Integral Global Optimization

This book treats the subject of global optimization with minimal restrictions on the behavior on the objective functions. In particular, optimal conditions were developed for a class of noncontinuous functions characterized by their having level sets that are robust. The integration-based approach contrasts with existing approaches which require some degree of convexity or differentiability of the objective function. Some computational results on a personal computer are presented.

I Preliminary.- §1 Introduction.- 1.1 Statement of the Problem.- 1.2 Examples.- 1.3 Outline.- §2 An Appropriate Concept of Measure.- 2.1 Q-Measure.- 2.2 Lemmata.- II Integral Characterizations of Global Optimality.- §1 Mean Value Conditions.- 1.1 Mean Value over Level Sets.- 1.2 A Limit-Based Definition.- 1.3 Mean Value Conditions.- §2 Variance and Higher Moment Conditions.- 2.1 Variance over Level Sets.- 2.2 A Limit-Based Definition.- 2.3 Variance Conditions.- 2.4 Higher Moments.- 2.5 Higher Moment Conditions.- §3 The Constrained Cases.- 3.1 Rejection Conditions.- 3.2 Reduction Conditions.- 3.3 Linear Equality Constraints.- §4 Penalty Global Optimality Conditions.- 4.1 Penalty Mean Value.- 4.2 Penalty Mean Value Conditions.- 4.3 Penalty Variance and Higher Moment Conditions.- §5 Convex Programming.- 5.1 Optimality Conditions for Differentiable Convex Functions.- 5.2 Optimality Lemmas.- 5.3 Optimality Conditions for Convex Minimization.- 5.4 Generalized Gradient.- §6 Optimality Conditions for Differentiable Functions.- 6.1 General Discussion: the Unconstrained Case.- 6.2 The Inequality Constrained Case in ?n.- 6.3 Equality and Inequality Constrained Cases in ?n.- §7 Integer and Mixed Programming.- 7.4 Integer Minimization Problems.- 7.5 Optimality Conditions.- 7.6 Mixed Minimization Problems.- §8 Optimality Conditions for a Class of Discontinuous Functions.- 8.3 Robust Sets.- 8.4 The Structure of a Robust Set on the Real Line ?.- 8.5 Robust Continuity.- 8.6 Optimality Conditions.- III Theoretical Algorithms and Techniques.- §1 The Mean Value-Level Set (M-L) Method.- 1.1 Algorithm.- 1.2 Convergence.- 1.3 The Actual Descent Property.- 1.4 The Influence of Errors.- §2 The Rejection and Reduction Methods.- 2.1 The Rejection Method.- 2.2 The Reduction Method.- 2.3 The Reduction Method for Linear Equality Constrained Cases in ?n.- §3 Global SUMT and Discontinuous Penalty Functions.- 3.1 SUMT and the Set of Global Minima.- 3.2 Discontinuous Penalty Functions.- §4 The Nonsequential Penalty Method.- 4.1 Construction.- 4.2 Convergence.- §5 The Technique of Adaptive Change of Search Domain.- 5.1 A Simple Model.- 5.2 Convergence.- 5.3 Optimality Conditions of the Simple Model.- 5.4 The General Model.- §6 Stability of Global Minimization.- 6.1 Continuity of Mean Value.- 6.2 Stability of Global Minima.- §7 Lower Dimensional Approximation.- 7.1 Approximation of Global Minimum.- 7.2 Estimation of Degree of Approximation.- IV Monte Carlo Implementation.- §1 A Simple Model of Implemention.- 1.1 The Model.- 1.2 Monte Carlo Implementation.- 1.3 The Flow Chart.- §2 Statistical Analysis of the Simple Model.- 2.1 Estimators of the Search Domains.- 2.2 The Probability of Escape and the Sample Size.- 2.3 Asymtotic Estimation of the Amount of Computation.- §3 Strategies of Adaptive Change of Search Domains.- 3.1 Strategies.- 3.2 The Change of Domain Theorem.- 3.3 Reduction of the Skew Rate.- §4 Remarks on Other Models.- 4.1 Rejection and Reduction Models.- 4.2 Integer and Mixed Programming.- 4.3 The Multi-Solution Model.- §5 Numerical Tests.- V Applications.- §1 Unconstrained Problems.- 1.1 Automatic Design of Optical Thin Films.- 1.2 Optimal Design of an Equalizer Network.- §2 Applications of the Rejection Method.- 2.1 Optimal Design of Optical Phase Filters.- 2.2 Optimal Design of an Automatic Transmission Line Attenuation Compensation Network.- §3 Applications of the Reduction Method.- 3.1 Optimal Design of a Turbine Wheel.- 3.2 Nonlinear Observation and Identification.- §4 An Application of the Penalty Method.- 4.1 Weight Minimization of a Speed Reducer.- §5 An Application of Integer and Mixed Programming.- 5.1 Optimal Design of an Optical Thin Film System.