Methods for solving the bilevel optimization problems

1. Introduction. Nowadays, the bilevel optimization problems, arising in various applications [1, 2], seem to be one of the most attractive fields for many experts [1, 3, 4, 5]. Bilevel problems are such optimization problems, which – side by side with ordinary constraints such as equalities and inequalities [6] – include a constraint described as an optimization subproblem, called the lower-level problem or follower problem. In course of investigation of bilevel optimization problems the difficulty arises already at the stage of defining the concept of solution. The optimistic and pessimistic (guaranteed) definitions of a solution are known to be the most popular [1, 3, 4]. During the three decades of intensive investigation of bilevel optimization problems there were proposed various methods for finding an optimistic solution by different authors (see the survey [5]). But there was a few for finding pessimistic solutions [7, 8]. From the computational point of view difficulties with numerical solving bilevel problems arise due to the fact that even the simpliest linear bilevel problem (with linear goal functions on both levels of hierarchy) in its optimistic setting is an equivalent to a nonconvex optimization problem (in terms of global solution) [4]. So, only results on finding optimistic solutions in bilevel problems of dimension up to 220 (at both upper and lower level variables) are known from literature [9, 10, 11]. As for more difficult problem of finding guaranteed solutions to bilevel optimization problems, only results on numerical solving of illustrative examples of dimension up to 4 are known [8]. In this paper we deal with new methods for finding optimistic and pessimistic solutions to linear and quadratic-linear (where the upper-level goal