Nonlinear Interval Optimization for Uncertain Problems

Contents
Abbreviations
1 Introduction
	1.1 The Research Significance of Uncertain Optimization
	1.2 Stochastic Programming and Fuzzy Programming
		1.2.1 Stochastic Programming
		1.2.2 Fuzzy Programming
		1.2.3 Troubles and Difficulties in Stochastic Programming and Fuzzy Programming
	1.3 Uncertain Optimization Based on Non-probabilistic Modeling
		1.3.1 Convex Model Optimization
		1.3.2 Interval Optimization
	1.4 Current Problems in Interval Optimization
	1.5 The Research Target and Framework of This Book
	References
2 The Basic Principles of Interval Analysis
	2.1 The Origin of Interval Number
	2.2 The Basic Conceptions of Interval Mathematics
	2.3 The Basic Arithmetic Operations of Interval Number
	2.4 The Overestimation Problem in Interval Arithmetic
	2.5 Summary
	References
3 Mathematical Transformation Models of Nonlinear Interval Optimization
	3.1 The Description of a General Nonlinear Interval Optimization Problem
	3.2 Possibility Degree of Interval Number and Transformation of Uncertain Constraints
		3.2.1 An Improved Possibility Degree of Interval Number
		3.2.2 Transformation of Uncertain Constraints Based on Possibility Degree of Interval Number
	3.3 The Mathematic Transformation Model Based on Order Relation of Interval Number
		3.3.1 Order Relation of Interval Number and Transformation of Uncertain Objective Function
		3.3.2 The Transformed Deterministic Optimization
	3.4 The Mathematic Transformation Model Based on Possibility Degree of Interval Number
	3.5 A Two-Layer Optimization Algorithm Based on IP-GA
		3.5.1 A Brief Introduction of IP-GA
		3.5.2 Procedure of the Algorithm
	3.6 Numerical Example and Discussions
		3.6.1 By Using the Mathematic Transformation Model Based on Order Relation of Interval Number
		3.6.2 By Using the Mathematic Transformation Model Based on Possibility Degree of Interval Number
	3.7 Summary
	References
4 Interval Optimization Based on Hybrid Optimization Algorithm
	4.1 The Nonlinear Interval Optimization with Uniformly Expressed Constraints
	4.2 The ANN Model
	4.3 Construction of the Hybrid Optimization Algorithms
		4.3.1 The Hybrid Optimization Algorithm with Multiple Networks
		4.3.2 The Hybrid Optimization Algorithm with a Single Network
	4.4 Engineering Applications
		4.4.1 The Variable Binder Force Optimization in U-Shaped Forming
		4.4.2 The Locator Optimization in Welding Fixture
	4.5 Summary
	References
5 Interval Optimization Based on Interval Structural Analysis
	5.1 Interval Set Theory and Interval Extension
	5.2 The Interval Structural Analysis Method
		5.2.1 Interval Structural Analysis for Small Uncertainties
		5.2.2 Interval Structural Analysis for Large Uncertainties
		5.2.3 Numerical Example and Discussions
	5.3 An Efficient Interval Optimization Method
		5.3.1 Algorithm Description
		5.3.2 Engineering Applications
	5.4 Summary
	References
6 Interval Optimization Based on Sequential Linear Programming
	6.1 Formulation of the Algorithm
		6.1.1 Solution of the Linear Interval Optimization Problems
		6.1.2 Iteration Mechanism
		6.1.3 Calculation of the Intervals of the Actual Objective Function and Constraints in Each Iteration
	6.2 Testing of the Proposed Method
		6.2.1 Test Function 1
		6.2.2 Test Function 2
	6.3 Discussions on Convergence of the Proposed Method
	6.4 Application to the Design of a Vehicle Occupant Restraint System
	6.5 Summary
	References
7 Interval Optimization Based on Approximation Models
	7.1 Nonlinear Interval Optimization Based on the Approximation Model Management Strategy
		7.1.1 Quadratic Polynomial Response Surface
		7.1.2 Design of Experiment Method
		7.1.3 The Method by Using the Transformation Model Based on Order Relation of Interval Number
		7.1.4 The Method by Using the Transformation Model Based on Possibility Degree of Interval Number
		7.1.5 Test Functions
		7.1.6 Discussions on the Convergence
		7.1.7 Engineering Applications
	7.2 Nonlinear Interval Optimization Based on the Local-Densifying Approximation Technique
		7.2.1 Radial Basis Function
		7.2.2 Algorithm Flow
		7.2.3 Test Functions
		7.2.4 Application to the Crashworthiness Design of a Thin-Walled Beam of Vehicle Body
	7.3 Summary
	References
8 Interval Multidisciplinary Design Optimization
	8.1 An Interval MDO Model
	8.2 Decoupling the Multidisciplinary Analysis
	8.3 Transformation of the Interval Optimization Problem
	8.4 Numerical Example and Engineering Application
		8.4.1 Numerical Example
		8.4.2 Application to the Aerial Camera Design
	8.5 Summary
	References
9 A New Type of Possibility Degree of Interval Number and Its Application in Interval Optimization
	9.1 Three Existing Possibility Degree Models of Interval Number and Their Disadvantages
	9.2 The Reliability-Based Possibility Degree of Interval Number
	9.3 Interval Optimization Based on RPDI
		9.3.1 Linear Interval Optimization
		9.3.2 Nonlinear Interval Optimization
	9.4 Numerical Example and Engineering Applications
		9.4.1 Numerical Example
		9.4.2 Application to a 10-bar Truss
		9.4.3 Application to the Design of an Automobile Frame
	9.5 Summary
	References
10 Interval Optimization Considering the Correlation of Parameters
	10.1 Multidimensional Parallelepiped Interval Model
		10.1.1 Two-Dimensional Problem
		10.1.2 Multidimensional Problem
		10.1.3 Construction of the Uncertainty Domain
	10.2 Interval Optimization Based on the Multidimensional Parallelepiped Interval Model
		10.2.1 Affine Coordinate Transformation
		10.2.2 Conversion to a Deterministic Optimization
	10.3 Numerical Example and Engineering Applications
		10.3.1 Numerical Example
		10.3.2 Application to a 25-bar Truss
		10.3.3 Application to the Crashworthiness Design of Vehicle Side Impact
	10.4 Summary
	References
11 Interval Multi-objective Optimization
	11.1 An Interval Multi-objective Optimization Model
	11.2 Conversion to a Deterministic Multi-objective Optimization
	11.3 Algorithm Flow
	11.4 Numerical Example and Engineering Application
		11.4.1 Numerical Example
		11.4.2 Application to the Design of an Automobile Frame
	11.5 Summary
	References
12 Interval Optimization Considering Tolerance Design
	12.1 An Interval Optimization Model Considering Tolerance Design
	12.2 Conversion to a Deterministic Optimization
	12.3 Numerical Example and Engineering Applications
		12.3.1 Numerical Example
		12.3.2 Application to a Cantilever Beam
		12.3.3 Application to the Crashworthiness Design of Vehicle Side Impact
	12.4 Summary
	References
13 Interval Differential Evolution Algorithm
	13.1 Fundamentals of the Differential Evolution Algorithm
		13.1.1 Initial Population Generation Strategy
		13.1.2 Mutation Strategy
		13.1.3 Crossover Strategy
		13.1.4 Selection Strategy
	13.2 Formulation of the Interval Differential Evolution Algorithm
		13.2.1 Satisfaction Value of Interval Possibility Degree and Treatment of Uncertain Constraints
		13.2.2 Selection Strategy Based on an Interval Preferential Rule
		13.2.3 Algorithm Flow
	13.3 Numerical Examples and Engineering Application
		13.3.1 Numerical Examples
		13.3.2 Application to the Design of Augmented Reality Glasses
	13.4 Summary
	Appendix: Numerical Examples
	References