Alternative methods for representing the inverse of linear programming basis matrices

Methods for representing the inverse of Linear Programming (LP) basis matrices are closely related to techniques for solving a system of sparse unsymmetric linear equations by direct methods. It is now well accepted that for these problems the static process of reordering the matrix in the lower block triangular (LBT) form constitutes the initial step. We introduce a combined static and dynamic factorisation of a basis matrix and derive its inverse which we call the partial elimination form of the inverse (PEFI). This factorization takes advantage of the LBT structure and produces a sparser representation of the inverse than the elimination form of the inverse (EFI). In this we make use of the original columns (of the constraint matrix) which are in the basis. To represent the factored inverse it is, however, necessary to introduce special data structures which are used in the forward and the backward transformations (the two major algorithmic steps) of the simplex method. These correspond to solving a system of equations and solving a system of equations with the transposed matrix respectively. In this paper we compare the nonzero build up of PEFI with that of EFI. We have also investigated alternative methods for updating the basis inverse in the PEFI representation. The results of our experimental investigation are presented in this pape

A program to reorder and solve sparse unsymmetric linear systems using the envelope method

The envelope data structure and the Choleski based (bordering) method for the solution of symmetric sparse systems of linear equations have been extended by the authors to solve unsymmetric systems of linear equations. The data structures used in this general linear equation Solver and a set of FORTRAN 77 subroutines are described. Some test data (extracted from LP problems as basis matrices) together with experimental results are presented

Experimental investigation of an interior search method within a simple framework

A steepest gradient method for solving Linear Programming (LP) problems, followed by a procedure for purifying a non-basic solution to an improved extreme point solution have been embedded within an otherwise simplex based optimiser. The algorithm is designed to be hybrid in nature and exploits many aspects of sparse matrix and revised simplex technology. The interior search step terminates at a boundary point which is usually non-basic. This is then followed by a series of minor pivotal steps which lead to a basic feasible solution with a superior objective function value. It is concluded that the procedures discussed in this paper are likely to have three possible applications, which are (i) improving a non-basic feasible solution to a superior extreme point solution, (iii) an improved starting point for the revised simplex method, and (iii) an efficient implementation of the multiple price strategy of the revised simplex method

Experimental investigations in combining primal dual interior point method and simplex based LP solvers

The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or the stalling problem of the SSX, indeed it reaches the 'near optimum' solution very quickly. The SSX algorithm, in contrast, is not affected by the boundary conditions which slow down the convergence of the PD. If the solution to the LP problem is non unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby cross over from PD to SSX can take place at any stage of the PD optimization run. The cross over to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery

Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration

An investigation of pricing strategies within simplex

The PRICE step within the revised simplex method for the LP problems is considered in this report. Established strategies which have proven to be computationally efficient are first reviewed. A method based on the internal rate of return is then described. The implementation of this method and the results obtained by experimental investigation are discussed

An efficient application of goal programming to tackle multiobjective problems with recurring fitness landscapes

Many real-world applications require decision-makers to assess the quality of solutions while considering multiple conflicting objectives. Obtaining good approximation sets for highly constrained many objective problems is often a difficult task even for modern multiobjective algorithms. In some cases, multiple instances of the problem scenario present similarities in their fitness landscapes. That is, there are recurring features in the fitness landscapes when searching for solutions to different problem instances. We propose a methodology to exploit this characteristic by solving one instance of a given problem scenario using computationally expensive multiobjective algorithms to obtain a good approximation set and then using Goal Programming with efficient single-objective algorithms to solve other instances of the same problem scenario. We use three goal-based objective functions and show that on benchmark instances of the multiobjective vehicle routing problem with time windows, the methodology is able to produce good results in short computation time. The methodology allows to combine the effectiveness of state-of-the-art multiobjective algorithms with the efficiency of goal programming to find good compromise solutions in problem scenarios where instances have similar fitness landscapes

Role of Caustic Addition in Bitumen-Clay Interactions

Coating of bitumen by clays, known as slime coating, is detrimental to bitumen recovery from oil sands using the warm slurry extn. process. Sodium hydroxide (caustic) is added to the extn. process to balance many competing processing challenges, which include undesirable slime coating. The current research aims at understanding the role of caustic addn. in controlling interactions of bitumen with various types of model clays. The interaction potential was studied by quartz crystal microbalance with dissipation monitoring (QCM-D). After confirming the slime coating potential of montmorillonite clays on bitumen in the presence of calcium ions, the interaction of kaolinite and illite with bitumen was studied. To represent more closely the industrial applications, tailings water from bitumen extn. tests at different caustic dosage was used. At caustic dosage up to 0.5 wt % oil sands ore, a negligible coating of kaolinite on the bitumen was detd. However, at a lower level of caustic addn., illite was shown to attach to the bitumen, with the interaction potential decreasing with increasing caustic dosage. Increasing concn. of humic acids as a result of increasing caustic dosage was identified to limit the interaction potential of illite with bitumen. This fundamental study clearly shows that the crit. role of caustics in modulating interactions of clays with bitumen depends upon the type of clays. Thus, clay type was identified as a key operational variable