Lagrangian Penalization Scheme with Parallel Forward–Backward Splitting

We propose a new iterative algorithm for the numerical approximation of the solutions to convex optimization problems and constrained variational inequalities, especially when the functions and operators involved have a separable structure on a product space, and exhibit some dissymmetry in terms of...

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Veröffentlicht in:Journal of optimization theory and applications Jg. 177; H. 2; S. 413 - 447
Hauptverfasser: Molinari, Cesare, Peypouquet, Juan
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.05.2018
Springer Nature B.V
Schlagworte:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Computational fluid dynamics
Computational geometry
Convergence
Convex analysis
Convexity
Engineering
Iterative algorithms
Iterative methods
Mathematical models
Mathematics
Mathematics and Statistics
Newtonian fluids
Non Newtonian fluids
Operations Research/Decision Theory
Optimization
Theory of Computation
Forward–backward
49M37
Penalization
Lagrange multipliers
90C25
Convex programming
ISSN:0022-3239, 1573-2878
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Zusammenfassung:We propose a new iterative algorithm for the numerical approximation of the solutions to convex optimization problems and constrained variational inequalities, especially when the functions and operators involved have a separable structure on a product space, and exhibit some dissymmetry in terms of their component-wise regularity. Our method combines Lagrangian techniques and a penalization scheme with bounded parameters, with parallel forward–backward iterations. Conveniently combined, these techniques allow us to take advantage of the particular structure of the problem. We prove the weak convergence of the sequence generated by this scheme, along with worst-case convergence rates in the convex optimization setting, and for the strongly non-degenerate monotone operator case. Implementation issues related to the penalization of the constraint set are discussed, as well as applications in image recovery and non-Newtonian fluids modeling. A numerical illustration is also given, in order to prove the performance of the algorithm.
Bibliographie:ObjectType-Article-1
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-018-1265-x