A convex approach to a class of minimum norm problems

2007 ◽  
pp. 359-372 ◽  
Author(s):  
Graziano Chesi ◽  
Alberto Tesi ◽  
Antonio Vicino ◽  
Roberto Genesio
Keyword(s):  
Minimum Norm ◽  
Minimum Norm Problems

Solving minimum norm problems using penalty functions and the gradient method

Automatica ◽  
1995 ◽  
Vol 31 (1) ◽  
pp. 115-124 ◽  
Author(s):  
Stefen Hui ◽  
Walter E. Lillo ◽  
Stanislaw H. Żak
Keyword(s):  
Gradient Method ◽  
Penalty Functions ◽  
Minimum Norm ◽  
Minimum Norm Problems

scholarly journals Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

2020 ◽  
Vol 10 (1) ◽  
pp. 450-476
Author(s):  
Radu Ioan Boţ ◽  
Sorin-Mihai Grad ◽  
Dennis Meier ◽  
Mathias Staudigl
Keyword(s):  
Dynamical Systems ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Minimum Norm ◽  
Monotone Inclusion ◽  
Inclusion Problems ◽  
Monotone Inclusion Problem ◽  
Maximally Monotone Operators

Abstract In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

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