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Optimization Algorithms on Matrix Manifolds
Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.
eBook
A survey on evolutionary computation for complex continuous optimization
Complex continuous optimization problems widely exist nowadays due to the fast development of the economy and society. Moreover, the technologies like Internet of things, cloud computing, and big data also make optimization problems with more challenges including Many-dimensions, Many-changes, Many-optima, Many-constraints, and Many-costs. We term these as 5-M challenges that exist in large-scale optimization problems, dynamic optimization problems, multi-modal optimization problems, multi-objective optimization problems, many-objective optimization problems, constrained optimization problems, and expensive optimization problems in practical applications. The evolutionary computation (EC) algorithms are a kind of promising global optimization tools that have not only been widely applied for solving traditional optimization problems, but also have emerged booming research for solving the above-mentioned complex continuous optimization problems in recent years. In order to show how EC algorithms are promising and efficient in dealing with the 5-M complex challenges, this paper presents a comprehensive survey by proposing a novel taxonomy according to the function of the approaches, including reducing problem difficulty, increasing algorithm diversity, accelerating convergence speed, reducing running time, and extending application field. Moreover, some future research directions on using EC algorithms to solve complex continuous optimization problems are proposed and discussed. We believe that such a survey can draw attention, raise discussions, and inspire new ideas of EC research into complex continuous optimization problems and real-world applications.
Journal Article
Robust Optimization
Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and up-to-date account of the subject. Robust optimization is designed to meet some major challenges associated with uncertainty-affected optimization problems: to operate under lack of full information on the nature of uncertainty; to model the problem in a form that can be solved efficiently; and to provide guarantees about the performance of the solution. The book starts with a relatively simple treatment of uncertain linear programming, proceeding with a deep analysis of the interconnections between the construction of appropriate uncertainty sets and the classical chance constraints (probabilistic) approach. It then develops the robust optimization theory for uncertain conic quadratic and semidefinite optimization problems and dynamic (multistage) problems. The theory is supported by numerous examples and computational illustrations. An essential book for anyone working on optimization and decision making under uncertainty, Robust Optimization also makes an ideal graduate textbook on the subject.
eBook
Performance assessment of the metaheuristic optimization algorithms: an exhaustive review
The simulation-driven metaheuristic algorithms have been successful in solving numerous problems compared to their deterministic counterparts. Despite this advantage, the stochastic nature of such algorithms resulted in a spectrum of solutions by a certain number of trials that may lead to the uncertainty of quality solutions. Therefore, it is of utmost importance to use a correct tool for measuring the performance of the diverse set of metaheuristic algorithms to derive an appropriate judgment on the superiority of the algorithms and also to validate the claims raised by researchers for their specific objectives. The performance of a randomized metaheuristic algorithm can be divided into efficiency and effectiveness measures. The efficiency relates to the algorithm’s speed of finding accurate solutions, convergence, and computation. On the other hand, effectiveness relates to the algorithm’s capability of finding quality solutions. Both scopes are crucial for continuous and discrete problems either in single- or multi-objectives. Each problem type has different formulation and methods of measurement within the scope of efficiency and effectiveness performance. One of the most decisive verdicts for the effectiveness measure is the statistical analysis that depends on the data distribution and appropriate tool for correct judgments.
Journal Article
Set-Based Particle Swarm Optimisation: A Review
The set-based particle swarm optimisation algorithm is a swarm-based meta-heuristic that has gained popularity in recent years. In contrast to the original particle swarm optimisation algorithm, the set-based particle swarm optimisation algorithm is used to solve discrete and combinatorial optimisation problems. The main objective of this paper is to review the set-based particle swarm optimisation algorithm and to provide an overview of the problems to which the algorithm has been applied. This paper starts with an examination of previous attempts to create a set-based particle swarm optimisation algorithm and discusses the shortcomings of the existing attempts. The set-based particle swarm optimisation algorithm is established as the only suitable particle swarm variant that is both based on true set theory and does not require problem-specific modifications. In-depth explanations are given regarding the general position and velocity update equations, the mechanisms used to control the exploration–exploitation trade-off, and the quantifiers of swarm diversity. After the various existing applications of set-based particle swarm optimisation are presented, this paper concludes with a discussion on potential future research.
Journal Article
Practical methods of optimization
Fully describes optimization methods that are currently most valuable in solving real-life problems.Since optimization has applications in almost every branch of science and technology, the text emphasizes their practical aspects in conjunction with the heuristics useful in making them perform more reliably and efficiently.
eBook
Survey of Optimization Algorithms in Modern Neural Networks
The main goal of machine learning is the creation of self-learning algorithms in many areas of human activity. It allows a replacement of a person with artificial intelligence in seeking to expand production. The theory of artificial neural networks, which have already replaced humans in many problems, remains the most well-utilized branch of machine learning. Thus, one must select appropriate neural network architectures, data processing, and advanced applied mathematics tools. A common challenge for these networks is achieving the highest accuracy in a short time. This problem is solved by modifying networks and improving data pre-processing, where accuracy increases along with training time. Bt using optimization methods, one can improve the accuracy without increasing the time. In this review, we consider all existing optimization algorithms that meet in neural networks. We present modifications of optimization algorithms of the first, second, and information-geometric order, which are related to information geometry for Fisher–Rao and Bregman metrics. These optimizers have significantly influenced the development of neural networks through geometric and probabilistic tools. We present applications of all the given optimization algorithms, considering the types of neural networks. After that, we show ways to develop optimization algorithms in further research using modern neural networks. Fractional order, bilevel, and gradient-free optimizers can replace classical gradient-based optimizers. Such approaches are induced in graph, spiking, complex-valued, quantum, and wavelet neural networks. Besides pattern recognition, time series prediction, and object detection, there are many other applications in machine learning: quantum computations, partial differential, and integrodifferential equations, and stochastic processes.
Journal Article
Population-based optimization in structural engineering: a review
Structural engineering is focused on the safe and efficient design of infrastructure. Projects can range in size and complexity, many requiring massive amounts of materials and expensive construction and operational costs. Therefore, one of the primary objectives for structural engineers is a cost-effective design. Incorporating optimality criteria into the design procedure introduces additional complexities that result in problems that are nonlinear, nonconvex, and have a discontinuous solution space. Population-based optimization algorithms (known as metaheuristics) have been found to be very efficient approaches to these problems. Many researchers have developed and applied state-of-art metaheuristics to automate and optimize the design of real-world civil engineering problems. While there is a large body of published papers in this area, there are few comprehensive reviews that list, summarize, and categorize metaheuristic optimization in structural engineering. This paper provides an extensive survey of a wide range of metaheuristic techniques to structural engineering optimization problems. Also, information is provided on available structural engineering benchmark problems, the formulation of different objective functions, and the handling of various types of constraints. The performance of different optimization techniques is compared for many benchmark problems.
Journal Article
Nonsmooth differential geometry– An approach tailored for spaces with Ricci curvature bounded from below
We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see
that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian,
covariant/exterior derivatives and Ricci curvature.
eBook