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Recursive trust-region methods for multiscale nonlinear optimization. (English) Zbl 1163.90024
Summary: A class of trust-region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a means of speeding up the computation of the step. This use is recursive, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations. A simple algorithm of the class is then described and its numerical performance is shown to be numerically promising. This observation then motivates a proof of global convergence to first-order stationary points on the fine grid that is valid for all algorithms in the class.
MSC:
90C30Nonlinear programming
65K05Mathematical programming (numerical methods)
90C26Nonconvex programming, global optimization
90C06Large-scale problems (mathematical programming)