In this paper, a new class of nonconvex differentiable multiobjective Programming problems involving n-set functions with both inequality and equality constraints is considered. Then, under V-r-convexity and/or Generalized V-r-convexity hypotheses, several suffcient optimality conditions, saddle point criteria and various mixed duality theorems are proved for such not necessarily convex vector optimization problems involving n-set functions.

In this paper, a new class of nonconvex optimization problem is considered, namely  $(h,\varphi)$-$(b,F,\rho)$-convexity is defined for $(h,\varphi)$-differentiable mathematical Programming problem. The sufficiency of the so-called Karush-Kuhn-Tucker optimality conditions are established for the considered $(h,\varphi)$-differentiable mathematical Programming problem under (Generalized) $(h,\varphi)$-$(b,F,\rho)$-convexity hypotheses. Further, the so-called  Mond-Weir $(h,\varphi)$-dual problem is defined for the considered $(h,\varphi)$-differentiable mathematical Programming problem and several duality theorems in the sense of Mond-Weir are derived under appropriate (Generalized) $(h,\varphi)$-$(b,F,\rho)$-convex assumptions.

The linear systems are one of the most important tools for modeling real-world phenomena. Because the real-world phenomena are always associated with uncertainty, solving the fuzzy linear system have a great importance. One of the proposed methods to find the exact and approximate solutions of a fuzzy linear system is using the least squares method. In this method, by choosing an arbitrary meter and solving a quadratic Programming, they provide an approximate (or exact) solution for the fuzzy linear system. In this paper, at first, we prove that under some conditions and not depending on the selected meter the quadratic Programming is convex. Therefore, by considering three different meters and solving several examples, we compare the obtained approximate solutions.

This paper deals with linear Programming problem with interval numbers as coefficients to exhibit with uncertainty. Since, the set of common intervals is not a field, we define Generalized interval numbers to produce an algebraic interval field and on this field, we propose principle of uncertainty traverse instead of extension principle which permits to define operators on intervals exactly similar to the same operators on real numbers. In addition, we apply a total order on this field to transform interval linear Programming into a traditional problem. The proposed order can be extended either pessimistically or optimistically. The numerical experiments are given to demonstrate the efficiency of the proposed scheme in comparison with the previous established works. The approach in this paper can be Generalized to fuzzy linear Programming problems taking the fuzzy cuts into account.

Presented here is a generalization of the implicit enumeration algorithm that can be applied when the objective function is being maximized and can be rewritten as the difference of two non-decreasing functions. Also developed is a computational algorithm, named linear speedup, to use whatever explicit linear constraints are present to speed up the search for a solution. The method is easy to understand and implement, yet very effective in dealing with many integer Programming problems, including knapsack problems, reliability optimization, and spare allocation problems. To see some application of the Generalized algorithm, we notice that the branch-and-bound is the popular method to solve integer linear Programming problems. But branch-and bound cannot efficiently solve all integer linear Programming problems. For example, De Loera et al. in their 2005 paper discuss some knapsack problems that CPLEX cannot solve in hours. We use our Generalized algorithm to find a global or near global optimal solutions for those problems, in less than 100 seconds. The algorithm is based on function values only; it does not require continuity or differentiability of the problem functions. This allows its use on problems whose functions cannot be expressed in closed algebraic form. The reliability and efficiency of the proposed algorithm has been demonstrated on some integer optimization problems taken from the literature.

The aim of present paper is to study a constrained Programming with Generalized α − univex fuzzy mappings. In this paper we introduce the concepts of α − univex, α − preunivex, pseudo α − univex and α − unicave fuzzy mappings, and we discover that α − univex fuzzy mappings are more general than univex fuzzy mappings. Then, we discuss the relationships of Generalized α − univex fuzzy mappings and get some properties. In the last, we derive necessary and suffcient Karush-Kuhn-Tucker conditions and its dual problems with Generalized differentiable α − univex fuzzy mappings for fuzzy constrained Programming problem.

One of the issues of reliable performance in the power grid is the existence of electromechanical oscillations between interconnected generators. The number of generators participating in each electromechanical oscillation mode and the frequency oscillation depends on the structure and function of the power grid. In this paper, to improve the transient nature of the network and damping electromechanical fluctuations, a decentralized robust adaptive control method based on dynamic Programming has been used to design a stabilizing power system and a complementary static var compensator (SVC) controller. By applying a single line to ground fault in the network, the robustness of the designed control systems is demonstrated. Also, the simulation results of the method used in this paper are compared with controllers whose parameters are adjusted using the PSO algorithm. The simulation results show the superiority of the decentralized robust adaptive control method based on dynamic Programming for the stabilizing design of the power system and the complementary SVC controller. The performance of the control method is tested using the IEEE 16-machine, 68-bus, 5-area is verified with time domain simulation.