In this paper, we propose a split nonconvex variational inequality problem which is a natural extension of split convex variational inequality problem in two different Hilbert spaces. Relying on the prox-regularity notion, we introduce and establish the convergence of an iterative method for the new split nonconvex variational inequality problem. Further, we also establish the convergence of an iterative method for the split convex variational inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) variational inequality problems. These results also generalize, unify, and improve the previously known results of this area.

Newton method is one of the most famous numerical methods among the line search methods to minimize functions. It is well known that the search direction and step length play important roles in this class of methods to solve optimization problems. In this investigation, a new modi cation of the Newton method to solve uncon-strained optimization problems is presented. The significant merit of the proposed method is that the step length  k at each iteration is equal to 1. Additionally, the convergence analysis for this iterative algorithm is established under suitable conditions. Some illustrative examples are provided to show the validity and applicability of the presented method and a comparison is made with several other existing methods.

We present an effective algorithm for minimization of locally nonconvex Lipschitz functions based on mollifier functions approximating the Clarke generalized gradient. To this aim, first we approximate the Clarke generalized gradient by mollifier subgradients.To construct this approximation, we use a set of averaged functions gradients. Then, we show that the convex hull of this set serves as a good approximation for the Clarke generalized gradient.Using this approximation of the Clarke generalized gradient, we establish an algorithm for minimization of locally Lipschitz functions. Based on mollifier subgradient approximation, we propose a dynamic algorithm for finding a direction satisfying the Armijo condition without needing many subgradient evaluations. We prove that the search direction procedure terminates after finitely many iterations and show how to reduce the objective function value in the obtained search direction. We also prove that the first order optimality conditions are satisfied for any accumulation point of the sequence constructed by the algorithm. Finally, we implement our algorithm with MATLAB codes and approximate averaged functions gradients by the Monte-Carlo method. The numerical results show that our algorithm is effectively more efficient and also more robust than the GS algorithm, currently perceived to be a competitive algorithm for minimization of nonconvex Lipschitz functions.

In this paper, real-coded genetic algorithm with smart mutation (RCGA-SM) is proposed to solve the economic dispatch (ED) problem. In the proposed method, the required controllingprocess is accomplished on the total amount of chromosomes and consequently there is no need to use penalty cost function for controlling sum of variables in solving economic dispatch problem. This method will begin to explore the optimal answer just within the logic and acceptable zone in addition to its capability in reducing the search range. In order to show the performance and the efficiency of the proposed method, the ED problem considering several constraints is solved in 6, 15and 40 units systems through the proposed technique. The proposed coding could effectively escape from infeasible solutions. Thereby search efficiency and solution quality are dramatically improved.The obtained results are compared with other advanced technical algorithms, which well depict the superiority of the RCGA-SM technique over the other compared methods.

In this paper, we present a new hybrid conjugate gradient method for unconstrained optimization that possesses sufficient descent property independent of any line search. In our method, a convex combination of the Hestenes-Stiefel (HS) and the Fletcher-Reeves (FR) methods, is used as the conjugate parameter and the hybridization parameter is determined by minimizing the distance between the hybrid conjugate gradient direction and direction of the three-term HS method proposed by M. Li (\emph{A family of three-term nonlinear conjugate gradient methods close to the memoryless BFGS method,} Optim. Lett. \textbf{12} (8) (2018) 1911--1927). Under some standard assumptions, the global convergence property on general functions is established. Numerical results on some test problems in the CUTEst library illustrate the efficiency and robustness of our proposed method in practice.

In this paper, we present a method for charaterizing the solution set of nonconvex optimization problems via their dual problems. In fact, the constrainted optimization problem which is considerd has pseudoconvex and locally Lipschitz functions, which are not necessarily convex and smooth, and include a wide class of non-convex non-smooth functions. In the proposed method, a dual problem is formulated to characterizations of the solution set of the primal problem in a mixed type of Wolfe type and Mond-Weir type. First, we introduce some of the properties of the Lagrangian functions associated to these problems and then we explain the proof of the characterization of their solution sets.