Reduction In Solution Time Research Articles

Modeling languages and Condor: metacomputing for optimization

A generic framework is postulated for utilizing the computational resources provided by a metacomputer to concurrently solve a large number of optimization problems generated by a modeling language. An example of the framework using the Condor resource manager and the AMPL and GAMS modeling languages is provided. A mixed integer programming formulation of a feature selection problem from machine learning is used to test the mechanism developed. Due to this application’s computational requirements, the ability to perform optimizations in parallel is necessary in order to obtain results within a reasonable amount of time. Details about the simple and easy to use tool and implementation are presented so that other modelers with applications generating many independent mathematical programs can take advantage of it to significantly reduce solution times.

Optimal instruction scheduling using integer programming

This paper presents a new approach to local instruction scheduling based on integer programming that produces optimal instruction schedules in a reasonable time, even for very large basic blocks. The new approach first uses a set of graph transformations to simplify the data-dependency graph while preserving the optimality of the final schedule. The simplified graph results in a simplified integer program which can be solved much faster. A new integer-programming formulation is then applied to the simplified graph. Various techniques are used to simplify the formulation, resulting in fewer integer-program variables, fewer integer-program constraints and fewer terms in some of the remaining constraints, thus reducing integer-program solution time. The new formulation also uses certain adaptively added constraints (cuts) to reduce solution time. The proposed optimal instruction scheduler is built within the Gnu Compiler Collection (GCC) and is evaluated experimentally using the SPEC95 floating point benchmarks. Although optimal scheduling for the target processor is considered intractable, all of the benchmarks' basic blocks are optimally scheduled, including blocks with up to 1000 instructions, while total compile time increases by only 14%.

A Column Generation Approach for Large-Scale Aircrew Rostering Problems

This article describes a method for solving the crew rostering problem in air transportation. This problem consists of constructing personalized schedules that assign pairings, days off, and other activities to airline crew members. A generalized set partitioning model and a method using column generation have been used. This method has been adapted in a number of ways to take advantage of the nature of the problem and to accelerate solution. Numerical tests on problems from Air France have demonstrated that this method is capable of solving very large scale problems with thousands of constraints and hundreds of subproblems. The tests have also shown that these adaptations are capable of reducing solution time by a factor of about a thousand. Finally, results from this method are compared with those obtained with the method currently used at Air France.

An Interior-Point Algorithm for Nonconvex Nonlinear Programming

The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.

Projection techniques for iterative solution of [formula omitted] with successive right-hand sides

Projection techniques are developed for computing approximate solutions to linear systems of the form A x n = b n , for a sequence n = 1, 2,…, e.g. arising from time discretization of a partial differential equation. The approximate solutions are based upon previous solutions, and can be used as initial guesses for iterative solution of the system, resulting in significantly reduced computational expense. Examples of two- and three-dimensional incompressible Navier-Stokes calculations are presented in which x n represents the pressure at time level t n , and A is a consistent discrete Poisson operator. In flows containing significant dynamic activity, these projection techniques lead to as much as a two-fold reduction in solution time.

Solution procedures for the dynamic facility layout problem

Existing optimal solution procedures for the dynamic facility layout problem require the repeated solution of quadratic assignment problems within the framework of a dynamic program. The computational requirements for this problem necessitate the development of efficient algorithms for special cases of the problem and strong bounds for the general case of the problem. The concept of incomplete dynamic programming is applied to the dynamic facility layout problem to find the optimal solution to the problem with fixed rearrangement costs at exceptionally reduced solution times. Heuristics are also developed to provide a solution methodology for larger problems. A lower bound is developed for the general problem that dominates all existing bounds, and it is shown how the bound can be used as an initial test for optimality before the dynamic program is solved. Finally, the incomplete dynamic programming methodology is extended to find an upper bound for the general problem that dominates all previous bounds.

Secondary log breakdown optimization with dynamic programming

Log breakdown can be viewed as a two-stage process with logs sawn into slabs of wood known as flitches during the primary stage and flitches further processed during the secondary stage to produce edged (cut lengthwise) and trimmed (cut widthwise) pieces. This paper addresses the secondary problem and describes some procedures for determining the optimal cutting of flitches into graded dimensional boards. The problem is formulated as a set packing problem with the objective being to maximise total value. Extensions to the basic formulation include constraints which restrict the number of saws and which disallow waste between adjacent edged pieces. The problem is solved using dynamic programming techniques and the algorithms incorporated into a sawing simulation system. Comparisons with existing edging and trimming procedures show that substantial reductions in solution time (to as little as 1/25th of the time required for an enumerative search) can be achieved.

Computation-free preconditioners for the parallel solution of power system problems

Solution of a set of linear equations Ax=b is a recurrent problem in power system analysis. Because of computational dependencies, direct methods have proven to be of limited value in both parallel and highly vectorized computing environments. The preconditioned conjugate gradient method has been suggested as a better alternative to direct methods. The preconditioning step itself is not particularly well suited to parallel processing. Partitioned inverse representations of A are better suited to high performance computation. However, obtaining the partitioned inverse matrices can be expensive. This paper describes two techniques for preconditioning based on the partitioned inverses where the preconditioner matrix is obtained directly from an incomplete factorization without the need for additional numerical computation. Experiments indicate a 50% reduction in solution time in a parallel environment.

New MILP Models for Scheduling of Multiproduct Batch Plants under Zero-Wait Policy

This paper presents a new mixed-integer linear programming (MILP) model in which makespan is minimized for the multiproduct batch plants. In particular, the zero-wait (ZW) scheduling with transfer and setup times is analyzed. Not only idle times between the successive products but also heads and tails are used to select a possible production sequence with the minimum makespan. To determine the suitable head and tail, new binary variables are defined and aggregated to the assignment constraints of traveling salesman problems (TSPs). Although the number of binary variables increases, the mathematical formulation can yield a very compact MILP model. The effectiveness of the proposed model is demonstrated through several examples. The proposed model is extended to solve the scheduling problems for productions with single product campaigns (SPCs) and mixed product campaigns (MPCs) explained by Birewar and Grossmann (Ind. Eng. Chem. Res. 1989b, 28, 1333). The solutions by the proposed method are compared with the solutions obtained from the rigorous solution method by Birewar and Grossmann. The proposed model is formulated as a compact size and leads to a significant reduction in solution times.

Three-dimensional uniform grid modeling of electrical defibrillation on a data parallel computer

Finite element modeling has played an increasingly important role in the study of defibrillation. In order to model well the complex anatomical details, a large number of elements are required in the finite element grid, leading to a large set of equations that often cannot be solved effectively with the computational power of conventional computers. In this paper, we describe the use of a data parallel computer to provide the memory and reduction in solution time for solving these large finite element problems. Using a uniform grid and a nodal assembly technique, the discretized problem domain can be mapped efficiently to the parallel computer, allowing the solution of problems with over two million unknowns. The finite element algorithm for a three-dimensional inhomogeneous anisotropic body is described together with its parallel implementation. Test results for a canine torso model constructed from CT images are also presented.

Choosing the relaxation parameter for the solution of nonlinear magnetic field problems by the Newton-Raphson method

The use of the modified Newton method for the solution of nonlinear magnetostatic problems arising from the vector potential Finite Element analysis is investigated. In particular the optimum choice of the relaxation factor /spl alpha/ is investigated. A new method is developed for determining the relaxation factor which minimizes the energy functional in the direction along the solution update at each nonlinear iteration. This method is based on approximating the functional with a fourth order polynomial. In this way the optimum relaxation factor can be quickly determined with the minimum number of extra function evaluations. This choice of /spl alpha/ is compared to choosing the relaxation factor which minimizes the residual norm at each iteration. The modified Newton methods are compared to the standard Newton-Raphson method for the solution of 2D and 3D problems. For problems involving saturated iron parts convergence rates are greatly improved by use of the modified Newton methods. Of the two methods for choosing the relaxation factor the one which minimizes the functional is shown to be the better. Using the new algorithm to determine this relaxation factor results in substantial reductions in solution times. >

Numerical analysis of electrical defibrillation: The parallel approach

Numerical modeling offers a viable tool for studying electrical defibrilation, allowing the behavior of field quantities to be observed easily as the different system parameters are varied. One numerical technique, namely the finite-element method, has been found particularly effective for modeling complex thoracic anatomies. However, an accurate finite-element model of the thorax often requires a large number of elements and nodes, leading to a large set of equations that cannot be solved effectively with the computational power of conventional computers. This is especially true if many finite-element solutions need to be achieved within a reasonable time period (eg, electrode configuration optimization). In this study, the use of massively parallel computers to provide the memory and reduction in solution time for solving these large finite-element problems is discussed. Both the uniform and unstructured grid approaches are considered. Algorithms that allow efficient mapping of uniform and unstructured grids to data-parallel and message-passing parallel computers are discussed. An automatic iterative procedure for electrode configuration optimization is presented. The procedure is based on the minimization of an objective function using the parallel direct search technique. Computational performance results are presented together with simulation results.

A model and strategy for train pathing with choice of lines, platforms, and routes

Train pathing is concerned with assigning trains and train times for a set of rail links, stations stops, etc., so as to meet a system of constraints on headways, trip times, dwell times, etc. while minimizing delays or costs and meeting travel demands. In a previous paper we presented a model, algorithms, and strategy for pathing trains of different speeds and stopping patterns for a double track rail line dedicated to trains in one direction. Here we extend this to more general more complex rail networks, with choice of lines, station platforms, etc, as is more typical of the high density scheduled passenger railways in Britain and Europe. We apply the model to a small network and find acceptable solution times. Applying addition search strategies from the previous paper should reduce solution times by further orders of magnitude.

On the convergence behavior of the restarted GMRES algorithm for solving nonsymmetric linear systems

AbstractThe solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s>0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented.

A penalty for concave minimization derived from the tuy cutting plane

A wide variety of optimization problems have been approached with branch-and-bound methodology, most notably integer programming and continuous nonconvex programming. Penalty calculations provide a means to reduce the number of subproblems solved during the branch-and-bound search. We develop a new penalty based on the Tuy cutting plane for the nonconvex problem of globally minimizing a concave function over linear constraints and continuous variables. Computational testing with a branch-and-bound algorithm for concave minimization indicates that, for the problems solved, the penalty reduces solution time by a factor ranging from 1.2 to 7.2. © 1994 John Wiley & Sons, Inc.

Reducing solution time in monochromatic FDTD waveguide simulations

Rectangular waveguide is modelled using the Finite-Difference Time-Domain Method. It is shown that when using a monochromatic excitation that the steady state solution can take an undue amount of time to achieve. A simple technique is presented which minimises the duration of the resulting transient time enabling the steady state to be reached in a much shorter amount of time.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The waveform relaxation method in the concurrent dynamic process simulation

We investigate the concurrent solution of low-index differential-algebraic equations (DAEas) by the waveform relaxation (WR) method, an iterative method for system integration. The WR method (or Picard—Lindelöf iteration) is an operator-splitting approach to the solution of a system of DAEs partitioned into several lower-order systems (subsystems). Such a method, when efficiently implemented, results in algorithms with a highly parallelizable concurrent fraction and low sequential overhead, making them especially suitable for coarse- and medium-grain MIMD distributed-memory machines. Since problems represented by DAEs cannot be “scaled”, our performance measure is reduction in solution time for fixed problems, and the Amdahl's-law bottleneck therefore concerns us. We describe our new simulation code DAWRS (Differential—Algebraic—Waveform Relaxation Solver), written in C, to solve DAEs on parallel machines using the WR methods. We discuss the system partitioning, and describe new techniques to improve both the local and global convergence of such methods. We demonstrate the achievable concurrent performance when solving DAEs for a class of dynamic simulation applications of chemical engineering.

The waveform relaxation method in the concurrent dynamic process simulation

We investigate the concurrent solution of low-index differential-algebraic equations (DAEs) by the waveform relaxation (WR) method, an iterative method for system integration. The WR method (or Picard-Lindelöf iteration) is an operator-splitting approach to the solution of a system of DAEs partitioned into several lower-order systems (subsystems). Such a method, when efficiently implemented, results in algorithms with a highly parallelizable concurrent fraction and low sequential overhead, making them especially suitable for coarse- and medium-grain MIMD distributed-memory machines. Since problems represented by DAEs cannot be “scaled,” our performance measure is reduction in solution time for fixed problems, and the Amdahl's-law bottleneck therefore concerns us. We describe our new simulation code DAWRS (Differential — Algebraic -Waveform Relaxation Solver), written in C, to solve DAEs on parallel machines using the WR methods. We discuss the system partitioning, and describe new techniques to improve both the local and global convergence of such methods. We demonstrate the achievable concurrent performance when solving DAEs for a class of dynamic simulation applications in chemical engineering.

An efficient method for the free surface seepage flow problems

Based on the concept of ‘true air element’ and ‘true node’ proposed by the authors for the modelling of changing mesh problems in geotechnical problems, a simple but efficient finite element procedure for the solution of free surface seepage flow is developed. In the present method, the mesh used for the analysis will remain unchanged. Very rapid convergence with significant reduction in solution time and computer memory are achieved with the proposed method. Some interesting results in the seepage flow problems which are commonly overlooked are also discussed.

An Application of the AXIS Solution Framework to Multiple Objective Aggregate Production Planning

This paper provides data on the first application of a prototype of the AXIS solution framework. AXIS (algorithms combined with knowledge systems in an interactive sequence) is a framework for interactively combining structured algorithms that seek a best solution with knowledge-based expert systems that seek expert heuristic solutions. This paper tests the framework using an interactive multiple objective integer programming algorithm combined with heuristics taken from the domain of aggregate production planning. The results indicate the AXIS framework can be successful in generating high quality solutions, in vastly reduced solution times compared to the structured algorithms, at much lower costs compared to the expert heuristics working alone.