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This research monograph summarizes a line of research that maps certain classical problems of discrete mathematics and operations research - such as the Hamiltonian cycle and the Travelling Salesman problems – into convex domains where continuum analysis can be carried out.  Arguably, the inherent difficulty of these, now classical, problems stems precisely from the discrete nature of domains in which these problems are posed. The convexification of domains underpinning the reported results is achieved by assigning probabilistic interpretation to key elements of the original deterministic problems. In particular, approaches summarized here build on a technique that embeds Hamiltonian Cycle and Traveling Salesman problems in a structured singularly perturbed Markov decision process. The unifying idea is to interpret subgraphs traced out by deterministic policies (including Hamiltonian cycles, if any) as extreme points of a convex polyhedron in a space filled with randomized policies. The above, innovative, approach has now evolved to the point where there are many, both theoretical and algorithmic, results that exploit the nexus between graph theoretic structures and both probabilistic and algebraic entities of related Markov chains. The latter include moments of first return times, limiting frequencies of visits to nodes, or the spectra of certain matrices traditionally associated with the analysis of Markov chains. However, these results and algorithms are dispersed over more than fifteen research papers appearing in journals catering to disparate audiences such as:  MOR, Random Structures and Algorithms, SIAM J. on Discrete Mathematics, Optimization,  J. of Mathematical Analysis and Applications and some others. Furthermore, because of the evolution of this topic and specific orientation of these journals, the published manuscripts are often written in a very terse manner and use disparate notation.  As such it is difficult for new researchers to make use of the many advances reported in these papers. Hence the main purpose of this book is to present a concise and yet, well written, synthesis of the majority of the theoretical and algorithmic results obtained so far.  In addition the book will discuss numerous open questions and problems that arise from this body of work and which are yet to be fully solved.  The authors believe that their approach casts the Hamiltonian Cycle and Traveling Salesman problems in a mathematical framework that permits analytical concepts and techniques,  not used hitherto in their context, to be brought to bear to further clarify both the underlying difficulty of NP-completeness of these problems and the relative exceptionality of truly difficult instances. Finally, the material is arranged in such a manner that the introductory chapters require very little mathematical background and discuss instances of graphs with interesting structures that motivated a lot of the research in this topic.   More difficult results are introduced later but, unlike the research manuscripts where they were originally proved,  are illustrated with numerous examples.

Hamiltonian systems. --- Markov processes. --- Hamiltonian systems --- Markov processes --- Mathematics --- Management --- Physical Sciences & Mathematics --- Business & Economics --- Management Theory --- Geometry --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Business. --- Operations research. --- Decision making. --- Mathematical optimization. --- Management science. --- Probabilities. --- Business and Management. --- Operation Research/Decision Theory. --- Operations Research, Management Science. --- Optimization. --- Probability Theory and Stochastic Processes. --- Stochastic processes --- Differentiable dynamical systems --- Distribution (Probability theory. --- Operations Research/Decision Theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Quantitative business analysis --- Problem solving --- Statistical decision --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management decisions --- Choice (Psychology) --- Decision making

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This book serves as an advanced text for a graduate course on stochastic algorithms for the students of probability and statistics, engineering, economics and machine learning. This second edition gives a comprehensive treatment of stochastic approximation algorithms based on the ordinary differential equation (ODE) approach which analyses the algorithm in terms of a limiting ODE. It has a streamlined treatment of the classical convergence analysis and includes several recent developments such as concentration bounds, avoidance of traps, stability tests, distributed and asynchronous schemes, multiple time scales, general noise models, etc., and a category-wise exposition of many important applications. It is also a useful reference for researchers and practitioners in the field.

Statistical science --- Operational research. Game theory --- Mathematical statistics --- Probability theory --- Electrical engineering --- Planning (firm) --- Artificial intelligence. Robotics. Simulation. Graphics --- Recreation. Games. Sports. Corp. expression --- waarschijnlijkheidstheorie --- stochastische analyse --- automatisering --- mathematische modellen --- statistiek --- spellen --- econometrie --- KI (kunstmatige intelligentie) --- operationeel onderzoek --- kansrekening --- automatische regeltechniek --- statistisch onderzoek --- AI (artificiële intelligentie)

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Anàlisi estocàstica --- Stochastic approximation.

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This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines.

Convex sets. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Sets, Convex --- Convex domains --- Set theory --- Optimization. --- Conjunts convexos --- Optimització matemàtica

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This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines.

Numerical methods of optimisation --- Operational research. Game theory --- Computer. Automation --- automatisering --- wiskunde