Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bilevel programming problem

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Abstract

Particle Swarm Optimization (PSO) is a new optimization technique originating from artificial life and evolutionary computation. It completes optimization through following the personal best solution of each particle and the global best value of the whole swarm. PSO can be used to solve nonlinear programming problems for global optimal solutions efficiently, so a novel approach based on particle swarm optimization is proposed to solve nonlinear bilevel programming problem (NBLP). In the proposed approach, applying Karush–Kuhn–Tucker (KKT) condition to the lower level problem, we transform the NBLP into a regular nonlinear programming with complementary constraints, which is sequentially smoothed by Chen-Harker-Kanzow-Smale (CHKS) smoothing function. The PSO approach is then applied to solve the smoothed nonlinear programming for getting the approximate optimal solution of the NBLP problem. Simulations on 5 benchmark problems and practical example about watershed water trading decision-making problem are made and the results demonstrate the effectiveness of the proposed method for solving NBLP.

Introduction

The bilevel programming problem (BLP) is a nested optimizations problem with two levels in a hierarchy, the upper and lower level decision-making. Both of them have their own objective functions and constraints. The upper level maker makes his decision firstly, followed by the lower decision make. The objective function and constraint of the upper level programming rely not only on their own decision variables but also on the optimum solution of the lower level programming. The decision maker at the lower level has to optimize its own objective function under the given parameters from decision maker at the upper level, who, in return, with complete information on the possible reactions of the lower, selects the parameters so as to optimize its own objective function. Unlike the multiple objective mathematical programming techniques, the bilevel mathematical programming emphasizes the non-cooperative character of the system. The application of this hierarchical model can be used widely in such areas as resource allocation, decentralized control, network design problem, etc. [1].

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 classes 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 this 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.

Section snippets

Nonlinear bilevel programming problem and smoothing method

We consider the nonlinear bilevel programming (NBLP) formulated as follows [24]:(UP)minxF(x,y),t.h(x,y)0,(LP)minyf(x,y),s.t.g(x,y)0.where xXRn1,yYRn2.F,f:Rn1×Rn2R,h:Rn1×Rn2Rm1,g:Rn1×Rn2Rm2 are continuous differentiable functions. The term (UP) is called the upper-level problem and the term (LP) is called the lower-level problem and correspondingly the terms x,y are the upper-level variable and the lower-level variable respectively.

The notations are defined as follow:

  • (a) Constraint

Overview of particle swarm optimization

PSO is a population-based heuristic algorithm that simulates the social behavior as birds flocking to a promising position to achieve precise objectives in a multidimensional space. In PSO, the population is referred as a swarm and individuals are called particles. Like other evolutionary algorithms, PSO performs searches using a population of individuals that are updated from iteration to iteration. To find the optimal or approximately optimal solution, each particle changes its searching

Numerical examples

In this section, 5 benchmark problems were used for simulations to test the feasibility and efficiency of the proposed algorithm. For notational simplicity, denote x=(x1,,xn)T and y=(y1,,ym)T in these test problems. The details of these problems are as follows:

(1) Reference [5]: Let n=2 and m=2minx0F(x,y)=-x12-3x2-4y1+y22,s.t.x12+2x24,miny0f(x,y)=2x12+y12-5y2,s.t.x12-2x1+x22-2y1+y2-3,x2+3y1-4y24.

(2) Reference [9]: Let n=2 and m=2min0x50F(x,y)=2x1+2x2-3y1-3y2-60,s.t.x1+x2+y1-2y240,min-

Conclusions

Bilevel programming problem are intrinsically non-convex and it is difficult to solve for the global optimum solution. In this paper, we present a method based on particle swarm optimization to solve nonlinear bilevel programming problem. In this algorithm, KKT condition is used to the lower level problem to transform the NBLP into a single nonlinear programming problem with complementary constraints. And CHKS smoothing function is adopted to avoid the difficulty of dealing with the

Acknowledgements

This work was supported by the State Key Development Program for Basic Research of China (No. 2009CB421104) and the National Natural Science Foundation of China (Nos. 50809004 & 41071323).

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