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Speeding-Up a Random Search for the Global Minimum of a Non-Convex, Non-Smooth Objective Function

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Speeding-Up a Random Search for the Global Minimum of a Non-Convex, Non-Smooth Objective Function University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Spring 2021 SPEEDING-UP A RANDOM SEARCH FOR THE GLOBAL MINIMUM OF A NON-CONVEX, NON-SMOOTH OBJECTIVE FUNCTION Arnold C. Englander University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Englander, Arnold C., "SPEEDING-UP A RANDOM SEARCH FOR THE GLOBAL MINIMUM OF A NON- CONVEX, NON-SMOOTH OBJECTIVE FUNCTION" (2021). Doctoral Dissertations. 2569. https://scholars.unh.edu/dissertation/2569 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. SPEEDING-UP A RANDOM SEARCH FOR THE GLOBAL MINIMUM OF A NON-CONVEX, NON-SMOOTH OBJECTIVE FUNCTION By Arnold C. Englander M.S. in Engineering and Applied Science, Yale University, 1984 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Electrical and Computer Engineering May 2021 ii © 2021 Arnold C. Englander iii THESIS COMMITTEE PAGE This thesis has been examined and approved in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering by: Dr. Michael Carter, Thesis committee chair, Associate Professor emeritus, Electrical and Computer Engineering Dr. John Gibson, Associate Professor, Mathematics and Statistics Dr. Kyle Hughes, Aerospace Engineer, NASA Goddard Space Flight Center, Navigation and Mission Design Branch Dr. Richard Messner, Associate Professor, Electrical and Computer Engineering Dr. Se Young Yoon, Associate Professor, Electrical and Computer Engineering On April 13, 2021 Approval signatures are on file with the University of New Hampshire Graduate School. iv TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES vi DEDICATION xiii ACKNOWLEDGEMENTS xiv GLOSSARY xv ABSTRACT xviii INTRODUCTION 1 CHAPTERS: I. PRIOR WORK 8 II. MONOTONIC BASIN HOPPING (MBH) 18 III. THE BEHAVIOR OF f AND 푿픽 IMPACT MBH CONVERGENCE TIME 34 IV. SPEEDING-UP MBH BY ADAPTIVELY SHAPING THE DISTRIBUTION 42 OF HOP DISTANCES V. SPEEDING-UP MBH CONVERGENCE BY BIASING THE “HOP FROM” 76 LOCATION VI. THE PIONEER 11 TRAJECTORY OPTIMIZATION USE-CASE 107 VII. OPEN QUESTIONS AND HYPOTHESIS 120 VIII. SUMMARY 125 IX. REFERENCES 128 X. APPENDICES A. Asymptotic convergence proof by Baba et al., applicable to MBH 135 B. Justification for fitting a Gamma distribution to MBH FPTs as their FPTD 138 C. Probability of advancement on a 1-dimensional Gibsonian f 149 D. Python code used to generate simulated f 160 E. Details regarding the PyKep model 172 v LIST OF TABLES I: Three questions addressed in Chapter IV, their answers, and concepts and tools used II: Methods provided in Chapter V for speeding-up MBH by biasing the location from which each next hop is taken III.: Comparison of 푓* for various p using the PyKep model for Pioneer 11 vi LIST OF FIGURES 0.a Prototypical 1-dimensional non-convex objective function f having feasible domain X 0.b. Prototypical 1-dimensional non-convex f having disconnected, sparse feasible domain 푿픽 0.c Textured prototypical 1-dimensional non-convex f having disconnected, sparse feasible domain 푿픽 II.1: Prototypical 1-dimensional non-convex f and an example of its associated incumbent path {풙[t]; t = 1, 2, 3, …} in blue and candidate path {흃[t]; t = 1, 2, 3, …} in tan II.2: Prototypical 1-dimensional non-convex f, except with a disconnected, sparse 푿픽, and an example of its associated {풙[t]; t = 1, 2, 3, …} in blue and candidate path {흃[t]; t = 1, 2, 3, …} in tan II.3: Prototypical 1-dimensional non-convex f, except with a disconnected, sparse 푿픽 and texture, and an example of its associated {풙[t]; t = 1, 2, 3, …} in blue and candidate path {흃[t]; t = 1, 2, 3, …} in tan II.4: Histogram of { 풙 } drawn from the fixed non-Gaussian p used to drive the MBH search in the above figures II.1, II.2, and II.3. II.5: Prototypical 1-dimensional non-convex f, and the progress of the MBH as depicted by f[x[t]] II.6: Prototypical 1-dimensional non-convex f, except with a disconnected, sparse 푿픽, and the progress of the MBH as depicted by f[x[t]] II.7: Prototypical 1-dimensional non-convex f, except with a disconnected, sparse 푿픽 and texture, and the progress of the MBH as depicted by f[x[t]] II.8: g[d] II.9: Unimodal f for which for which 푿픽 is all of X, its corresponding g[d], and the and the progress of an MBH operating on it MBH as depicted by f[x[t]] II.10: Globally rugged f for which for which 푿픽 is all of X, its corresponding g[d], and the progress of an MBH operating on it MBH as depicted by f[x[t]] II.11: Prototypical 1-dimensional non-convex f and its associated g[d]. II.12: Prototypical 1-dimensional non-convex f, except with a disconnected, sparse 푿픽, and its associated g[d]. III.1: 3-dimensional “funnels” that are used as pedagogical models of energy landscapes in the literature of protein-folding vii III.2: 1-dimensional Gibsonian f for which 푿픽 is all of X, and the progress of an MBH operating on it MBH as depicted by f[x[t]]. Time-step axis on the lower panel scaled to span [0,650] III.3: 1-dimensional Gibsonian f for which 푿픽 is all of X, and the progress of an MBH operating on it MBH as depicted by f[x[t]]. Time-step axis on the lower panel scaled to span [0, 650000] IV.1: Unimodal f for which 푿픽 is all of X with x ~ 푞̂ collected from 64 concurrent but independent trials of MBH IV.2: Globally rugged f for which 푿픽 is all of X with x ~ 푞̂ collected from 64 concurrent but independent trials of MBH IV.3: Globally rugged f for which 푿픽 is all of X, with histograms of p (orange) and 푞̂ (blue) collected from a time-series comprised of “accepted” x from 64 concurrent but independent trials of MBH, using fixed p, showing DK-L(p, 푞̂), and with a Gamma distribution fit to the FPTs of 500 independent trials as their FPTD IV.4: A set of Gamma distributions fit to MBH FPTs, and their corresponding DK-L(p, 푞̂) IV.5: Globally rugged f for which 푿픽 is all of X, with , with histograms of p (orange) and 푞̂ (blue) collected from a time-series comprised of “accepted” x from 64 concurrent but independent trials of MBH, using p adapted to 푞̂, showing DK-L(p, 푞̂), and with a Gamma distribution fit to the FPTs of 500 independent trials as their FPTD IV.6: Histogram of { 풙 } drawn from a fixed Gaussian p IV.7: Prototypical f for which 푿픽 is all of X, upon which MBH operated using the fixed Gaussian p illustrated in Figure IV.6, with f[x[t,n]] for t being the first 2,000 of 50,000 MBH time-steps, n being the first 48 of 2500 trials, and with a Gamma distribution fit to the FPTs of 2,500 independent trials as their FPTD IV.8: Prototypical f for which 푿픽 is all of X, upon which MBH operated using the fixed non-Gaussian p illustrated in Figure II.4, with f[x[t,n]] for t being the first 2,000 of 50,000 MBH time-steps, n being the first 48 of 2500 trials, and with a Gamma distribution fit to the FPTs of 2,500 independent trials as their FPTD IV.9: Prototypical f for which 푿픽 is all of X, upon which MBH operated using an adaptive p, with f[x[t,n]] for t being the first 2,000 of 50,000 MBH time-steps, n being the first 48 of 2500 trials, and with a Gamma distribution fit to the FPTs of 2,500 independent trials as their FPTD IV.10: Textured prototypical f having a disconnected and sparse feasible domain, upon which MBH operated using a fixed Gaussian p, with f[x[t,n]] for t being the first 2,000 of 10,000 MBH time-steps, n viii being the first 48 of 2,500 trials, and with a Gamma distribution fit to the FPTs of 2,500 independent trials as their FPTD IV.11: Textured prototypical f having a disconnected and sparse feasible domain, upon which MBH operated using a fixed non-Gaussian p, with f[x[t,n]] for t being the first 2,000 of 10,000 MBH time-steps, n being the first 48 of 2500 trials, and with a Gamma distribution fit to the FPTs of 2,500 independent trials as their FPTD IV.12: Textured prototypical f having a disconnected and sparse feasible domain, upon which MBH operated using an adaptive p, with f[x[t,n]] for t being the first 2,000 of 10,000 MBH time-steps, n being the first 48 of 2500 trials, and with a Gamma distribution fit to the FPTs of 2,500 independent trials as their FPTD IV.13: MBH operating on an f using a fixed Laplace(0,1) p that is not similar to 푞̂ IV.14: MBH operating on the same f using a fixed 푝ℷ (fixed in the sense that in 푝ℷ(0, ) is time-invariant) IV.15: MBH operating on the same f using 푝ℷ(0, [t]) adapted to 푞̂[푡], where [t] was generated as described in the text V.1: Gibsonian f upon which MBH operated using fixed non-Gaussian p and SCLS for which a maximum of 32 local steps per MBH time-step was allowed, with f[x[t,n]] for t being the first 2,000 of 10,000 MBH time-steps, n being the first 48 of 1,000 trials, and a Gamma distribution fit to the FPTs of 1,000 independent trials as their FPTD V.2: Gibsonian f upon which MBH operated using fixed non-Gaussian p and SCLS for which a maximum of 64 local steps per MBH time-step was allowed, with f[x[t,n]] for t being the first 2,000 of 10,000 MBH time-steps, n being the first 48 of 1,000 trials, and a Gamma distribution fit to the FPTs of 1,000 independent trials as their FPTD V.3: Gibsonian f upon which MBH operated using
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