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Multiobjective bilevel optimization. (English) Zbl 1198.90347
This paper deals with nonlinear non-convex multi-objective bi-level optimization problems which are discussed using an optimistic approach. The author aims to obtain a good approximation of the feasible set of the upper level function by expressing it as the set of minimal solutions of a multi-objective optimization problem. To solve this problem he applies the scalarization approach of A. Pascoletti and P. Serafini [J. Optimization Theory Appl. 42, 499–524 (1984; Zbl 0505.90072)].
For generating the approximation mentioned above, the author uses sensitivity results for controlling the parameters of the corresponding scalarization problem adaptively. This sensitivity results are used again for solving the upper level problem in an iterative process. Thus, not only one minimal solution but an approximation of the whole efficient set of the multi-objective bilevel optimization problem is determined.
The proposed numerical method (without convexity assumptions) demands twice continuously differentiable functions and appropriate solvers for determining global solutions of the scalar problems.
Finally, an academic example and a topological problem arising in an application are solved with an algorithm designed for the case of a bi-criteria lower and upper level problem and a one-dimensional upper level variable.

MSC:
90C29 Multi-objective and goal programming
90C31 Sensitivity, stability, parametric optimization
90C59 Approximation methods and heuristics in mathematical programming
Software:
NBI
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