NIMBUS — Interactive Method for Nondifferentiable Multiobjective Optimization Problems

An interactive method, NIMBUS, for nondifferentiable multiobjective optimization problems is introduced. We assume that every objective function is to be minimized The idea of NIMBUS is that the decision maker can easily indicate what kind of improvements are desired and what kind of impairments are tolerable at the point considered.At each iteration, the decision maker is asked to classify the objectives into up to five different classes: those to be improved, those to be improved down to some aspiration level, those to be accepted as they are, those to be impaired till some upper bound and those allowed to change freely. The aspiration levels and the upper bounds are asked from the decision maker. The decision maker can also attach weighting coefficients to the objective functions. According to this classification, a new (possibly multiobjective) optimization problem is formed.An MPB (Multiobjective Proximal Bundle) method is employed to solve the new problem. The MPB method is a generalization of the Kiwiel’s proximal bundle approach for nondifferentiable single objective optimization into the multiobjective case. The multiple objectives are treated individually without employing any scalarization. The method is capable of handling several nonconvex Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints.