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Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization / von Bennet Gebken. Paderborn, 2022
Inhalt
Introduction
Basics of multiobjective optimization
Pareto optimality
Necessary optimality conditions and the Pareto critical set
Smooth case
Nonsmooth case
Existing solution methods
Scalarization methods
Generalizations of methods from scalar optimization
Set-based methods
Evolutionary methods
Box-continuation methods for smooth problems
Exact gradients
Covering via boxes
The algorithm
Examples
Extension to constrained MOPs
Obtaining the Pareto set
Inexact gradients
A tight superset of the Pareto critical set and its structure
Efficient computation
The algorithm
Examples
Application to PDE-constrained MOPs
An efficient descent method for nonsmooth problems
Theoretical descent direction
The Goldstein epsilon-subdifferential
Efficient computation of descent directions
The algorithm
Numerical experiments
Typical behavior
Comparison to the MPB
Combination with the subdivision algorithm
Hierarchical structure of Pareto critical sets
Topological and geometrical properties of the Pareto critical set
Classifying Pareto critical points via KKT multipliers
Tangent cones and the uniqueness of KKT vectors
The boundary of the Pareto critical set
Decomposing an MOP into lower-dimensional subproblems
Examples
Extension to constrained MOPs
Equality constraints
Inequality constraints
Examples
Extension to the nonsmooth case
Inferring objective vectors from Pareto critical data
Linearity of the inverse problem and its solution via SVD
Applications
Generating MOPs with prescribed properties
Inferring objectives of stochastic MOPs
Generation of surrogate models
Open problems of our approach
Conclusion and outlook
List of symbols
Bibliography
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