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Modification of the DIRECT Algorithm Abstract GABLONSKY, JORGÄ MAXIMILIAN XAVER. Modi¯cations of the DIRECT Algo- rithm. (Under the direction of Carl Timothy Kelley.) This work presents improvements of a global optimization method for bound con- straint problems along with theoretical results. These improvements are strongly biased towards local search. The global optimization method known as DIRECT was modi¯ed speci¯cally for small-dimensional problems with few global minima. The motivation for our work comes from our theoretical results regarding the behavior of DIRECT. Speci¯cally, we explain how DIRECT clusters its search near a global minimizer. An additional influence is our explanation of DIRECT's behavior for both constant and linear functions. We further improved the e®ectiveness of both DIRECT, and our modi¯cation, by combining them with another global optimization method known as Implicit Filtering. In addition to these improvements the methods were also extended to handle problems where the objective function is de¯ned solely on an unknown subset of the bounding box. We demonstrate the increased e®ectiveness and robustness of our modi¯cations using optimization problems from the natural gas transmission industry, as well as commonly used test problems from the literature. MODIFICATIONS OF THE DIRECT ALGORITHM by JorgÄ M. Gablonsky a thesis submitted to the graduate faculty of north carolina state university in partial fulfillment of the requirements for the degree of doctor of philosophy department of mathematics raleigh, north carolina 2001 approved by: C. T. Kelley J. C. Dunn chair of advisory committee P. A. Gremaud M. Shearer Dedication I want to dedicate this work to my parents Helga and Hans-Georg Gablonsky, my uncle Paul Gablonsky, my sisters Sigrid MÄuller and Doris Gablonsky, and my brother Peter Gablonsky. ii Biography JÄorg Gablonsky was born in Duisburg, Germany. He received his \Diplom" in \Wirtschaftsmathematik" (Master of Science in Mathematics and Business) in 1999 while already attending a PhD Program in Applied Mathematics at North Carolina State University. iii Acknowledgments I would like to express my appreciation for the support and guidance I received from my advisor Dr. C.T. Kelley. Furthermore I want to thank the members of my committee, Dr. J.C. Dunn, Dr. P.A. Gremaud, and Dr. M. Shearer for their support and advice. This work was supported by National Science Foundation grants #DMS-9700569, #DMS-9714811, and #DMS-0070641, and an allocation from the North Carolina Supercomputing Center. I would also like to thank Annie Win¯eld for her help with my writing and Dr. Susan Katz for her support. Furthermore I want to thank Dr. R.G. Carter for providing us with the problems from the gas pipeline industry, for answering all our questions in that context, and for pointing out ways to improve DIRECT. Finally, I want to thank the other students who helped me, either in the completion of this thesis, or by keeping me focused: B. Adams, Dr. A. Batterman, D.M. Bortz, I. Bredehoeft, C. Camalier, M. Campolattaro, Dr. T.D. Choi, K.A. and T. Co®ey, Dr. R. DelRosario, J. Duran, O.J. Esslinger, D. Finkel, A. Gruber, K. Hans, J.C. Haws, S. Jansen, Dr. E.W. Jenkins, K.R. Kavanagh, J. Lefeaux, B.M. Lewis, J.E. Massad, Dr. J.V. Matthews, M. Minimair, A. Patrick, S. and C. Quincke, D. Schurich, S. Shonek, K.A. Yokley, and M. Zager. iv Table of Contents List of Tables viii List of Figures xi 1 Introduction 1 1.1 Overview .................................. 6 1.2 Optimization of natural gas transmission networks ........... 7 2 The DIRECT Algorithm 11 2.1 Lipschitz Optimization Problem ..................... 11 2.2 Classical one-dimensional Lipschitz Optimization ........... 17 2.3 Classical N-dimensional Lipschitz Optimization ............ 22 2.4 The one-dimensional DIRECT Algorithm ................ 22 2.4.1 Area-Dividing Strategies ..................... 23 2.4.2 Potentially optimal intervals ................... 26 2.4.3 The one-dimensional DIRECT algorithm ............. 31 2.5 The general/multidimensional DIRECT algorithm ............ 33 2.5.1 Dividing in higher dimensions .................. 34 2.5.2 Potentially optimal hyperrectangles .............. 37 3 Extensions to the DIRECT algorithm 39 3.1 Changes to the original DIRECT algorithm ............... 39 3.1.1 Changes to DIRECT by other authors .............. 45 3.2 Stopping criteria ............................. 46 3.3 Choice of ² ................................. 48 3.4 Extensions to problems with hidden constraints ............ 49 3.4.1 Replace the function value at an infeasible point by a high value 49 3.4.2 Create an extra list of infeasible points ............. 50 3.4.3 Replace the function value at an infeasible point by the value of a nearby feasible point ..................... 51 3.5 Use of DIRECT as a starting point generator .............. 54 v 4 Analysis of DIRECT 57 4.1 Convergence Analysis ........................... 57 4.2 Local clustering .............................. 59 4.3 Iteration estimates ............................ 61 4.3.1 Worst case analysis ........................ 62 4.3.2 Constant functions ........................ 70 4.3.3 Linear functions .......................... 73 4.3.4 Hidden constraints ........................ 81 5 Numerical Results 83 5.1 Numerical Results in the Optimization of Gas Pipeline Networks . 84 5.1.1 The optimization Infrastructure ................. 84 5.1.2 Example problems ........................ 85 5.1.3 Accuracy of the model ...................... 90 5.1.4 Setup of the numerical Experiments ............... 90 5.1.5 Results of the optimization .................... 91 5.2 Numerical Results on Test Problems used by Jones et al. ....... 99 5.2.1 Description of the Test Problems ................ 100 5.2.2 Numerical results ......................... 105 6 Conclusions 110 List of References 112 A Implementation of the DIRECT and DIRECT-l algorithm 122 A.1 The general algorithm .......................... 122 A.1.1 The data structure ........................ 122 A.1.2 Details ............................... 125 A.2 Parallel Implementation ......................... 127 A.3 Extensions to problems with hidden constraints ............ 129 B Implicit Filtering 131 C DIRECT Version 2.0 User Guide 135 C.1 Introduction to DIRECT .......................... 136 C.1.1 Problem description ....................... 136 C.2 Using DIRECT ............................... 137 C.2.1 What is included in the package ................. 137 C.2.2 Calling DIRECT .......................... 139 C.2.3 Sample main program ...................... 144 C.3 A short overview of the DIRECT algorithm and our modi¯cations . 149 C.3.1 Dividing the domain ....................... 150 C.3.2 Potentially optimal hyperrectangles .............. 151 vi C.3.3 The DIRECT algorithm ...................... 153 C.3.4 Our modi¯cation to the DIRECT algorithm ........... 154 C.3.5 Extensions to the DIRECT algorithm .............. 155 C.4 The Test Problems ............................ 158 C.4.1 Elementary functions ....................... 159 C.4.2 Example for hidden constraints ................. 159 C.4.3 Test functions described in Jones et.al. [48] ........... 160 C.4.4 Main features of the Test Functions ............... 165 C.4.5 Numerical results ......................... 166 vii List of Tables 3.1 Summary of stopping criteria ...................... 48 4.1 The resulting hyperrectangles after the division of an N-dimensional hypercube with side length 1=3k. .................... 63 4.2 Number of hypercubes (stage 0) of side length 1=3k needed to cover the n-dimensional hypercube of length 1. ................ 64 4.3 Maximum number of iterations needed to branch all hypercubes of level k in dimension N........................... 69 5.1 Results for Problem A. The actual minimum value is 3233. ...... 92 5.2 Results for Problem B. The actual minimum value is 3204. ...... 93 5.3 Results for Problem C. The actual minimum value is 4041. ...... 95 5.4 Success rates for IFFCO with 500 randomly selected starting points. 99 5.5 Parameters for the Shekel's family of functions ............. 100 5.6 Parameters for the Hartman's family of functions ........... 101 5.7 Summary of the important features of the test functions. ....... 105 5.8 Numerical results with percentage termination criteria. ........ 106 5.9 Numerical results with a budget of 60N function evaluations ..... 107 5.10 Numerical results for DIRECT-IFFCO with a combined budget of 60N function evaluations. The budget is divided into 10N function evalua- tions for DIRECT and 50N function evaluations for IFFCO. ...... 108 5.11 Numerical results for DIRECT-l-IFFCO with a combined budget of 60N function evaluations. The budget is divided into 10N function evalua- tions for DIRECT-l and 50N function evaluations for IFFCO. ..... 108 A.1 Routines for the parallel implementation of DIRECT........... 128 C.1 Parameters for the Shekel's family of functions ............. 161 C.2 Parameters for the Hartman's family of functions ........... 162 C.3 Summary of the important features of the test functions. ....... 166 C.4 Results for the test functions. ...................... 167 viii List of Figures 1.1 View of a large compressor station. ................... 7 1.2 A schematic diagram of a simpli¯ed gas transmission network. .... 9 2.1 Example of a Lipschitz continuous function f and the lower bounding function f^.................................. 19 2.2 Example of the iteration of the Piyavskii algorithm. .......... 21 2.3 Example of a Lipschitz continuous function
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