Trilevel and multilevel optimization using monotone operator theory

We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a trilevel optimization problem, where the objective of the two lo...

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Published inMathematical methods of operations research (Heidelberg, Germany) Vol. 99; no. 1-2; pp. 77 - 114
Main Authors Shafiei, Allahkaram, Kungurtsev, Vyacheslav, Marecek, Jakub
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024
Springer Nature B.V
Springer Verlag
Subjects
Algorithms
Business and Management
Calculus of Variations and Optimal Control; Optimization
Convergence
Convexity
First order algorithms
Fixed points (mathematics)
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Original Article
Tri-level optimization
Non-expansive mappings
Bi-level optimization
Variational inequality
Multi-level minimization
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ISSN1432-2994
1432-5217
DOI10.1007/s00186-024-00852-5

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Summary:We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a trilevel optimization problem, where the objective of the two lower layers consists of a sum of a smooth and a non-smooth term. Based on fixed-point theory and related arguments, we present a natural first-order algorithm and analyze its convergence and rates of convergence in several regimes of parameters.
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ISSN:1432-2994
1432-5217
DOI:10.1007/s00186-024-00852-5