Special Issue Dedicated To Selected Surveys In Nonlinear Programming

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Inexact Restoration Approach For Minimization With Inexact Evaluation Of The Objective Function

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)A new method is introduced for minimizing a function that can be computed only inexactly, with different levels of accuracy. The challenge is to evaluate the (potentially very expensive) objective function with low accuracy as far as this does not interfere with the goal of getting high accuracy minimization at the end. For achieving this goal the problem is reformulated in terms of constrained optimization and handled with an Inexact Restoration technique. Convergence is proved and numerical experiments motivated by Electronic Structure Calculations are presented, which indicate that the new method overcomes current approaches for solving large-scale problems.8517751791Serbian Ministry of Education, Science, and Technological Development [174030]FAPESP (Fundacao de Amparo a Pesquisa do Estado de Sao Paulo under projects CEPID-Cemeai on Industrial Mathematics) [2013/07375-0, PT 2006/53768-0]CNPq [300933-2009-6, 400926-2013-0]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Inexact-Newton methods for semismooth systems of equations with block-angular structure

Systems of equations with block-angular structure have applications in evolution problems coming from physics, engineering and economy. Many times, these systems are time-stage formulations of mathematical models that consist of mathematical programming problems, complementarity, or other equilibrium problems, giving rise to nonlinear and nonsmooth equations. The final Versions of these dynamic models are nonsmooth systems with block-angular structure. If the number of state variables and equations is large, it is sensible to adopt an inexact-Newton strategy for solving this type of systems. In this paper we define two inexact-Newton algorithms for semismooth block-angular systems and we prove local and superlinear convergence. (C) 1999 Elsevier Science B.V. All rights reserved.103223924

A globally convergent Inexact-Newton method for solving reducible nonlinear systems of equations

A nonlinear system of equations is said to be reducible if it can be divided into m blocks (m > 1) in such a way that the i-th block depends only on the first i blocks of unknowns. Different ways of handling the different blocks with the aim of solving the system have been proposed in the literature. When the dimension of the blocks is very large, it can be difficult to solve the linear Newtonian equations associated to them using direct solvers based on factorizations. In this case, the idea of using iterative linear solvers to deal with the blocks of the system separately is appealing. In this paper, a local convergence theory that justifies this procedure is presented. The theory also explains the behavior of a Block-Newton method under different types of perturbations. Moreover, a globally convergent modification of the basic Block Inexact-Newton algorithm is introduced so that, under suitable assumptions, convergence can be ensured, independently of the initial point considered.131113

Globally convergent inexact quasi-Newton methods for solving nonlinear systems

Large scale nonlinear systems of equations can be solved by means of inexact quasi-Newton methods. A global convergence theory is introduced that guarantees that, under reasonable assumptions, the algorithmic sequence converges to a solution of the problem. Under additional standard assumptions, superlinear convergence is preserved.324173124926

Globally Convergent Inexact-newton Method For Solving Reducible Nonlinear Systems Of Equations

A nonlinear system of equations is said to be reducible if it can be divided into m blocks (m>1) in such a way that the i-th block depends only on the first i blocks of unknowns. Different ways of handling the different blocks with the aim of solving the system have been proposed in the literature. When the dimension of the blocks is very large, it can be difficult to solve the linear Newtonian equations associated to them using direct solvers based on factorizations. In this case, the idea of using iterative linear solvers to deal with the blocks of the system separately is appealing. In this paper, a local convergence theory that justifies this procedure is presented. The theory also explains the behavior of a Block-Newton method under different types of perturbations. Moreover, a globally convergent modification of the basic Block Inexact-Newton algorithm is introduced so that, under suitable assumptions, convergence can be ensured, independently of the initial point considered.1311134Crandall, M.G., (1978) Nonlinear Evolution Equations, , Academic PressDean, E.J., (1985) A Model Trust Region Modification of Inexact Newton's Method for Nonlinear Two Point Boundary Value Problems, , TR No 85-6, Department of Mathematical Sciences, Rice University, Houston, TexasDembo, R.S., Eisenstat, S.C., Steihaug, T., Inexact Newton methods (1982) SIAM Journal on Numerical Analysis, 19, pp. 400-408Dennis, J.E., Martínez, J.M., Zhang, X., Triangular decomposition methods for solving reducible nonlinear systems of equations (1994) SIAM Journal on Optimization, 4, pp. 358-382Dennis, J.E., Schnabel, R.B., (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations, , Prentice-Hall, Englewood CliffsEisenstat, S.C., Walker, H.F., Globally Convergent Inexact Newton Methods (1994) SIAM Journal on Optimization, 4, pp. 393-422Eisenstat, S.C., Walker, H.F., Choosing the forcing terms in an Inexact Newton method (1996) SIAM Journal on Scientific Computing, 17, pp. 16-32Feehery, W.F., Tolsma, J.E., Barton, P.I., Efficient sensitivity analysis of large - Scale differential - Algebraic systems (1997) Applied Numerical Mathematics, 25, pp. 41-54Lužanin, Z., Krejić, N., Herceg, D., Parameter selection for Inexact Newton method (1997) Nonlinear Analysis. Theory, Methods and Applications, 30, pp. 17-24Maly, T., Petzold, L.R., Numerical methods and software for sensitivity analysis of differential - Algebraic systems (1996) Applied Numerical Mathematics, 20, pp. 57-79Meis, T., Marcowitz, U., (1981) Numerical Solution of Partial Difference Equations, , Springer-Verlag, New York, Heidelberg, BerlinMyint, T.U., Debnath, L., (1987) Partial Differential Equations for Scientists and Engineers, , ElsevierOrtega, J.M., Rheinboldt, W.C., (1970) Iterative Solution of Nonlinear Equations in Several Variables, , Academic Press, N

An interior-point method for solving box-constrained underdetermined nonlinear systems

A method introduced recently by Bellavia, Macconi and Morini for solving square nonlinear systems with bounds is modified and extended to cope with the underdetermined case. The algorithm produces a sequence of interior iterates and is based on globalization techniques due to Coleman and Li. Global and local convergence results are proved and numerical experiments are presented. (C) 2004 Elsevier B.V. All rights reserved.1771678

Solution of bounded nonlinear systems of equations using homotopies with inexact restoration

Nonlinear systems of equations often represent mathematical models of chemical production processes and other engineering problems. Homotopic techniques (in particular, the bounded homotopies introduced by Paloschi) are used for enhancing convergence to solutions, especially when a good initial estimate is not available. In this paper, the homotopy curve is considered as the feasible set of a mathematical programming problem, where the objective is to find the optimal value of the homotopic parameter. Inexact restoration techniques can then be used to generate approximations in a neighborhood of the homotopy, the size of which is theoretically justified. Numerical examples are given.80221122

Iteration and evaluation complexity for the minimization of functions whose computation is intrinsically inexact

In many cases in which one wishes to minimize a complicated or expensive function, it is convenient to employ cheap approximations, at least when the current approximation to the solution is poor. Adequate strategies for deciding the accuracy desired at each stage of optimization are crucial for the global convergence and overall efficiency of the process. A recently introduced procedure [E. G. Birgin, N. Krejic, and J. M. Martinez, Math. Comp. 87 (2018), 1307-1326, 2018] based on Inexact Restoration is revisited, modified, and analyzed from the point of view of worst-case evaluation complexity in this work89321253278CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP309517/2014-1; 303750/2014-62013/03447-6; 2013/05475-7; 2013/07375-0; 2014/18711-3; 2016/01860-1This work was partially supported by theBrazilian agencies FAPESP (grants 2013/03447-6,2013/05475-7, 2013/07375-0, 2014/18711-3, and 2016/01860-1) and CNPq (grants 309517/2014-1 and 303750/2014-6) and by the Serbian Ministry of Education, Science, and TechnologicalDevelopment (grant 174030

Incomplete decomposition algorithms for discrete dynamic nonlinear models

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