Nonlinear Multiobjective Optimization

by Hillermeier, Claus

• ISBN13: 9783764364984
• ISBN10: 376436498X
• Format: Hardcover
• Copyright: 2001-02-01
• Publisher: Birkhauser

Summary

Arguably, many industrial optimization problems are of the multiobjective type. The present work, after providing a survey of the state of the art in multiobjective optimization, gives new insight into this important mathematical field by consequently taking up the viewpoint of differential geometry. This approach, unprecedented in the literature, very naturally results in a generalized homotopy method for multiobjective optimization which is theoretically well-founded and numerically efficient. The power of the new method is demonstrated by solving two real-life problems of industrial optimization.The book presents recent results obtained by the author and is aimed at mathematicians, scientists, students and practitioners interested in optimization and numerical homotopy methods.

Table of Contents

Introduction	3	(6)
Vector Optimization in Industrial Applications	9	(6)
The Design of a Combined-Cycle Power Plant	10	(2)
The Optimal Operating Point of a Recovery-Boiler	12	(3)
Principles and Methods of Vector Optimization	15	(30)
The Concept of Pareto Optimality	15	(4)
Survey of Methods	19	(11)
A New Stochastic Method for Unconstrained Vector Optimization	30	(15)
A Curve of Dominated Points	31	(6)
Notions from Probability Theory	37	(2)
A Special Stochastic Differential Equation	39	(3)
A Stochastic Algorithm for Vector Optimization	42	(3)
The Connection with Scalar-Valued Optimization	45	(20)
The Karush-Kuhn-Tucker (KKT) Condition for Pareto Optimality	45	(2)
Differential-Topological Notations	47	(6)
The Geometrical Meaning of the Weight Vector	53	(6)
Classification of Efficient Points	59	(6)
The Manifold of Stationary Points	65	(22)
Karush Kuhn Tucker Points as a Differentiable Manifold M	66	(2)
Criteria for the Rank Condition	68	(11)
A Necessary and Sufficient Criterion	68	(3)
Interpretation in View of Optimization	71	(4)
Variability of the Weight Vector	75	(4)
A Special Class of Local Charts	79	(8)
Homotopy Strategies	87	(22)
Method I: Local Exploration of M	88	(11)
Method Principle	88	(1)
Comparison with the Classical Homotopy Method	89	(4)
Homogeneous Discretization of the Efficient Set	93	(2)
Numerical Algorithm	95	(4)
Method II: Purposeful Change of the Weights	99	(10)
Significance of the Weight Vector for the User	99	(2)
Principle of the Procedure	101	(3)
Numerical Algorithm	104	(5)
Numerical Results	109	(20)
Example 1 (academic)	109	(6)
Example 2: Design of a Combined-Cycle Power Plant	115	(8)
Example 3: The Optimal Operating Point of a Recovery-Boiler	123	(6)
Bibliography	129	(4)
Index	133