Tropical Analogues of a Dempe-Franke Bilevel Optimization Problem

Abstract

We consider the tropical analogues of a particular bilevel optimization problem studied by Dempe and Franke [4] and suggest some methods of solving these new tropical bilevel optimization problems. In particular, it is found that the algorithm developed by Dempe and Franke can be formulated and its validity can be proved in a more general setting, which includes the tropical bilevel optimization problems in question. We also show how the feasible set can be decomposed into a finite number of tropical polyhedra, to which the tropical linear programming solvers can be applied.

Minimizing maximum lateness in two-stage projects by tropical optimization

“…In the two-stage optimization, the reduction of two-stage problem to a one-stage problem (if it exists) is often due to duality and complementary slackness conditions, which are lacking in tropical mathematics. Also, such reduction often introduces non-linear constraints, which was the case for the tropical bi-level optimization problems considered in [29], and this will very likely present a formidable challenge for the future development of two-stage and multi-stage optimization problems in tropical mathematics.…”

“…In view of the above mentioned development of tropical optimization problems applied to project scheduling, as well as successful development of algorithms to solve some other types of tropical optimization problems such as tropical linear programming and tropical linear-fractional programming [1,13], it seems to be a promising research direction. Note, however, that except for the tropical bi-objective problem of [27] and the tropical bi-level optimization problem of [29], we are not aware of any other tropical bi-objective or bi-level optimization problems considered in the existing literature.…”

Minimizing maximum lateness in two-stage projects by tropical optimization

“…In the bi-level optimization, the reduction of bi-level problem to a one-level problem (if it exists) is often due to duality and complementary slackness conditions, which are lacking in tropical mathematics. Also, such reduction often introduces non-linear constraints, which was the case for the tropical bi-level optimization problems considered in [28], and this will very likely present a formidable challenge for the future development of bi-level and multi-level optimization problems in tropical mathematics.…”

“…In view of the above mentioned development of tropical optimization problems applied to project scheduling, as well as successful development of algorithms to solve some other types of tropical optimization problems such as tropical linear programming and tropical linear-fractional programming [1,12], it seems to be a promising research direction. Note, however, that except for a particular tropical bi-level optimization problem [28], we are not aware of any other tropical bi-level optimization problems considered in the existing literature.…”