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Some variants of projection methods for large nonlinear optimization problems

Białoń P.

Journal of Telecommunications and Information Technology

2003

nr 3

43-49

Two ideas of modifying projection methods for the case of smooth nonlinear optimization are presented. Projection methods were originally successfully used in solving large- scale linear feasibility problems. The proposed instantiations of projection methods fall into two groups. One of them is a decomposition approach in which projections onto sets are realized as optimization problems which themselves involve much portions of original problem constraints. There are two subproblems: one build with linear constraints of the original problem and the second one build with original nonlinear constraints. These approaches use special accelerating cuts so that the separation of nonlinear and linear constraints can be effective and some problem sparsity preserved. The second group bases on penalty-shifting/multiplier methods and draws from the observation that unconstrained subproblems obtained there may solve very slowly due to their nonsmooth character. Thus it is proposed to solve them with modified projection methods which inherit from conjugate gradient methods a multi-dimensional subspace in one epoche.