Mathematcis

We propose a new version of potentially optimal intervals for the DIRECT algorithm. A two-points based sampling method is presented. The method starts from a distingished point (the peak point) by forming an initial triangle. The idea is to sample the midpoint of a specific interval: the basis of the resulting triangle. This specific interval is obtained by translating the initial interval towards the lowest function value : min{f(c i ),f(c i+1 )} and then overcoming the disadvantage if the global minimum lies at the boundaries. Two-dimensional version of our subdivision and sampling method is also discussed.

Drawing inspiration from the fact that each point sampled by the DIRECT algorithm will be a midpoint of the center subinterval, we present a one-dimensional version which considers two symmetric points, the one-third and two-third of the length of the considered interval. The center subinterval will be the region of interest. The interval is then bisected and two new points are added at every step. The two points sampled before became two-third and one-third, respectively. Two possibilities of definition of potentially optimal intervals are given. The proposed version predicts a fast convergence, and overcomes some disadvantages of the DIRECT in the case where the global minimum lies at the boundaries.

The DIRECT algorithm was motivated by a modification to Lipschitzian optimization. The algorithm begins its search by sampling the objective function at the midpoint of an interval, where this function attains its lowest value, and then divides this interval by trisecting it. One of its weakness is that if a global minimum lies at the boundaries, which can never be reached, the convergence of the algorithm will be unecessary slow. We present a one-dimensional variante of the DIRECT algorithm based on another strategy of sudividing the search domain. It consists of decreasing the number of intervals and increasing the number of sampling points, by interverting the roles of dividing and sampling at some steps of the DIRECT algorithm, and thus, overcoming this disadvantage.

In this paper we introduce some kinds of globally filled functions for solving global optimization problems. The functions we introduce have particular properties with respect to minimization problems. This transformation converges increasingly to the objective function when the parameter is large enough. We obtain a characterization of a sequence of minimizers for f. The reiterated application of this result quarantees the existence of the global minimum for the objective function.

In this study, a computational method for solving linear FredholmIntegro-Differential Equation (FIDE) of the first order under the mixed conditions using the Bernstein polynomials. First, we present some properties of these polynomials and the method is explained. These properties are then used to convert the integro-differential equation to a system of linear algebraic equations with unknown Bernstein coefficients. Using Galerkin method, we give an approximate solution. This method seems very attractive and simple to use. Illustrative examples show the efficiency and validity of the method we discuss the results using error analysis, the results are discussed.

In this paper, Bernstein polynomial method applied to the solutions of generalized Emden-Fowler equations as singular initial value problems is presented. Firstly, the singular differential equations are converted to Volterra integro-differential equations and then solved by the Bernstein polynomials method. The properties of Bernstein polynomials via Gauss-Legendre rule are used to reduce the integral equations to a system of algebraic equations which can be solved numerically. Some illustrative examples are discussed to demonstrate the validity and applicability of the present method.

In this note we show that a modification in the division procedure for the DIRECT algorithm can remove the sensitivity, which affects the smallest hypercube containing the low function value, for being potentially optimal. Mathematics Subject Classifications: 65K10, 90C30

In this paper we introduce some kinds of globally ...lled functions for solving global optimization problems. The function we introduce have particular properties with respect to minimization problems. This transformation converges increasingly to the objective function when the parameter is large enought. We obtain a characterization of a sequence of minimizers for f . The reiterated application of this result guarantees the existence of the global minimum for the objective function.

In this work, we introduce a new general procedure of finding operational matrices of integration P , differentiation D , and product Ĉ for the orthonormal Bernstein polynomials (OBPs) in the same way as in [12], S. A. Yousefi, M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int. J. Syst. Sci.41, 709-716, 2010. As an application, these matrices can be used to solve integral and integro-differential equations, differential equations, and in some problems in the calculus of variations and optimal control. The efficiency of this approach is shown by applying this procedure on some illustrative examples.

We propose a new version of potentially optimal intervals for the DIRECT algorithm. A two-points based sampling method is presented. The method starts from a distingished point (the peak point) by forming an initial triangle. The idea is to sample the midpoint of a specific interval: the basis of the resulting triangle. This specific interval is obtained by translating the initial interval towards the lowest function value : min{f(c i),f(c i+1)} and then overcoming the disadvantage if the global minimum lies at the boundaries. Two-dimensional version of our subdivision and sampling method is also discussed.

The DIRECT algorithm was motivated by a modification to Lip-schitzian optimization. The algorithm begins its search by sampling the ob-jective function at the midpoint of an interval, where this function attains its lowest value, and then divides this interval by trisecting it. One of its weak-ness is that if a global minimum lies at the boundaries, which can never be reached, the convergence of the algorithm will be unecessary slow. We present a one-dimensional variante of the DIRECT algorithm based on another strat-egy of sudividing the search domain. It consists of decreasing the number of intervals and increasing the number of sampling points, by interverting the roles of dividing and sampling at some steps of the DIRECT algorithm, and thus, overcoming this disadvantage.

It is shown that, contrary to a claim of [D. E. Finkel, C. T. Kelley, Additive scaling and the DIRECT algorithm, J. Glob. Optim. 36 (2006) 597-608], it is possible to divide the smallest hypercube which contains the low function value by considering hyperrectangles whose points are located on the diagonal of the center point of this hypercube using a division procedure which influences the slope to be below the the threshold fmin-ε |fmin| and thus reduces the influence of the parameter ε.

Abstract: In this paper the fractional order singular Emden-Fowler equation is introduced and a computational method is proposed for its numerical solution. For the approximation of the solutions we have used Boubaker polynomials for the first time and defined the formulation for its fractional derivative operational matrix. The operational matrix of the Caputo fractional derivative tool converts these problems to a system of algebraic equations whose solutions are simple and easy to compute. Numerical examples are examined to prove the validity and the effectiveness of the proposed method to find approximate and precise solutions.

We consider a global optimization problem for Lipschitz-continuous functions with an unknown Lipschitz constant. Our approach is based on the well-known DIRECT (DIviding RECTangles) algorithm and motivated by the diagonal partitioning strategy. One of the main advantages of the diagonal partitioning scheme is that the objective function is evaluated at two points at each hyper-rectangle and, therefore, more comprehensive information about the objective function is considered than using the central sampling strategy used in most DIRECT-type algorithms. In this paper, we introduce a new DIRECTtype algorithm, which we call BIRECT (BIsecting RECTangles). In this algorithm, a bisection is used instead of a trisection which is typical for diagonal-based and DIRECT-type algorithms. The bisection is preferable to the trisection because of the shapes of hyperrectangles, but usual evaluation of the objective function at the center or at the endpoints of the diagonal are not favorable for bisection. In the proposed algorithm the objective function is evaluated at two points on the diagonal equidistant between themselves and the endpoints of a diagonal. This sampling strategy enables reuse of the sampling points in descendant hyperrectangles. The developed algorithm gives very competitive numerical results compared to the DIRECT algorithm and its well know modifications.

This paper introduces novel enhancements to the most recent versions of DIRECT-type algorithms, especially when dealing with solutions located at the hyper-rectangle vertices. The BIRECT algorithm encounters difficulties in efficiently sampling points at the boundaries of the feasible region, leading to potential slowdowns in convergence. This issue is particularly pronounced when the optimal solution resides near the boundary. Our research explores diverse approaches, with a primary focus on incorporating a grouping strategy for hyper-rectangles of similar sizes. This categorization into different classes, constrained by a predefined threshold, aims to enhance computational efficiency, particularly involving a substantial number of hyper-rectangles of varying sizes. To further improve the new algorithm's efficiency, we implemented a mechanism to prevent oversampling and mitigate redundancy in sampling at shared vertices within descendant sub-regions. Comparisons with several DIRECT-type algorithms highlight the promising nature of the proposed algorithms as a global optimization solution. Statistical analyses, including Friedman and Wilcoxon tests, demonstrated the effectiveness of the improvements introduced in this new algorithm.