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Semi-Algebraic Techniques in Variational Analysis: Pseudospectra, Robustness, Generic Continuity, and Mountain Pass Algorithms

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Semi-Algebraic Techniques in Variational Analysis: Pseudospectra, Robustness, Generic Continuity, and Mountain Pass Algorithms SEMI-ALGEBRAIC TECHNIQUES IN VARIATIONAL ANALYSIS: PSEUDOSPECTRA, ROBUSTNESS, GENERIC CONTINUITY, AND MOUNTAIN PASS ALGORITHMS. A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Jeffrey Pang Chin How August 2009 c 2009 Jeffrey Pang Chin How ALL RIGHTS RESERVED SEMI-ALGEBRAIC TECHNIQUES IN VARIATIONAL ANALYSIS: PSEUDOSPECTRA, ROBUSTNESS, GENERIC CONTINUITY, AND MOUNTAIN PASS ALGORITHMS. Jeffrey Pang Chin How, Ph.D. Cornell University 2009 Variational Analysis is the modern theory of nonsmooth, nonconvex analysis built on the theory of convex and smooth optimization. While the general theory needs to handle pathologies, functions and sets appearing in applications are typically structured. Semi-algebraic functions and sets eliminates much of the pathological behavior, and still forms a broad class of constructs appearing in practice, making it an ideal setting for practical variational analysis. Chapter 1 is an introduction to the thesis and Chapter 2 reviews preliminaries. Chapters 3 to 5 describe various semi-algebraic techniques in variational anal- ysis. Chapter 3 gives equivalent conditions for the Lipschitz continuity of pseu- dospectra in the set-valued sense. As corollaries, we give formulas for the Lipschitz constants of the pseudospectra, pseudospectral abscissa and pseudospectral radius. We also study critical points of the resolvent function. Chapter 4 studies robust solutions of an optimization problem using the “-robust regularization” of a func- tion, and prove that it is Lipschitz at a point for all small > 0 for nice functions, and in particular semi-algebraic functions. This result generalizes some of the ideas in Chapter 3. Chapter 5 studies the continuity properties of set-valued maps. We prove that the set of points where a closed-valued semi-algebraic (and more gen- erally, tame) set-valued map is not strictly continuous is a set of lower dimension. As a by product of our analysis, we prove a Sard-type theorem for local (Pareto) minimums of scalar-valued and vector-valued functions. Chapter 6 departs from the theoretical bent of the rest of the thesis and is computational. We describe algorithms for computing critical points of mountain pass type. We prove that sub-level sets of a function coalesce at critical points of mountain pass type and discuss algorithmic implications. In particular, we propose a locally superlinearly convergent algorithm for smooth nondegenerate critical points of Morse index 1. We conclude this chapter by describing a strategy for the Wilkinson problem of finding the closest matrix with repeated eigenvalues. BIOGRAPHICAL SKETCH Chin How Jeffrey Pang was born in Singapore in 1979. He had his pre-university education in Singapore, and represented Singapore in the International Mathemati- cal Olympiads in 1995 (Canada), 1996 (India) and 1997 (Argentina). He graduated with a Bachelor degree in Mathematics with Honours from the National University of Singapore (NUS) in July 2003, and a Masters in the Singapore MIT Alliance in July 2004. He came to Cornell University in August 2004 to study for a PhD in Applied Mathematics under the direction of Adrian S. Lewis, and has been in Cornell for five years. In Fall 2008, he was a visiting scholar hosted by Aris Daniilidis at the Centre de Recerca Mathem`atica, Barcelona. In Fall 2009, he will be at the Fields Institute based in the University of Toronto as the Marsden Postdoctoral Fellow for the thematic program on the Foundations of Computational Mathematics. In Spring 2010, he will continue his postdoctoral work at the University of Waterloo. iii To all who have taught me Math iv ACKNOWLEDGEMENTS First and foremost, I wish to thank my adviser Adrian Lewis for his guidance throughout my PhD program. I benefitted much from his wisdom. I gratefully acknowledge him for introducing me to the field of variational analysis, for coaching me on all aspects of being a better mathematician, and for his financial support throughout the PhD program. Secondly, I thank Aris Daniilidis for hosting me at the Centre de Recerca Mathem`atica, Barcelona in Fall 2008. I had a wonderful time working with him and chatting with him about things mathematical or otherwise. The material in Chapter 5 is a result of our work during that time. I would like to thank my committee members, James Renegar and Charles Van Loan, for cheerfully agreeing to serve on my thesis committee and for the excellent classes I took with them. I thank the other people working in continuous optimization in Cornell for the pleasant academic environment in Cornell. In particular, I thank Mike Todd for his insightful comments (one of which led to a sharpening of a result in Chapter 4), and I thank Stefan Wild and Dennis Leventhal for organizing the continuous optimization seminars. I thank the director of CAM, Steve Strogatz, and the Admin Manager of CAM, Dolores Pendell, for keeping CAM functioning on a daily basis. I thank many of the professors in the National University of Singapore for guiding me through my Math education, especially those since the days I was involved in training for the IMO. I am especially grateful to those who gave me a chance to try out for the IMO team back then, even when it looked like I never had any chance at that time. This thesis was typed using LaTeX and LyX. Thanks to their creaters. v TABLE OF CONTENTS Biographical Sketch . iii Dedication . iv Acknowledgements . v Table of Contents . vi List of Tables . viii List of Figures . ix 1 Introduction 1 1.1 Variational Analysis of Pseudospectra . 3 1.2 Lipschitz behavior of the robust regularization . 5 1.3 Continuity of set-valued maps revisited in the light of tame geometry 8 1.4 Level set methods for finding critical points of mountain pass type . 9 2 Preliminaries 13 2.1 Variational analysis . 13 2.2 Semi-algebraic geometry . 19 3 Variational Analysis of Pseudospectra 22 3.1 Feasible-set mappings and continuity of pseudospectra . 25 3.2 General results . 32 3.3 Subdifferential calculus . 34 3.4 Main result . 45 3.5 Lipschitz continuity of pseudospectra . 49 3.6 Pseudospectral abscissa and pseudospectral radius . 52 3.7 Resolvent-critical points . 56 3.8 Acknowledgements . 61 4 Lipschitz behavior of the robust regularization 62 4.1 Calmness as an extension to Lipschitzness . 62 4.2 Calmness and robust regularization . 66 4.3 Robust regularization in general . 69 4.4 Semi-algebraic robust regularization . 74 4.5 1-peaceful sets . 85 5 Continuity of set-valued maps revisited in the light of tame ge- ometry 91 5.1 A Sard result for local (Pareto) minima . 94 5.2 Extending the Mordukhovich criterion and a critical value result . 98 5.3 Some preliminary results . 101 5.4 More on the structure of semi-algebraic maps . 103 5.5 Main result . 113 5.6 Applications in tame variational analysis . 114 vi 6 Level set methods for finding critical points of mountain pass type117 6.1 A level set algorithm . 117 6.2 A locally superlinearly convergent algorithm . 121 6.3 Superlinear convergence of the local algorithm . 123 6.4 Further properties of the local algorithm . 138 6.5 Saddle points and criticality properties . 148 6.6 Wilkinson’s problem: Background . 156 6.7 Wilkinson’s problem: Implementation and numerical results . 158 6.8 Non-Lipschitz convergence and optimality conditions . 163 Bibliography 168 vii LIST OF TABLES 3.1 Summary of definitions . 24 3.2 Summary of definitions . 25 3.3 Examples of Pseudospectra for Example 3.21. 42 6.1 Convergence data for Example 6.27. Significant digits are in bold. 161 viii LIST OF FIGURES 3.1 Equivalences of properties summarized in Theorem 3.26. 23 5.1 Linking sets (A, Γ0) and (A, Γ). 108 6.1 Illustration of Algorithm 6.1. 119 6.2 Local structure of saddle point. 126 2 2 6.3 lev≤0f for f(x) = (x2 − x1)(x1 − x2) . 145 6.4 Illustration of saddle point in Example 6.22. 151 6.5 Different types of critical points . 156 6.6 A sample run of Algorithm 6.4. 162 6.7 An example where the Voronoi diagram heuristic fails. 162 ix CHAPTER 1 INTRODUCTION Variational analysis is the modern theory of nonsmooth, nonconvex analysis built on the theory of convex and smooth optimization – see the text by Rockafellar- Wets [80] and others, for example, Aubin-Frankowska [7], Borwein-Zhu [16], Clarke [31], Clarke-Ledyaev-Stern-Wolenski [32], and Mordukhovich [72]. While the gen- eral theory of variational analysis needs to accommodate pathological examples, constructions typically appearing in practice enjoy some of the properties of smooth functions and convex functions. For this reason, a good part of variational analysis focuses on favorable classes of nonsmooth, nonconvex functions, like Clarke regular functions, prox-regular functions and amenable functions. A recent direction in variational analysis is the study of variational properties of semi-algebraic functions and sets. Semi-algebraic objects eliminate most of the pathologies in analysis, and still form a broad class of objects appearing in practice. Semi-algebraic objects are defined by a finite set of polynomials, and include, for example, piecewise polynomials, rational functions, and the mapping to eigenvalues. The pseudospectral mapping, which is the focus of Chapters 3 and 6, is semi-algebraic. The set of positive semidefinite matrices, a common set in optimization, is semi-algebraic. Semi-algebraicity in concrete problems can often be easily checked using the Tarski-Seidenberg principle. Semi-algebraic geometry can be studied in an ax- iomatic manner under the broader theory of definable functions and sets, and o-minimal structures. O-minimal structures enjoy a variety of favorable properties which are employed in parts of this thesis, like the local conic homeomorphism and various decomposition theorems.
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