Sharif Digital Repository

We explain a filter-trust-region algorithm for solving nonlinear optimization problems with simple bounds recently proposed by Sainvitu and Toint. The algorithm is shown to be globally convergent to at least one first-order critical point. We implement the algorithm and test the program on various problems. The results show the effectiveness of the algorithm  

Worst-case complexity guarantees for non-convex optimization algorithms is a topic that have received increasing attention. Here , we review trust-region Newton methods recently proposed in the literature . After a slight modification of the main model , two methods are proposed : one of them is based on the exact solution of the sub-problem , and the other is based on the inexact solution of the sub-problem , such as ``trust-region Newton-conjugate gradient " method with the complexity bounds corresponding to the best known bounds for this class of algorithms . We implement the proposed algorithms and test the programs in the Python software environment  

Considering the special importance of network flow problems in human life, as well as the complexity of solving these problems in very large scales, there are numerous methods to solve them and the interior point methods are the most important approaches among them. In a number of methods, a preconditioned conjugate gradient solver has been applied for the solution of the Karush-Kuhn-Tucker (KKT) system, in each interior point iteration; therefore, the selection of an appropriate preconditioner is a special issue. In spite of presenting different preconditioners in recent years, discussion and implementation of a particular class of triangulated graph-based preconditioners is our main... 

Design, analysis and practical implementation of the filter trust-region algorithms are investigated. First, we introduce two filter trust-region algorithms for solving the unconstrained optimization problem. These algorithms belong to two different class of optimization algorithms: (1) The monotone class, and (2) The non-monotone class. We prove the global convergence of the sequence of the iterates generated by the new algorithms to the first and second order critical points. Then, we propose a filter trust-region algorithm for solving bound constrained optimization problems and show that the algorithm converges to a first order critical point. Moreover, we address some well known... 

Nowadays, solving large-scale unconstrained optimization problems has wide applications in data science and machine learning. Therefore, the development and analysis of efficient algorithms for solving unconstrained optimization problems is of great interest. Line search and trust region are two general frameworks for guaranteeing the convergence of algorithms for solving unconstrained optimization problems. Conjugate gradient (CG) methods and the conjugate residual (CR) balance by Hestenes and Stiefel, have been presented for solving linear systems with symmetric and positive definite coefficient matrices. The basic feature of CR, that is, residual minimization, is important and can be used... 

Line search-based methods are known as a category of the most efficient iterative algo- rithms for solving unconstrained optimization problems. Among them, the conjugate gradient method is of particular importance in solving large-scale contemporary world problems due to its simplicity of structure, low memory requirement and strong convergence characteristics. In spite of the desirable numerical behavior of the conjugate gradient method, this method generally lacks the descent property even for uniformly convex objective functions. To overcome this defect, some effective modifications have been presented in the literature. Amidst, the three-term extension attracted the attention of many...