Integral Global Optimization Theory, Implementation and Applications [Paperback]

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 Theorl³"