Abstract and Applied Analysis
Volume 2015 (2015), Article ID 429595, 2 pages
http://dx.doi.org/10.1155/2015/429595

Editorial

Nonlinear Analysis: Optimization Methods, Convergence Theory, and Applications

Gonglin Yuan,¹ Gaohang Yu,² Neculai Andrei,³ Yunhai Xiao,⁴ and Li Zhang⁵

¹College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
²School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China
³Research Institute for Informatics (ICI), 8-10 Averescu Avenue, 71316 Bucharest 1, Romania
⁴College of Mathematics and Information Science, Henan University, Kaifeng 475000, China
⁵Department of Mathematics, Changsha University of Science and Technology, Changsha 410004, China

Received 1 March 2015; Accepted 1 March 2015

Copyright © 2015 Gonglin Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nonlinear analysis has been used in many practical application fields, such as nonlinear fitting, economics, optimization, convergence, engineering, hydrodynamics, parameter estimating, function approximating, and elasticity. There are many achievements on nonlinear analysis that have been obtained by authors. However, there still exist lots of challenging problems, such as the large-scale problems, fast algorithm, and convergence, since the complex of the nonlinear object function on its variables cannot be obviously determined in many cases. So the research and application space of nonlinear analysis are broad.

The issue invites investigators to contribute original research articles as well as review articles that will help in understanding the important new developments in nonlinear analysis and its applications with a particular emphasis on the following potential topics. There exist many special topics including the nonlinear analysis: optimization, variation analysis, economical models, fixed point theory, numerical methods, convergence, nonlinear equations, semidefinite programming, polynomial optimization, tensor computation, image processing, and so forth.

The research papers are welcome with new ideas or good numerical experiments. (1) New methods for nonlinear analysis are encouraged, such as the new formulas on conjugate gradient methods, quasi-Newton methods, limited memory quasi-Newton method, trust region methods, and SQP methods; convergence results of algorithms are established which is needed. (2) Numerical experiments should be done to improve the theory idea: for unconstrained optimization problems, the CUTEr problems should be tested [1, 2] in Table 1. For nonlinear equations problems, there are many problems [3–7] that are listed in Table 2.

Table 1

Problems names
Character

ARGLINA, ARGLINB, ARGLINC, BDQRTIC, BROWNAL, BROYDN7D, BRYBND CHAINWOO, CHNROSNB, COSINE, CRAGGLVY, CURLY10, CURLY20, DIXMAANA, DIXMAANB, DIXMAANC, DIXMAAND, DIXMAANE, DIXMAANF, DIXMAANG, DIXMAANH DIXMAANI, DIXMAANJ, DIXMAANL, DIXON3DQ, DQDRTIC, DQRTIC, EDENSCH EG2, ENGVAL1, ERRINROS, EXTROSNB, FLETCBV2, FLETCHCR, FREUROTH GENHUMPS, GENROSE, INDEF, LIARWHD, MANCINO, MSQRTALS, MSQRTBLS NONCVXU2, NONDIA, NONDQUAR, PENALTY1, PENALTY2, POWELLSG POWER, QUARTC, SCHMVETT, SENSORS, SINQUAD, SPARSINE, SPARSQUR SPMSRTLS, SROSENBR, TESTQUAD, TOINTGSS, TQUARTIC, TRIDIA VARDIM, VAREIGVL, and WOODS
Academic

DECONVU, FMINSRF2, FMINSURF, MOREBV, TOINTGOR, and TOINTQOR
Modeling

Table 2

Functions names
Optimization value

Exponential function 1, exponential function 2, trigonometric function, singular function, logarithmic function, Broyden tridiagonal function, trigexp function, strictly convex function 1, linear function-full rank, penalty function, variable dimensioned function, tridiagonal system, five-diagonal system, extended Freudenstein and Roth function, discrete boundary value problem, Troesch problem, and so forth
0

We hope that readers of this special issue will find not only convergence results and updated reviews on the common nonlinear analysis, but also important open problems to be resolved such as new formulas in optimization methods, new algorithms for variation analysis and new models for economic problems. Moreover, large-scale problems in nonlinear equations, semidefinite programming, and image processing are tested to turn out the performance of the new methods.

Gonglin Yuan
Gaohang Yu
Neculai Andrei
Yunhai Xiao
Li Zhang

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