PARALLEL OPTIMIZATION
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1 PARALLEL OPTIMIZATION Theory, Algorithms, and Applications YAIR CENSOR Department of Mathematics and Computer Science University of Haifa STAVROS A. ZENIOS Department of Public and Business Administration University of Cyprus New York Oxford Oxford University Press 1997
2 Contents Foreword, by George B. Dantzig Preface Acknowledgments Glossary of Symbols vii i x xvi xxvii Introduction Parallel Computers Taxonomy of parallel architectures Unifying concepts of parallel computing Control and data parallelism How Does Parallelism Affect Computing? A Classification of Parallel Algorithms Parallelism due to algorithm structure: Iterative projection algorithms Parallelism due to problem structure: Model decomposition and interior point algorithms Measuring the Performance of Parallel Algorithms Notes and References 24 PARTI THEORY Generalized Distances and Generalized Projections Bregman Functions and Generalized Projections Generalized Projections onto Hyperplanes Bregman Functions on the Whole Space Characterization of Generalized Projections Csiszar ip-divergences Notes and References 47 Proximal Minimization with D-Functions The Proximal Minimization Algorithm Convergence Analysis of the PMD Algorithm Special Cases: Quadratic and Entropic PMD Notes and References 57
3 XX Penalty Methods, Barrier Methods and Augmented Lagrangians Penalty Methods Barrier Methods The Primal-Dual Algorithmic Scheme Augmented Lagrangian Methods Notes and References 74 PART II ALGORITHMS Iterative Methods for Convex Feasibility Problems Preliminaries: Control Sequences and Relaxation Parameters The Method of Successive Orthogonal Projections The Cyclic Subgradient Projections Method The Relationship of CSP to Other Methods The method of successive orthogonal projections A remotest set controlled subgradient projections method The scheme of Oettli The linear feasibility problem: solving linear inequalities Kaczmarz's algorithm for systems of linear equations and its nonlinear extension The (6, ^-Algorithm The Block-Iterative Projections Algorithm Convergence of the BIP algorithm The Block-Iterative (S,??)-Algorithm The Method of Successive Generalized Projections The Multiprojections Algorithm The product space setup Generalized projections in the product space The simultaneous multiprojections algorithm and the split feasibility problem Automatic Relaxation for Linear Interval Feasibility Problems Notes and References 122
4 xxi 6 Iterative Algorithms for Linearly Constrained Optimization Problems The Problem, Solution Concepts, and the Special Environment ~ The problem Approaches and solution concepts The special computational environment Row-Action Methods, Bregman's Algorithm for Inequality Constrained Problems Algorithm for Interval-Constrained Problems Row-Action Algorithms for Norm Minimization The algorithm of Kaczmarz The algorithm of Hildreth ART4 - An algorithm for norm minimization over linear intervals Row-Action Algorithms for Shannon's Entropy Optimization Block-Iterative MART Algorithm Underrelaxation Parameters and Extension of the Family of Bregman Functions The Hybrid Algorithm: A Computational Simplification Hybrid algorithms for Shannon's entropy Algorithms for the Burg entropy function Renyi's entropy function Notes and References Model Decomposition Algorithms General Framework of Model Decompositions Problem modifiers Solution algorithms The Linear-Quadratic Penalty (LQP) Algorithm Analysis of the e-smoothed linear-quadratic penalty function e-exactness properties of the LQP function Notes and References Decompositions in Interior Point Algorithms The Primal-Dual Path Following Algorithm for Linear Programming Choosing the step lengths 222
5 xxii Choosing the barrier parameter The Primal-Dual Path Following Algorithm for Quadratic Programming Parallel Matrix Factorization Procedures for the Interior Point Algorithm The matrix factorization procedure for the dual step direction calculation Notes and References 232 PART III APPLICATIONS 9 Matrix Estimation Problems Applications of Matrix Balancing Economics: social accounting matrices (SAMs) Transportation: estimating Origin-Destination Matrices Statistics: estimating contingency tables Demography: modeling interregional migration Stochastic modeling: estimating transition probabilities Mathematical Models for Matrix Balancing Matrix estimation formulations Network structure of matrix balancing problems Entropy optimization models for matrix balancing Iterative Algorithms for Matrix Balancing The range-ras algorithm (RRAS) The RAS scaling algorithm The range-dss algorithm (RDSS) The diagonal similarity scaling (DSS) algorithm Notes and References Image Reconstruction from Projections Transform Methods and the Fully Discretized Model A Fully Discretized Model for Positron Emission Tomography The expectation-maximization algorithm A Justification for Entropy Maximization in Image
6 xxiii Reconstruction Algebraic Reconstruction Technique (ART) for Systems of Equations Iterative Data Refinement in Image Reconstruction The fundamentals of iterative data refinement Applications in medical imaging On the Selective Use of Iterative Algorithms for Inversion Problems Notes and References The Inverse Problem in Radiation Therapy Treatment Planning Problem Definition and the Continuous Model The continuous forward problem The continuous inverse problem Discretization of the Feasibility Problem Computational Inversion of the Data Consequences and Limitations Experimental Results Combination of Plans in Radiotherapy Basic definitions and mathematical modeling The feasible case The infeasible case Notes and References Multicommodity Network Flow Problems Preliminaries Problem Formulations Transportation problems Multicommodity network flow problems Sample Applications Example one: Covering positions in stock options Example two: Air-traffic control Example three: Routing of traffic Iterative Algorithms for Multicommodity Network Flow Problems Row-action algorithm for quadratic transportation- problems Extensions to generalized networks Row-action algorithm for quadratic
7 XXIV multicommodity transportation problems 12.5 A Model Decomposition Algorithm for Multicommodity Network Flow Problems The linear-quadratic penalty (LQP) algorithm 12.6 Notes and References 13 Planning Under Uncertainty 13.1 Preliminaries 13.2 The Newsboy Problem 13.3 Stochastic Programming Problems Anticipative models Adaptive models Recourse models 13.4 Robust Optimization Problems 13.5 Applications Robust optimization for the diet problem Robust optimization for planning capacity expansion Robust optimization for matrix balancing 13.6 Stochastic Programming for Portfolio Management Notation Model formulation 13.7 Stochastic Network Models Split-variable formulation of stochastic network models Component-wise representation of the stochastic network problem Iterative Algorithm for Stochastic Network Optimization Notes and References Decompositions for Parallel Computing Vector-Random Access Machine (V-RAM) Parallel prefix operations Mapping Data to Processors Mapping a dense matrix Mapping a sparse matrix Parallel Computing for Matrix Balancing Data parallel computing with RAS Control parallel computing with RAS Parallel Computing for Image Reconstruction 428
8 XXV Parallelism within a block Parallelism with independent blocks Parallelism between views Parallel Computing for Network-structured Problems Solving dense transportation problems Solving sparse transportation problems Solving sparse transshipment graphs Solving network-structured problems Parallel Computing with Interior Point Algorithms The communication schemes on a hypercube The parallel implementation on a hypercube An alternative parallel implementation Notes and References Numerical Investigations Reporting Computational Experiments on Parallel Machines Matrix Balancing Data parallel implementations Control parallel implementations Image Reconstruction Multicommodity Network Flows Row-action algorithm for transportation problems Row-action algorithm for multicommodity transportation problems Linear-quadratic penalty (LQP) algorithm for multicommodity network flow problems Planning Under Uncertainty Interior point algorithm Row-action algorithm for nonlinear stochastic networks Proximal Minimization with D-functions Solving linear network problems Solving linear stochastic network problems Description of Parallel Machines Alliant FX/ Connection Machine CM Connection Machine CM CRAY X-MP and Y-MP Intel ipsc/
9 xxvi 15.8 Notes and References 478 Bibliography 481 Index 527
Contents. I Basics 1. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.
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