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Numerical Algebra, Control & Optimization

2014 , Volume 4 , Issue 4

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Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces
B. S. Lee and Arif Rafiq
2014, 4(4): 287-293 doi: 10.3934/naco.2014.4.287 +[Abstract](138) +[PDF](283.8KB)
Abstract:
The purpose of this paper is to establish a strong convergence of an implicit iteration process to a common fixed point for a finite family of Lipschitz $\phi-$uniformly pseudocontractive mappings in real Banach spaces. The results presented here improve and extend the corresponding results in [2, 4, 6] and the consecutive remark explains in details about the facts.
Nonlinear scalarization with applications to Hölder continuity of approximate solutions
Lili Li and Chunrong Chen
2014, 4(4): 295-307 doi: 10.3934/naco.2014.4.295 +[Abstract](252) +[PDF](396.5KB)
Abstract:
In this article, firstly, some useful properties of the Gerstewitz scalarizing function are discussed, such as its globally Lipschitz property, concavity and monotonicity. Secondly, as an application of these properties, verifiable sufficient conditions for Hölder continuity of approximate solutions to parametric generalized vector equilibrium problems are established via Gerstewitz scalarizations. Moreover, some examples are provided to illustrate our main conclusions in the vector settings.
Topological properties of Henig globally efficient solutions of set-valued problems
Guolin Yu
2014, 4(4): 309-316 doi: 10.3934/naco.2014.4.309 +[Abstract](405) +[PDF](290.7KB)
Abstract:
This paper deals with the topological properties of Henig globally efficiency in vector optimization problem with set-valued mapping. The closednesst and compactness results are presented for Henig globally efficiency, Especially, under the assumption of ic-cone-convexlikeness, the connectedness of Henig globally efficient set is obtained. As the application of the results, the relevant topological properties of Benson proper efficiency are also presented.
A note on the stability of a second order finite difference scheme for space fractional diffusion equations
Wei Qu, Siu-Long Lei and Seak-Weng Vong
2014, 4(4): 317-325 doi: 10.3934/naco.2014.4.317 +[Abstract](275) +[PDF](380.6KB)
Abstract:
The unconditional stability of a second order finite difference scheme for space fractional diffusion equations is proved theoretically for a class of variable diffusion coefficients. In particular, the scheme is unconditionally stable for all one-sided problems and problems with Riesz fractional derivative. For problems with general smooth diffusion coefficients, numerical experiments show that the scheme is still stable if the space step is small enough.
Minimax problems for set-valued mappings with set optimization
Yu Zhang and Tao Chen
2014, 4(4): 327-340 doi: 10.3934/naco.2014.4.327 +[Abstract](200) +[PDF](357.3KB)
Abstract:
In this paper, we introduce a class of set-valued mappings with some set order relations, which is called uniformly same-order. For this sort of mappings, we obtain some existence results of saddle points and depict the structures of the sets of saddle points. Moreover, we obtain a minimax theorem and establish an equivalent relationship between the minimax theorem and a saddle point theorem for the scalar set-valued mappings, in which the minimization and the maximization of set-valued mappings are taken in the sense of set optimization.
Essential issues on solving optimal power flow problems using soft-computing
Kit Yan Chan, Changjun Yu, Kok Lay Teo and Sven Nordholm
2014, 4(4): 341-351 doi: 10.3934/naco.2014.4.341 +[Abstract](149) +[PDF](399.7KB)
Abstract:
Optimal power flow (OPF) problems are important optimization problems in power systems which aim to minimize the operation cost of generators so that the load demand can be met and the loadings are within the feasible operating regions of the generators. This brief paper emphasizes two essential issues related to solving the OPF problems and which are rarely addressed in recent research into power systems: 1) the necessity to validate operational constraints on OPF, which determine the feasibility of power systems designed for the OPF problems; and 2) and the necessity to develop conventional methods for solving OPF problems which can be more effective than the commonly-used heuristic methods.

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