Abstract:
Fifht order Runge – Kutta method (Bucher’s method) is used in solving initial value problems of the form by the aid of Excel Spread Sheet for different values of step size h, and the true relative percent error εt is calculated for every case.

Abstract:
The paper shows that how powerful of excel spreadsheet in solving mathematical problems numerically. The numerical solutions of many initial value problems with boundary conditions of partial differential equations using excel are given and the effect of changing the parameters of the equations is treated.

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The goal of this paper is to solve a class of high-order polynomial benchmark optimization problems, including the Goldstein-Price problem and the Three Hump Camel Back problem. By using a generalized canonical duality theory, we are able to transform the nonconvex primal problems to concave dual problems over convex domain(without duality gap), which can be solved easily to obtain global solutions.

Abstract:
A nondifferentiable multiobjective optimization problem with nonempty set constraints is considered, and the equivalence of weakly efficient solutions, the critical points for the nondifferentiable multiobjective optimization problems, and solutions for vector variational-like inequalities is established under some suitable conditions. Nonemptiness and compactness of the solutions set for the nondifferentiable multiobjective optimization problems are proved by using the FKKM theorem and a fixed-point theorem. 1. Introduction The weak minimum (weakly efficient, weak Pareto) solution is an essential concept in mathematical models, economics, decision theory, optimal control, and game theory. For readers’ reference, we refer to [1–11] and the references therein. In [5], Garzón et al. studied some relationships among the weakly efficient solutions, the critical points of optimization problems, and the solutions of vector variational-like inequalities with differentiable functions. In [12], Mishra and Wang extended the work of Garzón et al. [5] to nonsmooth case. In [9], Lee et al. investigated the existence of solutions of vector optimization problems with differentiable functions. In [7], Kazmi considered the relationship between the weakly efficient solutions of a vector optimization problem and the solutions of a vector variational-like inequality with preinvex and Frechet differentiable functions. For more related work in this interesting area, we refer to [4, 10]. Motivated and inspired by the works mentioned above, we consider nondifferentiable multiobjective optimization problems (MOPs) with nonempty set constraints. The relationship among weakly efficient solutions, critical points of (MOP), and solutions of the vector variational-like inequalities (for short, (VVLI)) is presented under subinvexity, strictly pseudosubinvexity, and pseudosubinvexity conditions. By using the FKKM theorem and a fixed-point theorem, we prove the nonemptiness and compactness of solutions set for (MOP). The results presented in this paper extend the corresponding results of [5, 7, 9, 12, 13]. 2. Preliminaries Throughout this paper, without other specifications, let be the -dimensional Euclidean space, and . Let be a nonempty convex subset of , let be a subset of , and let be the relative interior of to . Let , , and let such that, for each , is a closed convex cone, , , and . The multiobjective optimization problem (for short, (MOP)) is defined as follows: We first recall some definitions and lemmas which are needed in the main results of this paper. Definition 2.1. A

Abstract:
We examine a new optimization problem formulated in the tropical mathematics setting as an extension of certain known problems. The problem is to minimize a nonlinear objective function, which is defined on vectors over an idempotent semifield by using multiplicative conjugate transposition, subject to inequality constraints. As compared to the known problems, the new one has a more general objective function and additional constraints. We provide a complete solution in an explicit form to the problem by using an approach that introduces an additional variable to represent the values of the objective function, and then reduces the initial problem to a parametrized vector inequality. The minimum of the objective function is evaluated by applying the existence conditions for the solution of this inequality. A complete solution to the problem is given by the solutions of the inequality, provided the parameter is set to the minimum value. As a consequence, we obtain solutions to new special cases of the general problem. To illustrate the application of the results, we solve a real-world problem drawn from project scheduling, and offer a representative numerical example.

Abstract:
A definition of a special class of optimization problems with set functions is given. The existence of optimal solutions and first-order optimality conditions are proved. This case of optimal problems can be transformed to standard mixed problems of mathematical programming in Euclidean space. It makes possible the applications of various algorithms for these optimization problems for finding conditional extrema of set functions.

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A new method for obtaining sensitivity information for parametric vector optimization problems (VOP)v is presented, where the parameters in the objective functions and anywhere in the constraints. This method depends on using differential equations technique for solving multiobjective nonlinear programing problems which is very effective in finding many local Pareto optimal solutions. The behavior of the local solutions for slight perturbation of the parameters in the neighborhood of their chosen initial values is presented by using the technique of trajectory continuation. Finally some examples are given to show the efficiency of the proposed method.

Abstract:
We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors of a finite-dimensional semimodule over an idempotent semifield, and may have constraints in the form of linear equations and inequalities. The aim of the paper is twofold: first to give a broad overview of known tropical optimization problems and solution methods, including recent results; and second, to derive a direct, complete solution to a new constrained optimization problem as an illustration of the algebraic approach recently proposed to solve tropical optimization problems with nonlinear objective function.

Abstract:
We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.

Abstract:
We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.