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About this book

This book systematically discusses nonlinear interval optimization design theory and methods. Firstly, adopting a mathematical programming theory perspective, it develops an innovative mathematical transformation model to deal with general nonlinear interval uncertain optimization problems, which is able to equivalently convert complex interval uncertain optimization problems to simple deterministic optimization problems. This model is then used as the basis for various interval uncertain optimization algorithms for engineering applications, which address the low efficiency caused by double-layer nested optimization. Further, the book extends the nonlinear interval optimization theory to design problems associated with multiple optimization objectives, multiple disciplines, and parameter dependence, and establishes the corresponding interval optimization models and solution algorithms. Lastly, it uses the proposed interval uncertain optimization models and methods to deal with practical problems in mechanical engineering and related fields, demonstrating the effectiveness of the models and methods.

Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract
This chapter introduces the engineering background and research significance of uncertain optimization and analyzes the research status of several mainstream uncertain optimization methods, in which the research status and main technical problems of interval optimization method are emphasized. Finally, this chapter gives the framework for this book.
Chao Jiang, Xu Han, Huichao Xie

Chapter 2. The Basic Principles of Interval Analysis

Abstract
This chapter briefly introduces the origin of interval number, the basic conceptions of interval mathematics, the calculation rules of interval number, and the important interval overestimation problem, providing some necessary theoretical foundations for interval optimization.
Chao Jiang, Xu Han, Huichao Xie

Chapter 3. Mathematical Transformation Models of Nonlinear Interval Optimization

Abstract
This chapter proposes two types of mathematical transformation models for general nonlinear interval optimization problems, i.e., the transformation model based on order relation of interval number and the transformation model based on possibility degree of interval number. Therefore, the uncertain optimization problem is transformed into a deterministic optimization. Furthermore, a two-layer optimization algorithm is established to solve the transformed deterministic optimization problem.
Chao Jiang, Xu Han, Huichao Xie

Chapter 4. Interval Optimization Based on Hybrid Optimization Algorithm

Abstract
By combining the genetic algorithm (GA) with the artificial neural network (ANN), this chapter establish two hybrid optimization algorithms to solve the nested optimization problem after transformation, based on which two efficient nonlinear interval optimization methods are developed.
Chao Jiang, Xu Han, Huichao Xie

Chapter 5. Interval Optimization Based on Interval Structural Analysis

Abstract
This chapter first introduces the conventional interval structural analysis method with small uncertainties. Based on the interval set theory and the subinterval technology, it is then extended to problems with large uncertainties. Consequently, an efficient interval optimization method based on the interval structural analysis is established.
Chao Jiang, Xu Han, Huichao Xie

Chapter 6. Interval Optimization Based on Sequential Linear Programming

Abstract
By introducing the sequential linear programming technique, this chapter develops an efficient nonlinear interval optimization algorithm. An iterative mechanism is also provided to ensure the convergence of the proposed algorithm.
Chao Jiang, Xu Han, Huichao Xie

Chapter 7. Interval Optimization Based on Approximation Models

Abstract
This chapter creates two nonlinear interval optimization methods based on the approximation model management strategy and the local-densifying approximation technique, respectively. Both the methods can greatly improve the computational efficiency of interval optimization by using approximation models, but different strategies are selected to guarantee accuracy during the iteration.
Chao Jiang, Xu Han, Huichao Xie

Chapter 8. Interval Multidisciplinary Design Optimization

Abstract
This chapter introduces the interval model into the multidisciplinary design optimization (MDO) problem, and whereby constructs an interval MDO model for complex MDO problems with uncertainties. A solution strategy is also given to solve the interval MDO model.
Chao Jiang, Xu Han, Huichao Xie

Chapter 9. A New Type of Possibility Degree of Interval Number and Its Application in Interval Optimization

Abstract
This chapter first proposes a new possibility degree model of interval number to realize quantitative comparison for not only overlapping intervals but also separate intervals, and then applies it to the nonlinear interval optimization.
Chao Jiang, Xu Han, Huichao Xie

Chapter 10. Interval Optimization Considering the Correlation of Parameters

Abstract
This chapter first introduces a new type of interval model, i.e., the multidimensional parallelepiped interval model, based on which an interval optimization model for multisource uncertain problems is then established. A solution algorithm is also given for this interval optimization model.
Chao Jiang, Xu Han, Huichao Xie

Chapter 11. Interval Multi-objective Optimization

Abstract
By employing an interval approach to describe the uncertainty of parameters in multi-objective optimization, this chapter proposes an interval multi-objective optimization model and also an efficient solution algorithm for this optimization model.
Chao Jiang, Xu Han, Huichao Xie

Chapter 12. Interval Optimization Considering Tolerance Design

Abstract
This chapter proposes an interval optimization method considering tolerance design, which provides not only the optimal design but also the optimal tolerances of the design variables.
Chao Jiang, Xu Han, Huichao Xie

Chapter 13. Interval Differential Evolution Algorithm

Abstract
By introducing the interval model into the existing differential evolution, this chapter proposes a novel interval differential evolution algorithm, which can directly solve the original interval optimization problem rather than transforming it to a deterministic optimization problem first.
Chao Jiang, Xu Han, Huichao Xie
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