linear program is described. This problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application. A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables. Experience using the simplex method as implemented in CPLEX, an interior point method as implemented in OBI, and a hybrid interior point/simplex approach is reported. The resulting procedure illustrates the power of an interior point/simplex combination for solving very large-scale linear programs.

Feature Article—Interior Point Methods for Linear Programming: Computational State of the Art

A survey of the significant developments in the field of interior point methods for linear programming is presented, beginning with Karmarkars projective algorithm and concentrating on the many variants that can be derived from logarithmic barrier methods. Full implementation details of the primal-dual predictor-corrector code OB1 are given, including preprocessing, matrix orderings, and matrix factorization techniques. A computational comparison of OB1 with a state-of-the-art simplex code using eight large models is given. In addition, computational results are presented where OB1 is used to solve two very large models that have never been solved by any simplex code INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Feasibility issues in a primal-dual interior-point method for linear programming

A new method for obtaining an initial feasible interior-point solution to a linear program is presented. This method avoids the use of a “big-M”, and is shown to work well on a standard set of test problems. Conditions are developed for obtaining a near-optimal solution that is feasible for an associated problem, and details of the computational testing are presented. Other issues related to obtaining and maintaining accurate feasible solutions to linear programs with an interior-point method are discussed. These issues are important to consider when solving problems that have no primal or dual interior-point feasible solutions.

Multicommodity network flows: The impact of formulation on decomposition

This paper investigates the impact of problem formulation on Dantzig—Wolfe decomposition for the multicommodity network flow problem. These problems are formulated in three ways: origin-destination specific, destination specific, and product specific. The path-based origin-destination specific formulation is equivalent to the tree-based destination specific formulation by a simple transformation. Supersupply and superdemand nodes are appended to the tree-based product specific formulation to create an equivalent path-based product specific formulation. We show that solving the path-based problem formulations by decomposition results in substantially fewer master problem iterations and lower CPU times than by using decomposition on the equivalent tree-based formulations. Computational results on a series of multicommodity network flow problems are presented.

A primal partitioning solution for the arc-chain formulation of a multicommodity network flow problem

We present a new solution approach for the multicommodity network flow problem (MCNF) based upon both primal partitioning and decomposition techniques, which simplifies the computations required by the simplex method. The partitioning is performed on an arc-chain incidence matrix of the MCNF, similar within a change of variables to the constraint matrix of the master problem generated in a Dantzig-Wolfe decomposition, to isolate a very sparse, near-triangular working basis of greatly reduced dimension. The majority of the simplex operations performed on the partitioned basis are simply additive and network operations specialized for the nine possible pivot types identified. The columns of the arc-chain incidence matrix are generated by a dual network simplex method for updating shortest paths when link costs change.

Computational experience with a globally convergent primal-dual predictor-corrector algorithm for linear programming

Kojima, Megiddo, and Mizuno proved global convergence of a primal—dual algorithm that corresponds to methods used in practice. Here, the numerical efficiency of a predictor—corrector extension of that algorithm is tested. Numerical results are extremely positive, indicating that the safety of a globally convergent algorithm can be obtained at little computational cost. The algorithm is tested on infeasible problems with less success. Finally, the algorithm is applied to a warm started problem, with very encouraging preliminary results.

The interior-point method for linear programming

A robust, reliable, and efficient implementation of the primal-dual interior-point method for linear programs, which is based on three well-established optimization algorithms, is presented. The authors discuss the theoretical foundation for interior-point methods which consists of three crucial building blocks: Newtons method for solving nonlinear equations, Joseph Lagranges methods for optimization with equality constraints, and Fiacco and McCormicks barrier method for optimization with inequality constraints. The construction of the primal-dual interior-point method using these methods is described. An implementation of the primal-dual interior-point method, its performance, and a comparison to other interior-point methods are also presented.<<ETX>>

Separable Quadratic Programming via a Primal-Dual Interior Point Method and its Use in a Sequential Procedure

This paper extends a primal-dual interior point procedure for linear programs to the case of convex separable quadratic objectives. Included are efficient procedures for: attaining primal and dual feasibility, variable upper bounding, and free variables. A sequential procedure that invokes the quadratic solver is proposed and implemented for solving linearly constrained convex separable nonlinear programs. Computational results are provided for several large test cases from stochastic programming. The proposed methods compare favorably with MINOS, especially for the larger examples. The nonlinear programs range in size up to 8,700 constraints and 22,000 variables. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.