Creating advanced bases for large scale linear programs exploiting embedded network structure

In this paper, we investigate how an embedded pure network structure arising in many linear programming (LP) problems can be exploited to create improved sparse simplex solution algorithms. The original coefficient matrix is partitioned into network and non-network parts. For this partitioning, a decomposition technique can be applied. The embedded network flow problem can be solved to optimality using a fast network flow algorithm. We investigate two alternative decompositions namely, Lagrangean and Benders. In the Lagrangean approach, the optimal solution of a network flow problem and in Benders the combined solution of the master and the subproblem are used to compute good (near optimal and near feasible) solutions for a given LP problem. In both cases, we terminate the decomposition algorithms after a preset number of passes and active variables identified by this procedure are then used to create an advanced basis for the original LP problem. We present comparisons with unit basis and a well established crash procedure. We find that the computational results of applying these techniques to a selection of Netlib models are promising enough to encourage further research in this area

MINET: Fast Network LP Solver. Description and User's Guide for V2.00

This paper extends the WP-87-050 in the direction to a detailed discussion of pricing and anti-degeneracy strategies applicable in solutions of network LP solvers. Besides the theoretical part it also serves as a users guide to the existing software

MINET a fast Network LP Solver

In comparison with already existing software for the solution of network type linear programming problems. MINET gives a possibility of very flexible pricing that can be further fitted to the special structure of the network and using a suitable interface, it can reflect the need for changing the network structure

An enhanced piecewise linear dual phase-1 algorithm for the simplex method

A dual phase-1 algorithm for the simplex method that handles all types of vari- ables is presented. In each iteration it maximizes a piecewise linear function of dual infeasibilities in order to make the largest possible step towards dual feasibility with a selected outgoing variable. The algorithm can be viewed as a generalization of traditional phase-1 procedures. It is based on the multiple use of the expensively computed pivot row. By small amount of extra work per iteration, the progress it can make is equivalent to many iterations of the traditional method. While this is its most important feature, it possesses some additional favorable properties, namely, it can be efficient in coping with degeneracy and numerical difficulties. Both theo- retical and computational issues are addressed. Some computational experience is also reported which shows that the potentials of the method can materialize on real world problems. This paper is based on IC Departmental Technical Report 2000/13 and contains an enhancement of the main algorithm

A general phase-I method in linear programming

The basic technique for solving LP problems is still the simplex method [2]. It has many variants but in practice the primal simplex methods are considered the most important. Phase I of the primal methods serves for finding a basic feasible solution but the same procedure can also be used for generating feasible points of other problems with linear constraints, or even for checking the consistency of a system of linear equalities/inequalities. Since in Phase I the objective function is not or only slightly considered it usually does not move towards optimality. It would be advantageous if Phase I were as short as possible or if it could better take into account the real objective function. In this paper we try to contribute to both of these aspects positively

Computational study of the GMDPO dual phase-1 algorithm

Maros's GDPO algorithm for phase-1 of the dual simplex method possesses some theoretical features that have potentially huge computational advantages. This paper gives account of a computational analysis of GDPO. Experience of a systematic study involving 48 problems shows that the predicted performance advantages can materialize to a large extent making GDPO an indispensable tool for dual pahase-1

A piecewise linear dual phase-1 algorithm for the simplex method with all types of variable

A dual phase-1 algorithm for the simplex method that handles all types of variables is presented. In each iteration it maximizes a piecewise linear function of dual infeasibilities in order to make the largest possible step towards dual feasibility with a selected outgoing variable. The new method can be viewed as a generalization of traditional phase-1 procedures. It is based on the multiple use of the expensively computed pivot row. By small amount of extra work per iteration, the progress it can make is equivalent to many iterations of the traditional method. In addition to this main achievement it has some further important and favorable features, namely, it is very efficient in coping with degeneracy and numerical diffculties. Both theoretical and computational issues are addressed. Examples are also given that demonstrate the power and flexibility of the method

An enhanced piecewise linear dual phase-1 algorithm for the simplex method

A dual phase-1 algorithm for the simplex method that handles all types of vari- ables is presented. In each iteration it maximizes a piecewise linear function of dual infeasibilities in order to make the largest possible step towards dual feasibility with a selected outgoing variable. The algorithm can be viewed as a generalization of traditional phase-1 procedures. It is based on the multiple use of the expensively computed pivot row. By small amount of extra work per iteration, the progress it can make is equivalent to many iterations of the traditional method. While this is its most important feature, it possesses some additional favorable properties, namely, it can be e cient in coping with degeneracy and numerical di culties. Both theo- retical and computational issues are addressed. Some computational experience is also reported which shows that the potentials of the method can materialize on real world problems. This paper is based on IC Departmental Technical Report 2000/13 and contains an enhancement of the main algorithm

OSCP: Optimization Service Connectivity Protocol

Optimization software e.g. solvers and modelling systems, expose software and vendor specific interfacing mechanisms to client applications e.g. decision support systems, thus introducing close coupling. The ‘Optimization Service Connectivity Protocol’ (OSCP) is an abstraction of the interfaces to optimization software, which is aimed primarily at simplifying the process of integrating optimization systems into software solutions by providing an abstracted, uniform and easy to use interface to such systems, regardless of system or vendor specific requirements. This paper presents a high-level overview of OSCP including descriptions of its main interfaces, and illustrates its use via examples

An outer approximation based branch and cut algorithm for convex 0-1 MINLP problems

A branch and cut algorithm is developed for solving 0-1 MINLP problems. The algorithm integrates Branch and Bound, Outer Approximation and Gomory Cutting Planes. Only the initial Mixed Integer Linear Programming (MILP) master problem is considered. At integer solutions Nonlinear Programming (NLP) problems are solved, using a primal-dual interior point algorithm. The objective and constraints are linearized at the optimum solution of those NLP problems and the linearizations are added to all the unsolved nodes of the enumerations tree. Also, Gomory cutting planes, which are valid throughout the tree, are generated at selected nodes. These cuts help the algorithm to locate integer solutions quickly and consequently improve the linear approximation of the objective and constraints, held at the unsolved nodes of the tree. Numerical results show that the addition of Gomory cuts can reduce the number of nodes in the enumeration tree