A New Nonsmooth Trust Region Algorithm for Locally Lipschitz Unconstrained Optimization Problems

In this paper, a new nonsmooth trust region algorithm is proposed for solving unconstrained minimization problems with locally Lipschitz objective functions. At first, by using an approximation of the steepest descent direction, a local model is presented for locally Lipschitz functions. More precis...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2015-03, Vol.164 (3), p.733-754
Main Authors: Akbari, Z., Yousefpour, R., Reza Peyghami, M.
Format: Article
Language:English
Subjects:
Algorithms
Applications of Mathematics
Approximation
Calculus of Variations and Optimal Control
Optimization
Descent
Engineering
Linear programming
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matlab
Methods
Operations research
Operations Research/Decision Theory
Optimization
Studies
Theory of Computation
Vectors (mathematics)
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Summary:In this paper, a new nonsmooth trust region algorithm is proposed for solving unconstrained minimization problems with locally Lipschitz objective functions. At first, by using an approximation of the steepest descent direction, a local model is presented for locally Lipschitz functions. More precisely, in the quadratic model of classical trust region methods, the gradient vector is replaced by an approximation of the steepest descent direction. We then apply one of the efficient approaches of classical trust region methods in order to solve the obtained model. Using the BFGS updating formula for the Hessian approximation of the model, we show that the proposed algorithm is convergent under some mild and standard conditions on the objective function. Finally, the presented algorithm is implemented in the MATLAB environment and applied on some nonsmooth test problems.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-014-0534-6