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Multigrid methods in convex optimization. (English) Zbl 0593.65041

GMD-Stud. 110, 25-37 (1986).
One considers the building and the study of multigrid algorithms applied to convex minimization problems with or without constraints, issued from the discretisation of boundary problems. In the no constraint case, a class of new algorithms is proposed where the smoothing steps are descent iterations and the coarse grid correction is inspired by W. Hackbusch [Lect. Notes Math. 960, 177-219 (1982; Zbl 0505.65036)].
These methods are shown to be globally convergent and the convergence factor is evaluated. The algorithms are extended to take into account inequality constraints. In the particular case for which the admissible set is a positive cone (or a translated cone) a family of multigrid algorithms is building where the coarse grid correction is very close from those proposed by Hackbusch, J. Mandel [Appl. Math. Optimization 11, 77-95 (1984; Zbl 0539.65046)] and A. Brandt [GMD- Stud. 85, 183 p. (1984; Zbl 0581.76033)]. Finally, we propose some possible extensions of these algorithms associated to penalty or Lagrangian techniques.

MSC:

65K05 Numerical mathematical programming methods
90C25 Convex programming
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs