  This is an overview paper written in style of research proposal. In recent
years we introduced a general framework for large-scale unconstrained
optimization -- Sequential Subspace Optimization (SESOP) and demonstrated its
usefulness for sparsity-based signal/image denoising, deconvolution,
compressive sensing, computed tomography, diffraction imaging, support vector
machines. We explored its combination with Parallel Coordinate Descent and
Separable Surrogate Function methods, obtaining state of the art results in
above-mentioned areas. There are several methods, that are faster than plain
SESOP under specific conditions: Trust region Newton method - for problems with
easily invertible Hessian matrix; Truncated Newton method - when fast
multiplication by Hessian is available; Stochastic optimization methods - for
problems with large stochastic-type data; Multigrid methods - for problems with
nested multilevel structure. Each of these methods can be further improved by
merge with SESOP. One can also accelerate Augmented Lagrangian method for
constrained optimization problems and Alternating Direction Method of
Multipliers for problems with separable objective function and non-separable
constraints.
