Integral Global Optimization: Theory, Implementation and Applications (Lecture Notes in Economics and Mathematical Systems, 298)

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.