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Mathematical optimization or mathematical programming consists of two main categories: linear programming and nonlinear programming. In this chapter, we will first introduce linear programming briefly.
The necessity of an optimisation procedure in optical phased array (OPA) design has been demonstrated. Simulated annealing was used to perform the numerical simulation. The analysis studies the scheme of a two-dimensional fibre-type OPA with piezoelectric ceramics (PZT) phase shifters. Moreover, an experimental system has been set up to confirm the integrity of the design.
No abstract is provided for this article.
No abstract is provided for this article.
Efficiency of an optimisation process is largely determined by the search algorithm and its fundamental characteristics. In a given optimisation, a single type of algorithm is used in most applications. In this paper, we will investigate the eagle strategy recently developed for global optimisation, which uses a two-stage strategy by combing two different algorithms to improve the overall search efficiency. We will discuss this strategy with differential evolution and then evaluate their performance by solving real-world optimisation problems such as pressure vessel and speed reducer design. Results suggest that we can reduce the computing effort by a factor of up to ten in many applications.
Pressure solution is an important process in sedimentary basins, and its behaviour depends mainly on the sediment rheology and temperature distribution. The compaction relation of pressure solution is typically assumed to be a viscous one and is often written as a relationship between effective stress and strain rate. A new derivation of viscous compaction relation is formulated based on more realistic boundary conditions at grain contacts. A non-linear diffusion problem with a moving boundary is solved numerically and a simple asymptotic solution is given to compare with numerical simulations. Pressure solution is significantly influenced by the temperature gradient. Porosity reduction due to pressure solution is enhanced in an environment with a higher thermal gradient, while porosity decreases much slowly in the region where the thermal gradient is small. Pressure solution tends to complete more quickly at shallower depths and earlier time in higher temperature environment than that in a low one. These features of pressure solution in porous sediments are analysed using a perturbation method to get a solution for the steady-state. Comparison with real data shows a reasonably very good agreement.
Smectite‐illite reaction of sediment minerals in sedimentary basins is modelled as a one‐step dehydration reaction. The water released from the mineral reaction can affect compaction process so as to retard porosity reduction and thus may cause overpressuring, especially in the reaction region. The coupled nonlinear model equations have been solved numerically, and the asymptotic analysis has been used to get the analytical solutions of the nonlinear model equations in some regions of most practical interests. The comparison of the obtained asymptotical solutions with the numerical results and real data shows very good agreement.
Nature-inspired algorithms are among the most powerful algorithms for optimization. In this study, a new nature-inspired metaheuristic optimization algorithm, called bat algorithm (BA), is introduced for solving engineering optimization tasks. The proposed BA is based on the echolocation behavior of bats. After a detailed formulation and explanation of its implementation, BA is verified using eight nonlinear engineering optimization problems reported in the specialized literature. BA has been carefully implemented and carried out optimization for eight well-known optimization tasks. Then, a comparison has been made between the proposed algorithm and other existing algorithms. The optimal solutions obtained by the proposed algorithm are better than the best solutions obtained by the existing methods. The unique search features used in BA are analyzed, and their implications for future research are also discussed in detail.
Based on product market competition, this paper studies the heterogeneity of monetary policy in promoting firms’ innovation. Specifically, taking 1420 listed companies from 2013 to 2018 as samples, this paper estimates the impact of monetary policy and product market competition on R&D investment density, and the moderating effect of competition on policy effect by using the fixed effect model. This study shows that a loose monetary policy shock relaxes firms’ financing constraints and increases their R&D investment capacity. In response, firms facing greater competitive pressures in product markets choose to increase their R&D investment density, while firms with less competitive pressures lack incentives for R&D investment.
Compaction in reactive porous media is modelled as a reaction-diffusion process with a moving boundary. Asymptotic analysis is used to find solutions for the coupled nonlinear compaction equations, and a traveling wave solution is obtained above the reaction zone.
The calcium transport in biological systems is modelled as a reaction–diffusion process. Nonlinear calcium waves are then simulated using a stochastic cellular automaton whose rules are derived from the corresponding coupled partial differential equations. Numerical simulations show self-organized criticality in the complex calcium waves and patterns. Both the stochastic cellular automaton approach and the equation-based simulations can predict the characteristics of calcium waves and complex pattern formation. The implication of locality of calcium distribution with positional information in biological systems is also discussed.
An accessible introduction to metaheuristics and optimization, featuring powerful and modern algorithms for application across engineering and the sciences From engineering and computer science to economics and management science, optimization is a core component for problem solving. Highlighting the latest developments that have evolved in recent years, Engineering Optimization: An Introduction with Metaheuristic Applications outlines popular metaheuristic algorithms and equips readers with the skills needed to apply these techniques to their own optimization problems. With insightful examples from various fields of study, the author highlights key concepts and techniques for the successful application of commonly-used metaheuristc algorithms, including simulated annealing, particle swarm optimization, harmony search, and genetic algorithms. The author introduces all major metaheuristic algorithms and their applications in optimization through a presentation that is organized into three succinct parts: Foundations of Optimization and Algorithms provides a brief introduction to the underlying nature of optimization and the common approaches to optimization problems, random number generation, the Monte Carlo method, and the Markov chain Monte Carlo method Metaheuristic Algorithms presents common metaheuristic algorithms in detail, including genetic algorithms, simulated annealing, ant algorithms, bee algorithms, particle swarm optimization, firefly algorithms, and harmony search Applications outlines a wide range of applications that use metaheuristic algorithms to solve challenging optimization problems with detailed implementation while also introducing various modifications used for multi-objective optimization Throughout the book, the author presents worked-out examples and real-world applications that illustrate the modern relevance of the topic. A detailed appendix features important and popular algorithms using MATLAB and Octave software packages, and a related FTP site houses MATLAB code and programs for easy implementation of the discussed techniques. In addition, references to the current literature enable readers to investigate individual algorithms and methods in greater detail. Engineering Optimization: An Introduction with Metaheuristic Applications is an excellent book for courses on optimization and computer simulation at the upper-undergraduate and graduate levels. It is also a valuable reference for researchers and practitioners working in the fields of mathematics, engineering, computer science, operations research, and management science who use metaheuristic algorithms to solve problems in their everyday work.
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This chapter introduces the fundamentals of ordinary differential equations (ODEs). Both first-order ODEs and second-order ODEs are explained in detail, including finding their complementary functions and particular integrals. Higher-order linear ODEs can be transformed into a system of first-order ODEs, which can be solved using dynamical system approaches.
