A GENERAL DETERMINISTIC PIVOT METHOD FOR ORIENTED MATROID PROGRAMMING

A new pivot method for oriented matroid progiamming is given out. This mathod is deterministic by nature and is general in the sense that its flexible pivot selection rule allows a family of possible algorithms to be its special cases, including the so called criss-cross algorithm and the Edmonds-Fukuda algorithm as well. As an example of a special implementation of our general method, an extended version of the Edmonds-Fukuda algorithm is presented.     

Basis- and partition identification for quadratic programming and linear complementarity problems

Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximally complementary solutions. Maximally complementary solutions can be characterized by optimal partitions. On the other hand, the solutions provided by simplex–based pivot algorithms are given in terms of complementary bases. A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A partition identification algorithm is an algorithm which generates a maximally complementary solution (and its corresponding partition), starting from any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus. Received April 9, 1996 / Revised version received April 27, 1998? Published online May 12, 1999     

Decomposability in fixed point computation with applications and acceleration techniques

In the past decade, several complementary pivot algorithms have been developed to search for fixed points of certain functions and point to set maps on unbounded regions. This paper develops a structure (called decomposability), which, when present, enables one to work in a lower dimensional space when solving these problems. Several examples of where this structure arises in applications are presented. It is shown that under suitable circumstances, the general constrained optimization problem (that of optimizing an objective function subject to both equality and inequality constraints) may be formulated as a decomposèble fixed point problem. At the same time, an approximation technique is developed to potentially improve the efficiency of the complementary pivot algorithms.     

The s-monotone index selection rules for pivot algorithms of linear programming

In this paper we introduce the concept of s-monotone index selection rule for linear programming problems. We show that several known anti-cycling pivot rules like the minimal index, Last-In–First-Out and the most-often-selected-variable pivot rules are s-monotone index selection rules. Furthermore, we show a possible way to define new s-monotone pivot rules. We prove that several known algorithms like the primal (dual) simplex, MBU-simplex algorithms and criss-cross algorithm with s-monotone pivot rules are finite methods.     

Elimination with Noninvertible Pivots

A method for carrying out the Gauss elimination solution of linear systems is presented. The novelty arises from the fact that the pivot matrices are not required to be invertible, so that, for example, a scalar pivot may be zero, and a matrix pivot may be rectangular or singular. The need to execute the elimination algorithm in such circumstances arose in connection with a finite element solution of the Navier-Stokes equations in velocity-pressure variables. This application is briefly discussed, as is a method for the implementation of the algorithm.     

Residual reduction algorithms for nonsymmetric saddle point problems

In this paper, we introduce and analyze Uzawa algorithms for non-symmetric saddle point systems. Convergence for the algorithms is established based on new spectral results about Schur complements. A new Uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework. Numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state Navier-Stokes equations.     

Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems with General Constraints

In Ref. 1, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints was proposed. At each iteration, this new algorithm only needs to solve four systems of linear equations having the same coefficient matrix, which is much less than the amount of computation required for existing SQP algorithms. Moreover, unlike the quadratic programming subproblems of the SQP algorithms (which may not have a solution), the subproblems of the SSLE algorithm are always solvable. In Ref. 2, it is shown that the new algorithm can also be used to deal with nonlinear optimization problems having both equality and inequality constraints, by solving an auxiliary problem. But the algorithm of Ref. 2 has to perform a pivoting operation to adjust the penalty parameter per iteration. In this paper, we improve the work of Ref. 2 and present a new algorithm of sequential systems of linear equations for general nonlinear optimization problems. This new algorithm preserves the advantages of the SSLE algorithms, while at the same time overcoming the aforementioned shortcomings. Some numerical results are also reported.     

Criss-cross methods: A fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon.     

Computational aspects of simplex and MBU-simplex algorithms using different anti-cycling pivot rules

Abstract

Several variations of index selection rules for simplex-type algorithms for linear programming, like the Last-In-First-Out or the Most-Often-Selected-Variable are rules not only theoretically finite, but also provide significant flexibility in choosing a pivot element. Based on an implementation of the primal simplex and the monotonic build-up (MBU) simplex method, the practical benefit of the flexibility of these anti-cycling pivot rules is evaluated using public benchmark LP test sets. Our results also provide numerical evidence that the MBU-simplex algorithm is a viable alternative to the traditional simplex algorithm.     

CONTACT ALGORITHMS FOR CONTACT DYNAMICAL SYSTEMS

1.IntroductionContactstructureisananalogofasymplecticoneforodd-dimensionalmanifolds,itstemsfrommanifoldsofcontactelementsofconfigurationspacesinmechanicsand,therefore,itisalsoofbasicimportanceinphysicalandengineeringsciences.Weapplytinthispaperttheideasof…     

An iterative back substitution algorithm for the solution of tridiagonal matrix systems with fringes

For tridiagonal matrix systems, a simple direct algorithm giving the solution exists, but in the most general case of tridiagonal matrix with fringes, the direct solving algorithms are more complicated. For big systems, direct methods are not well fitted and iterative algorithms are preferable. In this paper a relaxation type iterative algorithm is presented. It is an extension of the backward substitution method used for simple tridiagonal matrix systems. The performances show that this algorithm is a good compromise between a direct method and other iterative methods as block SOR. Its nature suggests its use as inner solver in the solution of problems derived by application of a decomposition domain method. A special emphasis is done on the programming aspect. The solving Fortran subroutines implementing the algorithm have been generated automatically from their specification by using a computer algebra system technique.     

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.     

Extensions of Lemke's algorithm for the linear complementarity problem

Lemke's algorithm for the linear complementarity problem fails when a desired pivot is not blocked. A projective transformation overcomes this difficulty. The transformation is performed computationally by adjoining a new row to a schema of the problem and pivoting on the element in this row and the unit constant column. Two new algorithms result; some conditions for their success are discussed.This research was partially supported by National Science Foundation, Grant GK-42092.     

Reinversion with the preassigned pivot procedure

Mathematical programming computer systems using the product form in the inverse (PFI) must periodically resort to a reinversion with the current basis in order to reduce the amount of work to be done in the succeeding iterations.In this paper, we show the consequences of column, pivot selection and sequence upon the transformation vector (ETA) density and give an algorithm which will tend to minimize eta density and work done per iteration.The algorithm has been implemented and tested as a replacement for the previous inversion algorithm on the OPTIMA system for the CDC 6000 computers and on the MPS/III mathematical programming system for the IBM 360 computer. A comparative performance table is given.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.     

The local convergence of ABS methods for nonlinear algebraic equations

Summary In this paper we consider an extension to nonlinear algebraic systems of the class of algorithms recently proposed by Abaffy, Broyden and Spedicato for general linear systems. We analyze the convergence properties, showing that under the usual assumptions on the function and some mild assumptions on the free parameters available in the class, the algorithm is locally convergent and has a superlinear rate of convergence (per major iteration, which is computationally comparable to a single Newton's step). Some particular algorithms satisfying the conditions on the free parameters are considered.     

Parallel search paths for the simplex algorithm

It is well known that the simplex method is inherently a sequential algorithm with little scope for parallelization. Even so, during the last decades several attempts were made to parallelize it since it is one of the most important algorithms for solving linear optimization problems. Such parallelization ideas mostly rely on iteration parallelism and overlapping. Since the simplex method goes through a series of basic solutions until it finds an optimal solution, each of them must be available before performing the next basis change. This phenomenon imposes a limit on the performance of the parallelized version of the simplex method which uses overlapping iterations. Another approach can be considered if we think about alternative paths on the n-dimensional simplex polyhedron. As the simplex method goes through the edges of this polyhedron it is generally true that the speed of convergence of the algorithm is not smooth. It depends on the actual part of the surface. If a parallel version of the simplex algorithm simultaneously goes on different paths on this surface a highly reliable algorithm can be constructed. There is no known dominating strategy for pivot selection. Therefore, one can try different pivot selection methods in parallel in order to guide the algorithm on different pathways. This approach can be used effectively with periodic synchronization on shared memory multi-core computing environments to speed up the solution algorithm and get around numerically and/or algorithmically difficult situations throughout the computations.     

A general projection algorithm for solving systems of linear equations

In this paper, we give a general projection algorithm for implementing some known extrapolation methods such as the MPE, the RRE, the MMPE and others. We apply this algorithm to vectors generated linearly and derive new algorithms for solving systems of linear equations. We will show that these algorithms allow us to obtain known projection methods such as the Orthodir or the GCR.     

New efficient shortest path simplex algorithm: pseudo permanent labels instead of permanent labels

We introduce a new network simplex pivot rule for the shortest path simplex algorithm. This new pivot rule chooses a subset of non-basic arcs to simultaneously enter into the basis. We call this operation a multiple pivot. We show that a shortest path simplex algorithm with this pivot rule performs O(n) multiple pivots and runs in O(nm) time. Our pivot rule is based on the new concept of a pseudo permanently labeled node, and it can be adapted to design a new label-correcting algorithm that runs in O(nm). Moreover, this concept lets us introduce new rules to identify negative cycles. Finally, we compare the network simplex algorithm with multiple pivots with other previously proposed efficient network simplex algorithm in a computational experiment.     

Parallel arc-allocation algorithms for optimizing generalized networks

Two parallel shared-memory algorithms are presented for the optimization of generalized networks. These algorithms are based on the allocation of arc-related operations in the (generalized) network simplex method. One method takes advantage of the multi-tree structure of basic solutions and performs pivot operations in parallel, utilizing locking to ensure correctness. The other algorithm utilizes only one processor for sequential pivoting, but parallelizes the pricing operation and overlaps this task with pivoting in a speculative manner (i.e. since pivoting and pricing involve data dependencies, a candidate for flow change generated by the pricing processors is not guaranteed to be acceptable, but in practice generally has this property). The relative performance of these two methods (on the Sequent Symmetry S81 multiprocessor) is compared and contrasted with that of a fast sequential algorithm on a set of large-scale test problems of up to 1,000,000 arcs.This research was supported in part by NSF grant CCR-8709952 and AFOSR grant AFOSR-86-0194.     

On the solution of linear equation/inequality systems

A pivotal algebra algorithm is given and finite convergence shown for finding a vector which satisfies an arbitrary system of linear equations and/or inequalities. A modified form of the algorithm, obtained by introducing a redundant equation, is then shown to be a way to describe phase I of the simplex method without reference to artificial variables or an artificial objective function.The hypothesis is introduced that in each pivot stage each row of the tableau has equal probability of being chosen as the pivot row. Under this assumption the expected value of the ratio of the number of pivot stages to the number of rows should grow with the natural Log of the number of rows.Use of the algorithm in proving theorems of the alternative is indicated.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.