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Algorithms for Non-Convex Global Optimization Based on the Statistical and Lipschitz Objective Function Models

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Algorithms for Non-Convex Global Optimization Based on the Statistical and Lipschitz Objective Function Models VILNIUS UNIVERSITY Gražina Gimbutiene˙ ALGORITHMS FOR NON-CONVEX GLOBAL OPTIMIZATION BASED ON THE STATISTICAL AND LIPSCHITZ OBJECTIVE FUNCTION MODELS Doctoral dissertation Physical sciences, informatics (09P) Vilnius, 2017 The dissertation work was carried out at Vilnius University from 2013 to 2017. Scientific supervisor: Prof. Dr. Habil. Antanas Žilinskas (Vilnius University, physical sciences, informatics - 09P). VILNIAUS UNIVERSITETAS Gražina Gimbutiene˙ STATISTINIAIS IR LIPŠICO TIKSLO FUNKCIJOS MODELIAIS PAGRI˛STI NEIŠKILOS GLOBALIOSIOS OPTIMIZACIJOS ALGORITMAI Daktaro disertacija Fiziniai mokslai, informatika (09P) Vilnius, 2017 Disertacija rengta 2013–2017 metais Vilniaus universitete. Mokslinis vadovas: prof. habil. dr. Antanas Žilinskas (Vilniaus universitetas, fiziniai mokslai, informatika - 09P). Abstract Global optimization problems arise in practice whenever there is a need to select a collection of variables corresponding to the best value of some objective function, e. g. the lowest price. In a typical "black-box" situation, when an analytic expression of the function is unavailable, algorithms based on statist- ical or Lipschitz objective function models can be applied. Statistical models are especially useful when function evaluations are expensive, therefore the efficiency of the algorithms has to be increased in terms of the number of tri- als. To this end, two approaches combining the statistical global search with local search techniques were suggested and their efficiency solving difficult multimodal problems was experimentally demonstrated. The optimization efficiency is influenced by the selected statistical model as well. Model selection problem was investigated experimentally and respective guidelines were for- mulated based on a priori information about the objective function complexity. In order to ensure not only efficient, but also theoretically justified search for optimal solutions, theoretical investigation of two algorithms was performed. First, asymptotic properties of a simplicial statistical model were investigated, allowing to relate a heuristic simplex selection criterion to the probability of improvement. Second, an optimal algorithm was derived, justifying the use of a certain trisection procedure in a set of Lipschitz optimization algorithms. v Santrauka Globaliosios optimizacijos uždaviniai iškyla praktikoje atsiradus poreikiui surasti kintamu˛ju˛ kombinacij ˛a,atitinkanˇci˛ageriausi ˛atam tikros tikslo funkcijos reikšm˛e, pvz. mažiausi ˛akain ˛a.Dažnoje "juodosios dež˙ es"˙ situacijoje, kai funkcija gali buti¯ apskaiˇciuojamaapibrežimo˙ srityje, taˇciaujos analitine˙ išraiška nežinoma, sekmingai˙ taikomi algoritmai, pagri˛sti statistiniais ar Lipšico tikslo funkcijos modeliais. Statistiniai modeliai taikytini, kai funkcijos i˛vertinimai brangus,¯ del˙ to svarbu didinti algoritmu˛ efektyvum ˛a,mažinant globaliojo minimumo paieškai sunaudot ˛aju˛ skaiˇciu˛. Tuo tikslu disertacijoje pasiulyti¯ du statistines˙ globaliosios optimizacijos derinimo su lokali ˛ajapaieška budai¯ bei eksperi- mentiškai pademonstruotas ju˛ efektyvumas sprendžiant sudetingus˙ neiškilosios optimizacijos uždavinius. Optimizacijos algoritmo efektyvum ˛alemia ir stat- istinio modelio pasirinkimas. Modelio pasirinkimo uždavinys disertacijoje tirtas remiantis pasiulyta¯ eksperimentine metodika, suformuluotos rekomendacijos atsižvelgiant i˛ numatom ˛atikslo funkcijos sudetingum˙ ˛a.Siekiant, kad optimaliu˛ sprendimu˛ paieška but¯ u˛ ne tik efektyvi, taˇciauir teoriškai pagri˛sta, atlikti du teoriniai algoritmu˛ savybiu˛ tyrimai. Pirma, ištyrus simpleksinio statistinio mod- elio asimptotik ˛a,euristinis simplekso išrinkimo kriterijus susietas su pagerinimo tikimybe. Antra, išvestas optimalus algoritmas, pagrindžiantis staˇciakampiu˛ posriˇciu˛dalijim ˛ai˛tris lygias dalis Lipšico optimizacijos algoritmuose. vi Table of Contents Notation xiii 1 Introduction 1 1.1 Research Context . .1 1.2 Relevance of the Study . .4 1.3 Objectives and Tasks of the Thesis . .5 1.4 Scientific Novelty and Results . .6 1.5 Statements to Be Defended . .7 1.6 Structure of the Thesis . .8 2 Global Optimization Based on Statistical and Lipschitz Objective Func- tion Models 9 2.1 The Global Optimization Problem . .9 2.2 Global Optimization Based on Statistical Models . 10 2.2.1 Common Statistical Models . 12 2.2.2 Seminal Conceptual Algorithms . 19 2.2.3 Further Developments . 23 2.3 Global Optimization Based on Lipschitz Models . 24 2.3.1 The Univariate Case . 25 2.3.2 The Multivariate Case . 27 2.3.3 The DIRECT Algorithm and Its Extensions . 29 2.4 Chapter Summary and Conclusions . 32 3 Choice of Statistical Model 34 3.1 Introduction . 34 3.2 Statement of the Problem . 35 3.3 Considered Models . 35 3.4 Special Forms of the Algorithms . 39 3.5 Numerical Experiments . 40 3.5.1 The Objective Functions . 41 3.5.2 The Implementation of the Algorithms . 42 3.5.3 Experimental Setup . 44 3.5.4 Results and Discussion . 45 3.6 Chapter Summary and Conclusions . 50 vii TABLE OF CONTENTS 4 Extensions for Hyper-Rectangular Decomposition-Based Statistical Global Optimization 52 4.1 A Multivariate Statistical Global Optimization Algorithm . 53 4.1.1 Introduction . 53 4.1.2 Description . 54 4.1.3 Illustrated Example . 59 4.2 Globally-Biased Statistical Global Optimization Algorithm . 61 4.2.1 Introduction . 61 4.2.2 Description . 64 4.2.3 Illustrated Example . 67 4.3 Statistical Global Optimization Algorithm with Clustering-Based Local Refinement . 69 4.3.1 Introduction . 69 4.3.2 Description . 70 4.3.3 Illustrated Example . 76 4.4 Numerical Experiments . 77 4.4.1 Experiments with the 2- and 3-dimensional Test Suite . 78 4.4.2 Experiments with the GKLS Function Generator . 83 4.5 Chapter Summary and Conclusions . 86 5 Asymptotic Properties in Simplicial Statistical Global Optimization 87 5.1 Introduction . 88 5.2 Motivation of the Research . 89 5.3 Statement of the Problem . 91 5.4 Assessment of Approximation . 93 5.5 Chapter Summary and Conclusions . 97 6 Worst-Case Optimality in Univariate Bi-objective Lipschitz Optimiza- tion 98 6.1 Introduction . 99 6.2 One-Step Worst-Case Optimal Methods for Single-Objective Uni- variate Optimization . 100 6.2.1 Bisection in the Single-Objective Case . 100 6.2.2 Trisection in the Single-Objective Case . 102 6.3 One-Step Worst-Case Optimal Trisection for Bi-objective Univari- ate Optimization . 105 6.3.1 Definitions . 105 6.3.2 The One-Step Worst-Case Optimal Trisection Problem . 110 6.3.3 The Trisection Algorithm . 111 6.4 Numerical Experiments . 114 6.5 Chapter Summary and Conclusions . 119 Results and Conclusions 121 viii TABLE OF CONTENTS A Expressions of Conditional Characteristics of Stochastic Functions 124 B Statistical Model Parameter Estimation Statistics 126 C Proofs of Chapter 6 129 Bibliography 133 Publications by the Author 143 ix List of Figures 2.1 An example of assuming that the objective function is a realization of the Wiener process. 15 2.2 Illustration of the common concepts in the univariate Lipschitz optimization. 26 2.3 Illustration of the selection of hyper-rectangles in DIRECT..... 30 3.1 Examples of realizations of the 1-dimensional Gaussian random processes. 36 3.2 Examples of realizations of the 2-dimensional stationary isotropic Gaussian random fields. 37 3.3 The percentage of cases when the global minimum was not found. 47 4.1 A 2-dimensional GKLS test function. 59 4.2 An illustration of the Rect algorithm operation. 60 4.3 An illustration of the GB algorithm operation. 68 4.4 An illustration of the Cluster algorithm operation. 76 4.5 The number of trials for the GKLS test classes. 85 6.1 One-step worst-case optimal bisection and trisection in the single- objective Lipschitz optimization. 102 6.2 Illustration of the definition of ∆ndp¨q................. 106 6.3 Illustration of the definition of ∆dp¨q.................. 107 6.4 Generated subintervals when a dominating trial exists. 108 6.5 The results of the Trisection algorithm for the problem Rastr.... 115 6.6 The results of the Trisection algorithm for the problem Fo & Fle.. 116 6.7 The results of the Trisection algorithm for the problem Schaf.... 117 1 ¯ ˚ C.1 The surface, formed by C ∆ p¨q..................... 132 x List of Tables 3.1 Characteristics of 1-dimensional random processes. 38 3.2 Characteristics of stationary isotropic Gaussian random fields. 39 3.3 n strategies for the 1-dimensional P-algorithm............ 44 3.4 n strategies for the 2-dimensional P-algorithm............ 45 3.5 The P-algorithm optimization results: the univariate case. 46 3.6 The MEI algorithm optimization results: the univariate case. 46 3.7 The P-algorithm optimization results: the bivariate case. 46 3.8 The MEI algorithm optimization results: the bivariate case. 48 3.9 Summary of the guidelines for the choice of the assumed model. 49 4.1 Parameter settings of the algorithms. 78 4.2 Comparison of different Rect algorithm implementations in 2D. 80 4.3 Comparison of different algorithms in 2D. 81 4.4 Comparison of different Rect algorithm implementations in 3D. 81 4.5 Comparison of different algorithms in 3D. 82 4.6 The parameters of the GKLS test function classes. 83 4.7 The maximum number of trials for the GKLS classes. 85 4.8 The average number of trials for the GKLS classes. 85 6.1 Univariate bi-objective test problems used. 114 6.2 Comparison of the algorithms when the maximum tolerance of the intervals falls below “ 0:1..................... 116 6.3 Comparison of the optimization algorithms, using a fixed number of function evaluations Nmax “ 100.................
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