Abstract
Interval bilevel programming problem is hard to solve due to its hierarchical structure as well as the uncertainty of coefficients. This paper is focused on a class of interval linear bilevel programming problems, and an evolutionary algorithm based on duality bases is proposed. Firstly, the objective coefficients of the lower level and the right-hand-side vector are uniformly encoded as individuals, and the relative intervals are taken as the search space. Secondly, for each encoded individual, based on the duality theorem, the original problem is transformed into a single level program simply involving one nonlinear equality constraint. Further, by enumerating duality bases, this nonlinear equality is deleted, and the single level program is converted into several linear programs. Finally, each individual can be evaluated by solving these linear programs. The computational results of 7 examples show that the algorithm is feasible and robust.
Highlights
The bilevel programming problem (BLPP) is a hierarchical optimization problem, involving two optimization problems located at upper and lower levels
S.t. g (x, y) ≤ 0, where x ∈ Rn are the upper level variables, which are controlled by the leader, and y ∈ Rm are called the lower level variables and are controlled by the follower; F, f : Rn+m → R are the leader and the follower objective functions, respectively, while G (x, y) ≤ 0 and g (x, y) ≤ 0 are called the upper level and the lower level constraints which consist of some inequalities or equalities
In order to obtain the optimal value range, the interval programming problem was further converted into two BLPPs with determined coefficients
Summary
The bilevel programming problem (BLPP) is a hierarchical optimization problem, involving two optimization problems located at upper and lower levels. Calvete and Gale [18] addressed a linear bilevel problem in which only objective coefficients were assumed to lie between specific bounds and developed two enumerative algorithms to compute the optimal value range based on the kth best method. In order to obtain the optimal value range, the interval programming problem was further converted into two BLPPs with determined coefficients In these approaches, one has to execute two different algorithms to compute the upper bound and the lower bound of optimal objective values, respectively. We concentrate on the linear bilevel programming problem in which the coefficients of two objective functions are intervals as well as the right-hand-side values of constraints and present an efficient evolutionary algorithm based on duality bases for solving the problem.
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