Zhongping Wan

Genetic algorithm based on simplex method for solving linear-quadratic bilevel programming problem

The bilevel programming problems are useful tools for solving the hierarchy decision problems. In this paper, a genetic algorithm based on the simplex method is constructed to solve the linear-quadratic bilevel programming problem (LQBP). By use of Kuhn-Tucker conditions of the lower level programming, the LQBP is transformed into a single level programming which can be simplified to a linear programming by the chromosome according to the rule. Thus, in our proposed genetic algorithm, only the linear programming is solved by the simplex method to obtain the feasibility and fitness value of the chromosome. Finally, the feasibility of the proposed approach is demonstrated by the example.

A hybrid intelligent algorithm by combining particle swarm optimization with chaos searching technique for solving nonlinear bilevel programming problems

Abstract In this paper, a hybrid intelligent algorithm by combining the particle swarm optimization (PSO) with chaos searching technique (CST) is presented for solving nonlinear bilevel programming problems. The bilevel programming is transformed into a single level programming problem by use of the KKT conditions of the lower level problem. Then, the hybrid intelligent algorithm is proposed to solve the transformed problem. Our approach embeds the CST into PSO. Firstly, the algorithm is initialized by a set of random particles which travel through the search space. Secondly, an optimization problem is solved by CST to judge whether the particle is feasible or not. In each iteration, all the feasible particles are ranked in ascending order. Particles in the front of list are updated by PSO, while particles in the end of list are updated by CST. The CST used here is not only to enhance the particles but also to improve the diversity of the particle swarm so as to avoid PSO trapping the local optima. Finally, the hybrid intelligent algorithm is commented by illustrating the numerical results on several benchmark problems from the references.

A penalty function method based on Kuhn–Tucker condition for solving linear bilevel programming☆

Abstract Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the linear bilevel programming problem into a corresponding single level programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we decompose the linear bilevel programming into a series of linear programming problems and get the optimal solution of the linear bilevel programming using linear programming method.

Estimation of distribution algorithm for a class of nonlinear bilevel programming problems

In this paper, a novel evolutionary algorithm called estimation of distribution algorithm (EDA) is proposed for solving a special class of nonlinear bilevel programming problems (BLPPs) in which the lower level problem is a convex programming problem for each given upper level decision. This special type of BLPP is transformed into a equivalent single-level constrained optimization problem using the Karush-Kuhn-er conditions of the lower level problem. Then, we propose an EDA based on the statistical information of the superior candidate solutions to solve the transformed problem. We stress that the new population of individuals is sampled from the probabilistic distribution of those superior solutions. Thus, one of the main advantages of EDA over most other meta-heuristics is its ability to adapt the operators to the structure of the problem, although adaptation in EDA is usually limited by the initial choice of the probabilistic model. In addition, two specific rules are established in the initialization procedure to make use of the hierarchical structure of BLPPs and to handle the constraints. Moreover, without requiring the differentiability of the objective function, or the convexity of the search space of the equivalent problem, the proposed algorithm can address nonlinear BLPPs with non-differentiable or non-convex upper level objective function and upper level constraint functions. Finally, the proposed algorithm has been applied to 16 benchmark problem; in five of these problems, all of the upper level variables and lower level variables are 10-dimensional. The numerical results compared with those of other methods reveal the feasibility and effectiveness of the proposed algorithm.

Bilevel newsvendor models considering retailer with CVaR objective

In this paper, the bilevel newsvendor model in two-echelon systems is presented. In this model, CVaR (conditional value-at-risk) measure is considered as the retailers objective. We obtain an analytical solution when the products demand is uniformly distribution. Several examples are given to illustrate our model and for comparable with the basic two-echelon newsvendor model. For the circumstance when the manufacturer knows the retailers objective, we get a counter-intuitive result: the retailers CVaR measure is larger when his objective is minimizing CVaR than that when his objective is maximizing the expected profit. The reason is that the manufacturer changes the wholesale price to maximize his own profit when he knows the retailers objective is CVaR. Some effects of parameters (such as unit shortage penalty) on the optimal wholesale price and order quantity are also revealed in the paper.

A neural network for solving a convex quadratic bilevel programming problem

A neural network is proposed for solving a convex quadratic bilevel programming problem. Based on Lyapunov and LaSalle theories, we prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the equilibrium, which corresponds to the optimal solution of a convex quadratic bilevel programming problem. Numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem.

A neural network approach for solving nonlinear bilevel programming problem

A neural network model is presented for solving nonlinear bilevel programming problem, which is a NP-hard problem. The proposed neural network is proved to be Lyapunov stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem. The asymptotic properties of the neural network are analyzed and the condition for asymptotic stability, solution feasibility and solution optimality are derived. The transient behavior of the neural network is simulated and the validity of the network is verified with numerical examples.

A fuzzy interactive method for a class of bilevel multiobjective programming problem

In this paper, we address a class of bilevel multiobjective programming problem where the lower level is a linear multiobjective optimization problem. We use the concepts of satisfactoriness as well as multiobjective optimization at the upper level and a measurement function at the lower level, to develop an interactive method for solving such a problem. The final solution of the proposed method is always efficient to the upper level when the lower level achieves satisfaction. It may be more significant in practice since bilevel programming problem is hierarchical, and the upper level decision making is in the dominant position. Finally, an illustrative numerical example is given to demonstrate the feasibility of the proposed method.

An improved artificial bee colony algorithm for solving constrained optimization problems

The artificial bee colony (ABC) algorithm is a global stochastic optimization algorithm inspired by simulating the foraging behavior of honey bees. It has been successfully applied to solve the constrained optimization problems (COPs) with a constraint handling technique (Deb’s rules). However, it may also lead to premature convergence. In order to improve this problem, we propose an improved artificial bee colony (I-ABC) algorithm for COPs. In I-ABC algorithm, we firstly relax the Deb’s rules by introducing the approximate feasible solutions to suitably utilize the information of the infeasible solutions with better objective function value and small violation. Next, we construct a selection strategy based on rank selection and design a search mechanism using the information of the best-so-far solution to balance the exploration and the exploitation at different stages. In addition, periodic boundary handling mode is used to repair invalid solutions. To verify the performance of I-ABC algorithm, 24 benchmark problems are employed and two comparison experiments have been carried out. The numerical results show that the proposed I-ABC algorithm has an outstanding performance for the COPs.

A globally convergent algorithm for a class of bilevel nonlinear programming problem

Bilevel programming, one of the multilevel programming, is a class of optimization with hierarchical structure. This paper proposes a globally convergent algorithm for a class of bilevel nonlinear programming. In this algorithm, by use of the dual theory, the bilevel nonlinear programming is transformed into a traditional programming problem, which can be turned into a series of programming problem without constraints. So we can solve the infinite nonlinear programming in parallelism to obtain the globally convergent solution of the original bilevel nonlinear programming. And the example illustrates the feasibility and efficiency of the proposed algorithm.