The successful application of this hierarchical model depend

The successful application of this hierarchical model depends on how well it is solved in handling realistic complications. A significant amount of effort have been devoted to solving bilevel mathematical programming and many efficient algorithms have been proposed. To date a few algorithms exist to solve BLP, which can be classified into four types: approach of using the Karush–Kuhn–Tucker (K-K-T) condition [2], [3], [4], [5], [6], penalty function approach [7], [8], [9], [10], descent approach [11], [12] and evolutionary approach [13]. Recently, the evolutionary algorithms are widely used to solve different problems in optimal areas and become an alternative for solving bilevel programming for its good characteristics. In 1994, Mathieu etc. [14] firstly developed a genetic algorithm based bilevel programming algorithm. In 1998, Kemal etc. [15] proposed a dual temperature simulated annealing approach for solving bilevel programming problems. In this method, the lower level problem is stochastically relaxed with a parameter that can be used as a temperature scale in simulated annealing. Oduguwa etc. [16] proposed a bilevel genetic algorithm, which is an elitist optimization algorithm developed to encourage limited asymmetric cooperation between the two players, to solve different Mechlorethamine HCl of the bilevel problems within a single framework. Wang etc. [17] proposed an evolutionary algorithm for solving nonlinear bilevel programming problem. A specific optimization problem is constructed with two objectives firstly, which then is solved by a new evolutionary algorithm. By solving the specific problem, they decrease the upper objective value, identify the quality of any feasible solution from infeasible solutions, force the infeasible solutions moving toward the feasible region and improve the feasible solutions gradually. In modern science and technology, many optimization problems need to be solved in real time, while these classical methods cannot render real-time solutions to these optimization problems, especially large-scale problems. As a new metaheuristic, particle swarm optimization (PSO) [18], [19] has proved to be a competitive algorithm for optimization problems compared with other algorithms such as genetic algorithm (GA) and simulating algorithm (SA). It can converge to the optimal solution rapidly [20], [21], and leaf primordia advantage has been attracting researchers to solve BLP problem using PSO approach. [22], [23] proposed a hierarchical particle swarm optimization for solving BLP problem. In this paper, for a class of nolinear bilevel programming (NBLP) problem, replaced the lower level problem by its Kraush-Kuhn-Tucker optimality conditions, the NBLP problem is reduced into a regular nonlinear programming with complementary constraints. It is then smoothed by CHKS smoothing function. Finally, a particle swarm optimization approach is proposed to solve the smoothed nonlinear programming for getting the approximate optimal solution of the NBLP problem. This paper is organized as follows: Section 2 introduces the formulation and basic definitions of bilevel nonlinear programming, and also introduces the smoothing method for nonlinear complementarity problem. Section 3 introduces a particle swarm optimization for solving the smoothed programming problem. Numerical examples are reported in Section 4. And the conclusion is given in Section 5.