Trust Region Newton Method With Two Dimensional Subspace Minimization
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Date
2018-05-03
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Addis Ababa University
Abstract
The trust region method, minimization of a real valued function f by using its quadratic
model subject to Euclidean norm trust region constraint, occurs in many trust region
algorithms. In most cases, it is inexpensive and even unnecessary to _nd an exact solution
of a trust region problem unless the number of variables is relatively small. In order to
alleviate this di_culty, in this thesis, we emphasized on a trust region method that takes
Hessian of the model to be the true Hessian of a twice continuously di_erentiable real
valued objective function f on Rn , at a given current iterate point xk using the so called
two dimensional subspace minimization strategy. As the name indicates, the two
dimensional subspace minimization method de_nes an approximate minimizer s to lie
in a subspace Sk of Rn spanned by two reasonably chosen directions. We constructed
such subspace using Lanczos iterative method. This subspace method _rst reduces the
n-dimensional trust region problem to 2-dimensional constrained problem, and then to
_nding roots of a fourth degree polynomial with the help of Lagrangian approach for
constrained optimization problems. We compared the performance of the two dimensional
subspace minimization method with that of an alternative trust region method, namely,
the dogleg method, and the other common methods such as the steepest descent and
Newton's method using MATLAB and we obtain that the two dimensional subspace
minimization method is more e_cient.
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Keywords
Optimization Problems, Lagrangian Method For Equality, Lagrangian Method For Inequality