Constrained composite optimization and augmented Lagrangian methods

We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling framework for a variety of applications. We study stationarity an...

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Bibliographic Details
Published in:Mathematical programming Vol. 201; no. 1-2; pp. 863 - 896
Main Authors: De Marchi, Alberto, Jia, Xiaoxi, Kanzow, Christian, Mehlitz, Patrick
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2023
Springer
Subjects:
Algorithms
Analysis
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical Analysis
Theoretical
65K05
90C30
49J53
Nonlinear optimization
Augmented Lagrangian methods
Nonsmooth optimization
Composite nonconvex optimization
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling framework for a variety of applications. We study stationarity and regularity concepts, and propose a flexible augmented Lagrangian scheme. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. It is demonstrated how the inner subproblems can be solved by off-the-shelf proximal methods, notwithstanding the possibility to adopt any solvers, insofar as they return approximate stationary points. Finally, we describe our matrix-free implementation of the proposed algorithm and test it numerically. Illustrative examples show the versatility of constrained composite programs as a modeling tool and expose difficulties arising in this vast problem class.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01922-4