Abstract
In this paper, we focus on a class of nonlinear bilevel programming problems where the follower’s objective is a function of the linear expression of all variables, and the follower’s constraint functions are convex with respect to the follower’s variables. First, based on the features of the follower’s problem, we give a new decomposition scheme by which the follower’s optimal solution can be obtained easily. Then, to solve efficiently this class of problems by using evolutionary algorithm, novel evolutionary operators are designed by considering the best individuals and the diversity of individuals in the populations. Finally, based on these techniques, a new evolutionary algorithm is proposed. The numerical results on 20 test problems illustrate that the proposed algorithm is efficient and stable.
Highlights
The bilevel programming problem (BLPP) involves two optimization problems at different levels, in which the feasible region of one optimization problem is implicitly determined by the other
We focus on a class of nonlinear bilevel programming problems where the follower’s objective is a function of the linear expression of all variables, and the follower’s constraint functions are convex with respect to the follower’s variables
BLPP is widely used to lots of fields such as economy, control, engineering and management [1,2], and more and more practical problems can be formulated as the bilevel programming models, so it is important to design all types of effective algorithms to solve different types of BLPPs
Summary
The bilevel programming problem (BLPP) involves two optimization problems at different levels, in which the feasible region of one optimization problem (leader’s problem/upper level problem) is implicitly determined by the other (follower’s problem/lower level problem). It is the simplest one among the family of BLPPs, and the optimal solutions can occurs at vertices of feasible region Based on these properties, lots of algorithms are proposed to solve this kind of problems [2,3,4,5]. In order to develop efficient algorithms, most algorithmic research to date has focused on some special cases of bilevel programs with convex/concave functions[1,2,6,7,8,9], especially, convex follower’s functions [10,11,12,13,14] In these algorithms, the convexity of functions guarantees that the globally optimal solutions can be obtained.
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