Intro
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