This brief aims to merge the theories of fractional calculus and discrete calculus in a concise but comprehensive manner. It is designed for graduate students, but will be useful for any researcher interested in the theory of discrete fractional calculus and fractional difference equations.

In the last decades, algorithmic advances as well as hardware and software improvements have provided an excellent environment to create and develop solving methods to hard optimization problems. Modern exact and heuristic techniques are dramatically enhancing our ability to solve significant practical problems. This monograph sets out state-of-the-art methodologies for solving combinatorial optimization problems, illustrating them with two well-known problems. This second edition of the book extends the first one by adding to the 'linear ordering problem' (LOP), included in the first edition, the 'maximum diversity problem' (MDP). In this way, we provide the reader with the background, elements and strategies to tackle a wide range of different combinatorial optimization problems. The exact and heuristic techniques outlined in these pages can be put to use in any number of combinatorial optimization problems. While the authors employ the LOP and the MDP to illustrate cutting-edge optimization technologies, the book is also a tutorial on how to design effective and successful implementations of exact and heuristic procedures alike. This monograph provides the basic principles and fundamental ideas that will enable students and practitioners to create valuable applications based on both exact and heuristic technologies. Specifically, it is aimed at engineers, scientists, operations researchers, and other applications specialists who are looking for the most appropriate and recent optimization tools to solve particular problems. The book provides a broad spectrum of advances in search strategies with a focus on its algorithmic and computational aspects.

This book deals with critical infrastructure safety analysis based on reliability modelling of multistate ageing system. It shows how changes of the operation process as well as climate-weather changes in the operating area of the critical infrastructure do influence the safety parameters of its assets. Building upon previous authors' research, the book formulates an integrated modeling approach where the multistate critical infrastructure safety model is combined with semi-Markov models for its operation process and for the climate-weather change process. This approach is shown to be successful in determining basic critical infrastructure safety, risk and resilience indicators, regardless of the number of assets and the number of their safety states. Besides the theory, the book reports on a successful application to the safety analysis of a real critical infrastructure, such as a port oil terminal. All in all, this book proposes a comprehensive and timely review of cutting-edge mathematical methods for safety identification, prediction and evaluation of critical infrastructures. It demonstrates that these methods can be applied in practice for analyzing safety of critical infrastructure under time-varying operation and climate-weather change processes.

This book highlights new and original contributions on Graph Theory and Combinatorial Optimization both from the theoretical point of view and from applications in all fields. The book chapters describe models and methods based on graphs, structural properties, discrete optimization, network optimization, mixed-integer programming, heuristics, meta-heuristics, math-heuristics, and exact methods as well as applications. The book collects selected contributions from the CTW2020 international conference (18th Cologne-Twente Workshop on Graphs and Combinatorial Optimization), held online on September 14-16, 2020. The conference was organized by IASI-CNR with the contribution of University of Roma Tre, University Roma Tor Vergata, and CNRS-LIX and with the support of AIRO. It is addressed to researchers, PhD students, and practitioners in the fields of Graph Theory, Discrete Mathematics, Combinatorial Optimization, and Operations Research.

This book describes applications of Jaya and Rao algorithms on real case studies concerning different renewable energy sources. In the last few decades, researchers have focused on renewable energy resources like solar energy, bio-energy, wave energy, ocean thermal energy, tidal energy, geothermal energy, and wind energy. This has resulted in the development of new techniques and tools that could harvest energy from renewable energy sources. Many researchers and scientists have focused on developing and optimizing the energy systems to extract and utilize renewable energy more efficiently. In this book, recently developed Jaya and Rao (Rao-1, Rao-2, and Rao-3) algorithms are introduced for single- and multi-objective optimization of selected renewable energy systems. The results of applications of the different versions of Jaya and Rao algorithms are compared with the other optimization techniques like GA, NSGA-II, PSO, MOPSO, ABC, etc., and the performance of the Jaya and Rao algorithms is highlighted compared to other optimization algorithms in the case of renewable energy systems. The book also includes the validation of different versions of the Jaya and Rao algorithms through the application to complex single- and multi-objective unconstrained benchmark functions. The algorithms and computer codes of different version of Jaya and Rao algorithms are included in the book that will be very much useful to readers in industry and academic research.

This book presents best practices involving applications of decision sciences, business tactics and behavioral sciences for COVID-19. Addressing concrete problems in these vital fields, it focuses on theoretical and methodological investigations of managerial decisions that drive production and service enterprises' productivity and success. Moreover, it presents optimization techniques and tools that can also be adopted for other applications in various research areas after a thorough analysis of the specific problem. The book is intended for researchers and practitioners seeking optimum solutions to real-life problems in various application areas concerning COVID-19, helping them make scientifically founded decisions.

This book presents new techniques and methods for distributed control and optimization of networked microgrids. Distributed consensus issues under network-based and event-triggered mechanisms are first addressed in a multi-agent system framework, which can explicitly characterize the relationship between communication resources and the control performance. Then, considering the effects of network uncertainties, multi-agent system-based distributed schemes are tailored to solve the fundamental issues of networked microgrids such as distributed frequency regulation, voltage regulation, active power sharing/load sharing, and energy management. The monograph will contribute to stimulating extensive interest of researchers in electrical and control fields.

Many of the most challenging problems in the applied sciences involve non-differentiable structures as well as partial differential operators, thus leading to non-smooth distributed parameter systems. This edited volume aims to establish a theoretical and numerical foundation and develop new algorithmic paradigms for the treatment of non-smooth phenomena and associated parameter influences. Other goals include the realization and further advancement of these concepts in the context of robust and hierarchical optimization, partial differential games, and nonlinear partial differential complementarity problems, as well as their validation in the context of complex applications. Areas for which applications are considered include optimal control of multiphase fluids and of superconductors, image processing, thermoforming, and the formation of rivers and networks. Chapters are written by leading researchers and present results obtained in the first funding phase of the DFG Special Priority Program on Nonsmooth and Complementarity Based Distributed Parameter Systems: Simulation and Hierarchical Optimization that ran from 2016 to 2019.

This book presents a modern perspective on the modelling, analysis, and synthesis ideas behind convex-optimisation-based control of nonlinear systems: it embeds them in models with convex structures. Analysis and Synthesis of Nonlinear Control Systems begins with an introduction to the topic and a discussion of the problems to be solved. It then explores modelling via convex structures, including quasi-linear parameter-varying, Takagi-Sugeno models, and linear fractional transformation structures. The authors cover stability analysis, addressing Lyapunov functions and the stability of polynomial models, as well as the performance and robustness of the models. With detailed examples, simulations, and programming code, this book will be useful to instructors, researchers, and graduate students interested in nonlinear control systems.

This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot-Caratheodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.

This volume provides a timely survey of interactions between the calculus of variations and theoretical and applied mechanics. Chapters have been significantly expanded since preliminary versions appeared in a special issue of the Journal of Optimization Theory and Applications (184(1), 2020) on "Calculus of Variations in Mechanics and Related Fields". The variety of topics covered offers researchers an overview of problems in mechanics that can be analyzed with variational techniques, making this a valuable reference for researchers in the field. It also presents ideas for possible future areas of research, showing how the mastery of these foundational mathematical techniques can be used for many exciting applications. Specific topics covered include: Topology optimization Identification of material properties Optimal control Plastic flows Gradient polyconvexity Obstacle problems Quasi-monotonicity Variational Views in Mechanics will appeal to researchers in mathematics, solid-states physics, and mechanical, civil, and materials engineering.

Since Additive Manufacturing (AM) techniques allow the manufacture of complex-shaped structures the combination of lightweight construction, topology optimization, and AM is of significant interest. Besides the established continuum topology optimization methods, less attention is paid to algorithm-driven optimization based on linear optimization, which can also be used for topology optimization of truss-like structures. To overcome this shortcoming, we combined linear optimization, Computer-Aided Design (CAD), numerical shape optimization, and numerical simulation into an algorithm-driven product design process for additively manufactured truss-like structures. With our Ansys SpaceClaim add-in construcTOR, which is capable of obtaining ready-for-machine-interpretation CAD data of truss-like structures out of raw mathematical optimization data, the high performance of (heuristic-based) optimization algorithms implemented in linear programming software is now available to the CAD community.

This highly informative and carefully presented textbook introduces the general principles involved in system design and optimization as applicable to thermal systems, followed by the methods to accomplish them. It introduces contemporary techniques like Genetic Algorithms, Simulated Annealing, and Bayesian Inference in the context of optimization of thermal systems. There is a separate chapter devoted to inverse problems in thermal systems. It also contains sections on Integer Programming and Multi-Objective optimization. The linear programming chapter is fortified by a detailed presentation of the Simplex method. A major highlight of the textbook is the inclusion of workable MATLAB codes for examples of key algorithms discussed in the book. Examples in each chapter clarify the concepts and methods presented and end-of-chapter problems supplement the material presented and enhance the learning process.

This book presents an introduction to variational analysis, a field which unifies theories and techniques developed in calculus of variations, optimization, and control, and covers convex analysis, nonsmooth analysis, and set-valued analysis. It focuses on problems with constraints, the analysis of which involves set-valued mappings and functions that are not differentiable. Applications of variational analysis are interdisciplinary, ranging from financial planning to steering a flying object. The book is addressed to graduate students, researchers, and practitioners in mathematical sciences, engineering, economics, and finance. A typical reader of the book should be familiar with multivariable calculus and linear algebra. Some basic knowledge in optimization, control, and elementary functional analysis is desirable, but all necessary background material is included in the book.

Features recent advances and new applications in graph edge coloring

Reviewing recent advances in the Edge Coloring Problem, "Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture" provides an overview of the current state of the science, explaining the interconnections among the results obtained from important graph theory studies. The authors introduce many new improved proofs of known results to identify and point to possible solutions for open problems in edge coloring.

The book begins with an introduction to graph theory and the concept of edge coloring. Subsequent chapters explore important topics such as:

Use of Tashkinov trees to obtain an asymptotic positive solution to Goldberg's conjecture

Application of Vizing fans to obtain both known and new results

Kierstead paths as an alternative to Vizing fans

Classification problem of simple graphs

Generalized edge coloring in which a color may appear more than once at a vertex

This book also features first-time English translations of two groundbreaking papers written by Vadim Vizing on an estimate of the chromatic class of a p-graph and the critical graphs within a given chromatic class.

Written by leading experts who have reinvigorated research in the field, "Graph Edge Coloring" is an excellent book for mathematics, optimization, and computer science courses at the graduate level. The book also serves as a valuable reference for researchers interested in discrete mathematics, graph theory, operations research, theoretical computer science, and combinatorial optimization.

This book is an up-to-date documentation of the state of the art in combinatorial optimization, presenting approximate solutions of virtually all relevant classes of NP-hard optimization problems. The well-structured wealth of problems, algorithms, results, and techniques introduced systematically will make the book an indispensible source of reference for professionals. The smooth integration of numerous illustrations, examples, and exercises make this monograph an ideal textbook.

This book addresses and disseminates state-of-the-art research and development of differential evolution (DE) and its recent advances, such as the development of adaptive, self-adaptive and hybrid techniques. Differential evolution is a population-based meta-heuristic technique for global optimization capable of handling non-differentiable, non-linear and multi-modal objective functions. Many advances have been made recently in differential evolution, from theory to applications. This book comprises contributions which include theoretical developments in DE, performance comparisons of DE, hybrid DE approaches, parallel and distributed DE for multi-objective optimization, software implementations, and real-world applications. The book is useful for researchers, practitioners, and students in disciplines such as optimization, heuristics, operations research and natural computing.

Fractional evolution inclusions are an important form of differential inclusions within nonlinear mathematical analysis. They are generalizations of the much more widely developed fractional evolution equations (such as time-fractional diffusion equations) seen through the lens of multivariate analysis. Compared to fractional evolution equations, research on the theory of fractional differential inclusions is however only in its initial stage of development. This is important because differential models with the fractional derivative providing an excellent instrument for the description of memory and hereditary properties, and have recently been proved valuable tools in the modeling of many physical phenomena. The fractional order models of real systems are always more adequate than the classical integer order models, since the description of some systems is more accurate when the fractional derivative is used. The advantages of fractional derivatization become evident in modeling mechanical and electrical properties of real materials, description of rheological properties of rocks and in various other fields. Such models are interesting for engineers and physicists as well as so-called pure mathematicians. Phenomena investigated in hybrid systems with dry friction, processes of controlled heat transfer, obstacle problems and others can be described with the help of various differential inclusions, both linear and nonlinear. Fractional Evolution Equations and Inclusions is devoted to a rapidly developing area of the research for fractional evolution equations & inclusions and their applications to control theory. It studies Cauchy problems for fractional evolution equations, and fractional evolution inclusions with Hille-Yosida operators. It discusses control problems for systems governed by fractional evolution equations. Finally it provides an investigation of fractional stochastic evolution inclusions in Hilbert spaces.

This volume offers a wealth of interdisciplinary approaches to artificial intelligence, machine learning and optimization tools, which contribute to the optimization of urban features towards forming smart, sustainable, and livable future cities. Special features include: New research on the design of city elements and smart systems with respect to new technologies and scientific thinking Discussions on the theoretical background that lead to smart cities for the future New technologies and principles of research that can promote ideas of artificial intelligence and machine learning in optimized urban environments The book engages students and researchers in the subjects of artificial intelligence, machine learning, and optimization tools in smart sustainable cities as eminent international experts contribute their research results and thinking in its chapters. Overall, its audience can benefit from a variety of disciplines including, architecture, engineering, physics, mathematics, computer science, and related fields.

This book collects chapters on contemporary topics on metric fixed point theory and its applications in science, engineering, fractals, and behavioral sciences. Chapters contributed by renowned researchers from across the world, this book includes several useful tools and techniques for the development of skills and expertise in the area. The book presents the study of common fixed points in a generalized metric space and fixed point results with applications in various modular metric spaces. New insight into parametric metric spaces as well as study of variational inequalities and variational control problems have been included.

This book contains selected papers presented at ICGEC 2021, the 14th International Conference on Genetic and Evolutionary Computing, held from October 21-23, 2021 in Jilin City, China. The conference was technically co-sponsored by Springer, Northeast Electric Power University Fujian University of Technology, Shandong University of Science and Technology, and Western Norway University of Applied Sciences. It is intended as an international forum for the researchers and professionals in all areas of genetic and evolutionary computing. And the readers may learn the up-to-date techniques of the mentioned topics, including swarm intelligence, artificial intelligence, information hiding and data mining techniques, which can help them to bring new ideas or apply the designed approaches from the collected papers to their professional jobs.

This book describes the development of innovative non-centralized optimization-based control schemes to solve economic dispatch problems of large-scale energy systems. Particularly, it focuses on communication and cooperation processes of local controllers, which are integral parts of such schemes. The economic dispatch problem, which is formulated as a convex optimization problem with edge-based coupling constraints, is solved by using methodologies in distributed optimization over time-varying networks, together with distributed model predictive control, and system partitioning techniques. At first, the book describes two distributed optimization methods, which are iterative and require the local controllers to exchange information with each other at each iteration. In turn, it shows that the sequence produced by these methods converges to an optimal solution when some conditions, which include how the controllers must communicate and cooperate, are satisfied. Further, it proposes an information exchange protocol to cope with possible communication link failures. Finally, the proposed distributed optimization methods are extended to the cases with random communication networks and asynchronous updates. Overall, this book presents a set of improved predictive control and distributed optimization methods, together with a rigorous mathematical analysis of each proposed algorithms. It describes a comprehensive approach to cope with communication and cooperation issues of non-centralized control schemes and show how the improved schemes can be successfully applied to solve the economic dispatch problems of large-scale energy systems.

This book constitutes the refereed proceedings of the 12th International Conference on Optimization and Applications, OPTIMA 2021, held in Petrovac, Montenegro, in September - October 2021. Due to the COVID-19 pandemic the conference was partially held online. The 19 revised full papers presented were carefully reviewed and selected from 38 submissions. The papers are organized in topical sections on mathematical programming; global optimization; stochastic optimization; optimal control; mathematical economics; optimization in data analysis; applications.

This book includes over 500 most challenging exercises and problems in calculus. Topical problems and exercises are discussed on set theory, numbers, functions, limits and continuity, derivative, integral calculus, Rolle's theorem, mean value theorem, optimization problems, sequences and series. All the seven chapters recall important definitions, theorems and concepts, making this book immensely valuable to undergraduate students of engineering, mathematics, statistics, computer science and basic sciences.

This book introduces readers to benchmarking techniques in the stochastic environment, primarily stochastic data envelopment analysis (DEA), and provides stochastic models in DEA for the possibility of variations in inputs and outputs. It focuses on the application of theories and interpretations of the mathematical programs, which are combined with economic and organizational thinking. The book's main purpose is to shed light on the advantages of the different methods in deterministic and stochastic environments and thoroughly prepare readers to properly use these methods in various cases. Simple examples, along with graphical illustrations and real-world applications in industry, are provided for a better understanding. The models introduced here can be easily used in both theoretical and real-world evaluations. This book is intended for graduate and PhD students, advanced consultants, and practitioners with an interest in quantitative performance evaluation.