Row generation techniques for approximate solution of linear programming problems

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2010.Thesis (Master's) -- Bilkent University, 2010.Includes bibliographical references leaves 69-77.In this study, row generation techniques are applied on general linear programming problems with a very large number of constraints with respect to the problem dimension. A lower bound is obtained for the change in the objective value caused by the generation of a specific row. To achieve row selection that results in a large shift in the feasible region and the objective value at each row generation iteration, the lower bound is used in the comparison of row generation candidates. For a warm-start to the solution procedure, an effective selection of the subset of constraints that constitutes the initial LP is considered. Several strategies are discussed to form such a small subset of constraints so as to obtain an initial solution close to the feasible region of the original LP. Approximation schemes are designed and compared to make possible the termination of row generation at a solution in the proximity of an optimal solution of the input LP. The row generation algorithm presented in this study, which is enhanced with a warm-start strategy and an approximation scheme is implemented and tested for computation time and the number of rows generated. Two efficient primal simplex method variants are used for benchmarking computation times, and the row generation algorithm appears to perform better than at least one of them especially when number of constraints is large.Paç, A BurakM.S

A Hybrid Multi-GPU Implementation of Simplex Algorithm with CPU Collaboration

The simplex algorithm has been successfully used for many years in solving linear programming (LP) problems. Due to the intensive computations required (especially for the solution of large LP problems), parallel approaches have also extensively been studied. The computational power provided by the modern GPUs as well as the rapid development of multicore CPU systems have led OpenMP and CUDA programming models to the top preferences during the last years. However, the desired efficient collaboration between CPU and GPU through the combined use of the above programming models is still considered a hard research problem. In the above context, we demonstrate here an excessively efficient implementation of standard simplex, targeting to the best possible exploitation of the concurrent use of all the computing resources, on a multicore platform with multiple CUDA-enabled GPUs. More concretely, we present a novel hybrid collaboration scheme which is based on the concurrent execution of suitably spread CPU-assigned (via multithreading) and GPU-offloaded computations. The experimental results extracted through the cooperative use of OpenMP and CUDA over a notably powerful modern hybrid platform (consisting of 32 cores and two high-spec GPUs, Titan Rtx and Rtx 2080Ti) highlight that the performance of the presented here hybrid GPU/CPU collaboration scheme is clearly superior to the GPU-only implementation under almost all conditions. The corresponding measurements validate the value of using all resources concurrently, even in the case of a multi-GPU configuration platform. Furthermore, the given implementations are completely comparable (and slightly superior in most cases) to other related attempts in the bibliography, and clearly superior to the native CPU-implementation with 32 cores.Comment: 12 page

Parallel solution of linear programs

The factors limiting the performance of computer software periodically undergo sudden shifts, resulting from technological progress, and these shifts can have profound implications for the design of high performance codes. At the present time, the speed with which hardware can execute a single stream of instructions has reached a plateau. It is now the number of instruction streams that may be executed concurrently which underpins estimates of compute power, and with this change, a critical limitation on the performance of software has come to be the degree to which it can be parallelised. The research in this thesis is concerned with the means by which codes for linear programming may be adapted to this new hardware. For the most part, it is codes implementing the simplex method which will be discussed, though these have typically lower performance for single solves than those implementing interior point methods. However, the ability of the simplex method to rapidly re-solve a problem makes it at present indispensable as a subroutine for mixed integer programming. The long history of the simplex method as a practical technique, with applications in many industries and government, has led to such codes reaching a great level of sophistication. It would be unexpected in a research project such as this one to match the performance of top commercial codes with many years of development behind them. The simplex codes described in this thesis are, however, able to solve real problems of small to moderate size, rather than being confined to random or otherwise artificially generated instances. The remainder of this thesis is structured as follows. The rest of this chapter gives a brief overview of the essential elements of modern parallel hardware and of the linear programming problem. Both the simplex method and interior point methods are discussed, along with some of the key algorithmic enhancements required for such systems to solve real-world problems. Some background on the parallelisation of both types of code is given. The next chapter describes two standard simplex codes designed to exploit the current generation of hardware. i6 is a parallel standard simplex solver capable of being applied to a range of real problems, and showing exceptional performance for dense, square programs. i8 is also a parallel, standard simplex solver, but now implemented for graphics processing units (GPUs)

Μελέτη και Αξιολόγηση Παράλληλων Συστημάτων στους Αλγορίθμους Γραμμικού Προγραμματισμού

Η εργασία διαπραγματεύεται την μελέτη και την αξιολόγηση μεθόδων παράλληλων συ-στημάτων στον τομέα του γραμμικού προγραμματισμού. Αρχικά, μελετήθηκε ο αλγόριθ-μος της μεθόδου Simplex και οι δυνατότητες παραλληλοποίησης του. Αφού προέκυψαν τα απαραίτητα συμπεράσματα μελετήθηκαν δύο διαφορετικές παραλληλοποίησεις, η πρώτη του Mohamed Esseghir Lalami και η δεύτερη χειροποίητη με πολλαπλά νήματα στην κεντρική μονάδα επεξεργασίας μέσω Open MP. Ακολούθησε σύγκριση μεταξύ του απαιτούμενου χρόνου επεξεργασίας σε δεδομένο μέγεθος προβλήματος. Παρουσιάζεται με αυτόν τον τρόπο η δύναμη και η επεξεργαστική δεινότητα σε μια άμεση εφαρμογή του Open MP .This paper is about examining and evaluating the effectiveness of parallel program-ming in linear programming. Initially, this thesis presents the Simplex method and its possibilities of parallelization. After some preliminary study two different ways were im-plemented, the first one based of Mohamed Esseghir Lalami and one custom executed on CPU on multiple threads. It was followed up by a comparison of time spent during the various stages. It is beyond evident that the Open Mp standard offers brute force and compute ability in a simple way

Field D* pathfinding in weighted simplicial complexes

Includes abstract.Includes bibliographical references.The development of algorithms to efficiently determine an optimal path through a complex environment is a continuing area of research within Computer Science. When such environments can be represented as a graph, established graph search algorithms, such as Dijkstra’s shortest path and A*, can be used. However, many environments are constructed from a set of regions that do not conform to a discrete graph. The Weighted Region Problem was proposed to address the problem of finding the shortest path through a set of such regions, weighted with values representing the cost of traversing the region. Robust solutions to this problem are computationally expensive since finding shortest paths across a region requires expensive minimisation. Sampling approaches construct graphs by introducing extra points on region edges and connecting them with edges criss-crossing the region. Dijkstra or A* are then applied to compute shortest paths. The connectivity of these graphs is high and such techniques are thus not particularly well suited to environments where the weights and representation frequently change. The Field D* algorithm, by contrast, computes the shortest path across a grid of weighted square cells and has replanning capabilites that cater for environmental changes. However, representing an environment as a weighted grid (an image) is not space-efficient since high resolution is required to produce accurate paths through areas containing features sensitive to noise. In this work, we extend Field D* to weighted simplicial complexes – specifically – triangulations in 2D and tetrahedral meshes in 3D

Massively Parallel Computation Using Graphics Processors with Application to Optimal Experimentation in Dynamic Control

The rapid increase in the performance of graphics hardware, coupled with recent improvements in its programmability has lead to its adoption in many non-graphics applications, including wide variety of scientific computing fields. At the same time, a number of important dynamic optimal policy problems in economics are athirst of computing power to help overcome dual curses of complexity and dimensionality. We investigate if computational economics may benefit from new tools on a case study of imperfect information dynamic programming problem with learning and experimentation trade-off that is, a choice between controlling the policy target and learning system parameters. Specifically, we use a model of active learning and control of linear autoregression with unknown slope that appeared in a variety of macroeconomic policy and other contexts. The endogeneity of posterior beliefs makes the problem difficult in that the value function need not be convex and policy function need not be continuous. This complication makes the problem a suitable target for massively-parallel computation using graphics processors. Our findings are cautiously optimistic in that new tools let us easily achieve a factor of 15 performance gain relative to an implementation targeting single-core processors and thus establish a better reference point on the computational speed vs. coding complexity trade-off frontier. While further gains and wider applicability may lie behind steep learning barrier, we argue that the future of many computations belong to parallel algorithms anyway