Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics

In Quest of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, and Summation Identities for Entire Functions

The title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used throughout. Also a brief historical recount is given on the origin of the Bernstein inequality, which dated back to the days of the discovery of the Periodic table by the Russian Chemist Dmitri Mendeleev. In Chapter 2, we narrow down the contents stated in Chapter 1 to the problems we were interested in working during the course of this dissertation. Henceforth, we present a problem formulation mainly for those results for which solutions or partial solutions are provided in the subsequent chapters. Over the years Bernstein inequality has been generalized and extended in several directions. In Chapter 3, we establish rational analogues to some Bernstein-type inequalities for restricted zeros and prescribed poles. Our inequalities extend the results for polynomials, especially which are themselves improved versions of the classical Erdös-Lax and Turán inequalities. In working towards proving our results, we establish some auxiliary results, which may be of interest on their own. Chapters 4 and 5 focus on the research carried out with the Askey-Wilson operator applied on polynomials and entire functions (of exponential type) respectively. In Chapter 4, we first establish a Riesz-type interpolation formula on the interval [−1, 1] for the Askey-Wilson operator. In consequence, a sharp Bernstein inequality and a Markov inequality are obtained when differentiation is replaced by the Askey-Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein-type inequality in L2−space. We conclude this chapter with an overconvergence result which is applied to characterize all q-differentiable functions of Brown and Ismail. Chapter 5 is devoted to an intriguing application of the Askey-Wilson operator. By applying it on the Sampling Theorem on entire functions of exponential type, we obtain a series representation formula, which is what we called an extended Boas’ formula. Its power in discovering interesting summation formulas, some known and some new will be demonstrated. As another application, we are able to obtain a couple of Bernstein-type inequalities. In the concluding chapter, we state some avenues where this research can progress

On Roth type conditions, duality and central Birkhoff sums for i.e.m

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.Comment: 45 pages, 5 figure

Robust estimation : limit theorems and their applications

This thesis is concerned with the asymptotic theory of general M-estimators and some minimal distance estimators. Particular attention is paid to uniform convergence theory which is used to prove limit theorems for statistics that are usually implicitly defined as solutions of estimating equations. The thesis is divided into eight chapters and into three main sections. In Section A the theory of convergence is studied as a prelude to validating the use of the particular M-estimators given in Section B and C. Section B initially covers the view of robustness of Hampel (1968) but places more emphasis on the application of the notions of differentiability of functionals and on M-estimators of a general parameter that are robust against "tail" contamination. Sections A and B establish a base for a comparison of robustness and application aspects of minimal distance estimators, particularly with regard to their application to estimating mixtures of normal distributions. An important application of this is illustrated for the analysis of seismic data. This constitutes Section C. Chapter 1 is devoted to the study of uniform convergence theorems over classes of functions and sets allowing also the possibility that the underlying probability mechanism may be from a specified family. A new Glivenko-Cantelli type theorem is proved which has applications later to weakening differentiability requirements for the convergence of loss functions used in this thesis. For implicitly defined estimators it is important to clearly identify the estimator. By uniform convergence, asymptotic uniqueness in regions of the parameter space of solutions to estimating equations can be established. This then justifies the selection of solutions through appropriate statistics, thus defining estimators uniquely for all samples. This comes under the discussion of existence and consistency in Chapter 2. Chapter 3 includes central limit theorems and the law of the iterated logarithm for the general M-estimator, established under various conditions, both on the loss function and on the underlying distribution. Uniform convergence plays a central role in showing the validity of approximating expansions. Results are shown for both univariate and multivariate parameters. Arguments for the univariate parameter are often simpler or require weaker conditions. Our study of robustness is both of a theoretical and quantitative nature. Weak continuity and also Frechet differentiability with respect to Prokhorov, Levy and Kolmogorov distance functions are established for multivariate M-functionals under similar but necessarily stronger conditions than those required for asymptotic normality. Relationships between the conditions imposed on the class of loss functions in order to attain Frechet differentiability and those necessary and sufficient conditions placed on classes of functions for which uniform convergence of measures hold can be shown. Much weaker conditions exist for almost sure uniform convergence and this goes part way to explaining the restrictive nature of this functional derivative approach to showing asymptotic normality. In Chapter 5 the notion of a set of null influence is emphasized. This can be used to construct M-functionals robust (in terms of asymptotic bias and variance) against contamination in the "tails" of a distribution. This set can depend on the parameter being estimated and in this sense the resulting estimator is adaptive. Its construction is illustrated in Chapter 6 for the estimation of scale. Robustness against "tail" contamination is illustrated by numerical comparison with other M-estimators. Particular applications are given to inference in the joint estimation of location and scale where it is important to identify the root to the M-estimating equations. Techniques justified by uniform convergence are used here. Uniform convergence also lends itself to the use of a graphical method of plotting "expectation curves". It can be used for either identifying the M-estimator from multiple solutions of the defining equations or in large samples (e.g. > 50) as a visual indication of whether the fitted model is a good approximation for the underlying mechanism. Theorems based on uniform convergence are given that show a domain of convergence (numerical analysis interpretation) for the Newton-Raphson iteration method applied to M-estimating equations for the location parameter when redescending loss functions are used. The M-estimator theory provides a common framework whereby some minimal distance methods can be compared. Two established L₂ minimal distance estimators are shown to be general M-estimators. In particular a Cramer-Von Mises type distance estimator is shown to be qualitatively robust and have good small sample properties. Its applicability to some new mixture data from geological recordings, which clearly requires robust methods of analysis is demonstrated in Chapters 7 and 8

Calculus of Variations on Time Scales and Discrete Fractional Calculus

We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler-Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems. We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities. In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations. Corresponding Euler-Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided.Comment: PhD thesis, University of Aveiro, 2010. Supervisor: Delfim F. M. Torres; co-supervisor: Martin Bohner. Defended 26/July/201

Indefinite Knapsack Separable Quadratic Programming: Methods and Applications

Quadratic programming (QP) has received significant consideration due to an extensive list of applications. Although polynomial time algorithms for the convex case have been developed, the solution of large scale QPs is challenging due to the computer memory and speed limitations. Moreover, if the QP is nonconvex or includes integer variables, the problem is NP-hard. Therefore, no known algorithm can solve such QPs efficiently. Alternatively, row-aggregation and diagonalization techniques have been developed to solve QP by a sub-problem, knapsack separable QP (KSQP), which has a separable objective function and is constrained by a single knapsack linear constraint and box constraints. KSQP can therefore be considered as a fundamental building-block to solve the general QP and is an important class of problems for research. For the convex KSQP, linear time algorithms are available. However, if some quadratic terms or even only one term is negative in KSQP, the problem is known to be NP-hard, i.e. it is notoriously difficult to solve. The main objective of this dissertation is to develop efficient algorithms to solve general KSQP. Thus, the contributions of this dissertation are five-fold. First, this dissertation includes comprehensive literature review for convex and nonconvex KSQP by considering their computational efficiencies and theoretical complexities. Second, a new algorithm with quadratic time worst-case complexity is developed to globally solve the nonconvex KSQP, having open box constraints. Third, the latter global solver is utilized to develop a new bounding algorithm for general KSQP. Fourth, another new algorithm is developed to find a bound for general KSQP in linear time complexity. Fifth, a list of comprehensive applications for convex KSQP is introduced, and direct applications of indefinite KSQP are described and tested with our newly developed methods. Experiments are conducted to compare the performance of the developed algorithms with that of local, global, and commercial solvers such as IBM CPLEX using randomly generated problems in the context of certain applications. The experimental results show that our proposed methods are superior in speed as well as in the quality of solutions

Levenberg-Marquardt Algorithms for Nonlinear Equations, Multi-objective Optimization, and Complementarity Problems

The Levenberg-Marquardt algorithm is a classical method for solving nonlinear systems of equations that can come from various applications in engineering and economics. Recently, Levenberg-Marquardt methods turned out to be a valuable principle for obtaining fast convergence to a solution of the nonlinear system if the classical nonsingularity assumption is replaced by a weaker error bound condition. In this way also problems with nonisolated solutions can be treated successfully. Such problems increasingly arise in engineering applications and in mathematical programming. In this thesis we use Levenberg-Marquardt algorithms to deal with nonlinear equations, multi-objective optimization and complementarity problems. We develop new algorithms for solving these problems and investigate their convergence properties. For sufficiently smooth nonlinear equations we provide convergence results for inexact Levenberg-Marquardt type algorithms. In particular, a sharp bound on the maximal level of inexactness that is sufficient for a quadratic (or a superlinear) rate of convergence is derived. Moreover, the theory developed is used to show quadratic convergence of a robust projected Levenberg-Marquardt algorithm. The use of Levenberg-Marquardt type algorithms for unconstrained multi-objective optimization problems is investigated in detail. In particular, two globally and locally quadratically convergent algorithms for these problems are developed. Moreover, assumptions under which the error bound condition for a Pareto-critical system is fulfilled are derived. We also treat nonsmooth equations arising from reformulating complementarity problems by means of NCP functions. For these reformulations, we show that existing smoothness conditions are not satisfied at degenerate solutions. Moreover, we derive new results for positively homogeneous functions. The latter results are used to show that appropriate weaker smoothness conditions (enabling a local Q-quadratic rate of convergence) hold for certain reformulations.Der Levenberg-Marquardt-Algorithmus ist ein klassisches Verfahren zur Lösung von nichtlinearen Gleichungssystemen, welches in verschiedenen Anwendungen der Ingenieur-und Wirtschaftswissenschaften vorkommen kann. Kürzlich, erwies sich das Verfahren als ein wertvolles Instrument für die Gewährleistung einer schnelleren Konvergenz für eine Lösung des nichtlinearen Systems, wenn die klassische nichtsinguläre Annahme durch eine schwächere Fehlerschranke der eingebundenen Bedingung ersetzt wird. Auf diese Weise, lassen sich ebenfalls Probleme mit nicht isolierten Lösungen erfolgreich behandeln. Solche Probleme ergeben sich zunehmend in den praktischen, ingenieurwissenschaftlichen Anwendungen und in der mathematischen Programmierung. In dieser Arbeit verwenden wir Levenberg-Marquardt- Algorithmus für nichtlinearere Gleichungen, multikriterielle Optimierung - und nichtlineare Komplementaritätsprobleme. Wir entwickeln neue Algorithmen zur Lösung dieser Probleme und untersuchen ihre Konvergenzeigenschaften. Für ausreichend differenzierbare nichtlineare Gleichungen, analysieren und bieten wir Konvergenzergebnisse für ungenaue Levenberg-Marquardt-Algorithmen Typen. Insbesondere, bieten wir eine strenge Schranke für die maximale Höhe der Ungenauigkeit, die ausreichend ist für eine quadratische (oder eine superlineare) Rate der Konvergenz. Darüber hinaus, die entwickelte Theorie wird verwendet, um quadratische Konvergenz eines robusten projizierten Levenberg-Marquardt-Algorithmus zu zeigen. Die Verwendung von Levenberg-Marquardt-Algorithmen Typen für unbeschränkte multikriterielle Optimierungsprobleme im Detail zu untersucht. Insbesondere sind zwei globale und lokale quadratische konvergente Algorithmen für multikriterielle Optimierungsprobleme entwickelt worden. Die Annahmen wurden hergeleitet, unter welche die Fehlerschranke der eingebundenen Bedingung für ein Pareto-kritisches System erfüllt ist. Wir behandeln auch nicht differenzierbare nichtlineare Gleichungen aus Umformulierung der nichtlinearen Komplementaritätsprobleme durch NCP-Funktionen. Wir zeigen für diese Umformulierungen, dass die bestehenden differenzierbaren Bedingungen nicht zufrieden mit degenerierten Lösungen sind. Außerdem, leiten wir neue Ergebnisse für positiv homogene NCP-Funktionen. Letztere Ergebnisse werden verwendet um zu zeigen, dass geeignete schwächeren differenzierbare Bedingungen (so dass eine lokale Q-quadratische Konvergenzgeschwindigkeit ermöglichen) für bestimmte Umformulierungen gelten

Robot Motion Planning Under Topological Constraints

My thesis addresses the the problem of manipulation using multiple robots with cables. I study how robots with cables can tow objects in the plane, on the ground and on water, and how they can carry suspended payloads in the air. Specifically, I focus on planning optimal trajectories for robots. Path planning or trajectory generation for robotic systems is an active area of research in robotics. Many algorithms have been developed to generate path or trajectory for different robotic systems. One can classify planning algorithms into two broad categories. The first one is graph-search based motion planning over discretized configuration spaces. These algorithms are complete and quite efficient for finding optimal paths in cluttered 2-D and 3-D environments and are widely used [48]. The other class of algorithms are optimal control based methods. In most cases, the optimal control problem to generate optimal trajectories can be framed as a nonlinear and non convex optimization problem which is hard to solve. Recent work has attempted to overcome these shortcomings [68]. Advances in computational power and more sophisticated optimization algorithms have allowed us to solve more complex problems faster. However, our main interest is incorporating topological constraints. Topological constraints naturally arise when cables are used to wrap around objects. They are also important when robots have to move one way around the obstacles rather than the other way around. Thus I consider the optimal trajectory generation problem under topological constraints, and pursue problems that can be solved in finite-time, guaranteeing global optimal solutions. In my thesis, I first consider the problem of planning optimal trajectories around obstacles using optimal control methodologies. I then present the mathematical framework and algorithms for multi-robot topological exploration of unknown environments in which the main goal is to identify the different topological classes of paths. Finally, I address the manipulation and transportation of multiple objects with cables. Here I consider teams of two or three ground robots towing objects on the ground, two or three aerial robots carrying a suspended payload, and two boats towing a boom with applications to oil skimming and clean up. In all these problems, it is important to consider the topological constraints on the cable configurations as well as those on the paths of robot. I present solutions to the trajectory generation problem for all of these problems