We present a line search multigridmethod for solving discretized versions of general unconstrained infinite-dimensional optimization problems. At each iteration on each level, the algorithm computes either a "di...
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We present a line search multigridmethod for solving discretized versions of general unconstrained infinite-dimensional optimization problems. At each iteration on each level, the algorithm computes either a "direct search" direction on the current level or a "recursive search" direction from coarser level models. Introducing a new condition that must be satisfied by a backtracking line search procedure, the "recursive search" direction is guaranteed to be a descent direction. Global convergence is proved under fairly minimal requirements on the minimization method used at all grid levels. Using a limited memory BFGS quasi-Newton method to produce the "direct search" direction, preliminary numerical experiments show that our line search multigrid approach is promising.
A new two-level subspace method is proposed for solving the general unconstrained minimization formulations discretized from infinite-dimensional optimization problems. At each iteration, the algorithm executes either...
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A new two-level subspace method is proposed for solving the general unconstrained minimization formulations discretized from infinite-dimensional optimization problems. At each iteration, the algorithm executes either a direct step on the current level or a coarse subspace correction step. In the coarse subspace correction step, we augment the traditional coarse grid space by a two-dimensional subspace spanned by the coordinate direction and the gradient direction at the current point. Global convergence is proved and convergence rate is studied under some mild conditions on the discretized functions. Preliminary numerical experiments on a few variational problems show that our two-level subspace method is promising.
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