Iranian Journal of Numerical Analysis and Optimization
Volume:7 Issue: 1, Winter and Spring 2017

ýIn this paper, we formulate the fourth order Sturm-Liouville problem (FSLP) as a Lie group matrix differential equation. By solving this ma- trix differential equation by Lie group Magnus expansion, we compute the eigenvalues of the FSLP. The Magnus expansion is an infinite series of multiple integrals of Lie brackets. The approximation is, in fact, the truncation of Magnus expansion and a Gaussian quadrature are used to evaluate the integrals. Finally, some numerical examples are given.

In this paper, we investigate stability analysis of fractional differential systems equipped with the conformable fractional derivatives. Some stability conditions of fractional differential systems are proposed by applying the fractional exponential function and the fractional Laplace transform. Moreover, we check the stability of conformable fractional Lotka-Volterra system with the multi-step homotopy perturbation method to demonstrate the efficiency and effectiveness of the proposed procedure.

In this paper, an analytical method called the parametric iteration method (PIM) is presented for solving the second-order singular IVPs of Lane-Emden type, and its local convergence is discussed. Since it is often useful to have an approximate analytical solution to describe the Lane-Emden type equa- tions, especially for ones where the closed-form solutions do not exist at all, therefore, an effective improvement of the PIM is further proposed that is ca- pable of obtaining an approximate analytical solution. The improved PIM is finally treated as an algorithm in a sequence of intervals for finding accurate approximate solutions of the nonlinear Lane-Emden type equations. Also, we show how to identify an approximate optimal value of the convergence accelerating parameter within the frame of the method. Some examples are given to demonstrate the efficiency and accuracy of the proposed method.

A problem that sometimes occurs in multiobjective optimization is the existence of a large set of fairly effcient solutions. Hence, the decision making based on selecting a unique preferred solution is diffcult. Considering models with fair B-effciency relieves some of the burden from the decision maker by shrinking the solution set, since the set of fairly B-efficient solutions is contained within the set of fairly effcient solutions for the same problem. In this paper, first some theoretical and practical aspects of fairly B- effcient solutions are discussed. Then, some scalarization techniques are developed to generate fairly B-effcient solutions.

In this paper, we consider a special case of Weber location problem which we call goal location problem. The Weber location problem asks to find location of a point in the plane such that the sum of weighted distances between this point and n existing points is minimized. In the goal location problem each existing point Pi has a relevant radius ri and it’s ideal for us to locate a new facility on the distance ri from Pi for i = 1, ..., n. Since in the most instances there does not exist the location of a new facility such that its distance to each point Pi be exactly equal to ri. So we try to minimize the sum of the weighted square errors. We consider the case that the distances in the plane are measured by the Euclidean norm. We propose a Weiszfeld like algorithm for solving the problem and also we use two modifications of particle swarm optimization method for solving this problem. Finally the results of these algorithms are compared with results of BSSS algorithm.

Genetic algorithm (GA) has been extensively used in recent decades to solve many optimization problems in various fields of science and engineering.
In most cases, the number of iterations is the only criterion which is used to stop the GA. In practice, this criterion will lead to prolong execution times to ensure proper solution. A novel approach is presented in this article as the approximate number of decisive iterations (ANDI ) which can be used to successfully terminate the GA optimization method with minimum execution time. Two simple correlations are presented which relate the new parameter (ANDI ) with approximate degrees of freedom (Adf ) of the merit function at hand. For complex merit functions, a linear smoother (such as Regularization network) can be used to estimate the required Adf. Four illustrative case studies are used to successfully validate the proposed approach by effectively finding the optimum point by using to the presented correlation. The linear correlation is more preferable because it is much simpler to use and the horizontal axis represents the approximate (not exact) degrees of freedom. It was also clearly shown that the Regularization Networks can successfully filter out the noise and mimic the true hyper-surface underlying a bunch of noisy data set.

The extended trust region subproblem has been the focus of several research recently. Under various assumptions, strong duality and certain SOCP/SDP relaxations have been proposed for several classes of it. Due to its importance, in this paper, without any assumption on the problem, we apply the widely used alternating direction method of multipliers (ADMM) to solve it. The convergence of ADMM iterations to the first order stationary conditions is established. On several classes of test problems, the quality of the solution obtained by the ADMM for medium scale problems is compared with the SOCP/SDP relaxation. Moreover, the applicability of the method for solving large scale problems is shown by solving several large instances.