Control and Optimization in Applied Mathematics
Volume:3 Issue: 2, Summer-Autumn 2018

In this paper‎, ‎we introduce and study some new single-valued gap functions for non-differentiable semi-infinite multiobjective optimization problems with locally Lipschitz data‎. ‎Since one of the fundamental properties of gap function for optimization problems is its abilities in characterizing the solutions of the problem in question‎, ‎then the essential properties of the newly introduced gap functions are established‎. ‎All results are given in terms of the Clarke subdifferential.

For a nonsmooth multiobjective mathematical programming problem governed by infinitely many constraints‎, ‎we define a new gap function that generalizes the definitions of this concept in other articles‎. ‎Then‎, ‎we characterize the efficient‎, ‎weakly efficient‎, ‎and properly efficient solutions of the problem utilizing this new gap function‎. ‎Our results are based on $(Phi,rho)-$invexity‎, ‎defined by Clarke subdifferential.

In this paper‎, ‎we decide to select the best center nodes‎ ‎of radial basis functions by applying the Multiple Criteria Decision‎ ‎Making (MCDM) techniques‎. ‎Two methods based on radial basis‎ ‎functions to approximate the solution of partial differential‎ ‎equation by using collocation method are applied‎. ‎The first is based‎ ‎on the Kansa's approach‎, ‎and the second is based on the Hermite‎ ‎interpolation‎. ‎In addition‎, ‎by choosing five sets of center nodes‎: ‎Uniform grid‎, ‎Cartesian‎, ‎Chebyshev‎, ‎Legendre and‎ ‎Legendre-Gauss-Lobato (LGL) as alternatives and achieving the error‎, ‎the condition number of interpolation matrix and memory time as‎ ‎criteria‎, ‎rating of cases with the help of PROMETHEE technique is‎ ‎obtained‎. ‎In the end‎, ‎the best center nodes and method is selected‎ ‎according to the rankings‎. ‎This ranking shows that Hermite‎ ‎interpolation by using non-uniform nodes as center nodes is more‎ ‎suitable than Kansa's approach with each center node.

This paper studies the convex multiobjective optimization problem with vanishing constraints‎. ‎We introduce a new constraint qualification for these problems‎, ‎and then a necessary optimality condition for properly efficient solutions is presented‎. ‎Finally by imposing some assumptions‎, ‎we show that our necessary condition is also sufficient for proper efficiency‎. ‎Our results are formulated in terms of convex subdifferential.

Following the setting of the Dai-Liao (DL) parameter in conjugate gradient (CG) methods‎, ‎we introduce two new parameters based on the modified secant equation proposed by Li et al‎. ‎(Comput‎. ‎Optim‎. ‎Appl‎. ‎202:523-539‎, ‎2007) with two approaches‎, ‎which use an extended new conjugacy condition‎. ‎The first is based on a modified descent three-term search direction‎, ‎as the descent Hestenes-Stiefel CG method‎. ‎The second is based on the quasi-Newton (QN) approach‎. ‎Global convergence of the proposed methods for uniformly convex functions and general functions is proved‎. ‎Numerical experiments are done on a set of test functions of the CUTEr collection and the results are compared with some well-known methods.

The purpose of this paper is to evaluate the revenue efficiency in the fuzzy network data envelopment analysis‎. ‎Precision measurements in real-world data are not practically possible‎, ‎so assuming that data is crisp in solving problems is not a valid assumption‎. ‎One way to deal with imprecise data is fuzzy data‎. ‎In this paper‎, ‎linear ranking functions are used to transform the full fuzzy efficiency model into a precise linear programming problem and‎, ‎assuming triangular fuzzy numbers‎, ‎the fuzzy revenue efficiency of decision makers is measured‎. ‎In the end‎, ‎a numerical example shows the proposed method.