Optimality and Sensitivity of Least-Distance and Avoidance Solutions in Multicriteria Optimization

This paper investigates sensitivity and optimality of distance scalarizing solutions in multicriteria problems with reference sets. When approaching reference values, this preference requirement results in solving an equivalent second-level scalar optimization problem with a certain distance measure to be minimized. In a situation where the reference set contains values to be avoided within the solution process, distance maximization with respect to this set is a natural technique. The reference sets may arise as relaxed criteria space constraints and are non-precisely defined. Therefore, we provide sufficient conditions ensuring the stability of solutions with respect to reference set perturbations. For various reasons, including trajectory safety, the above problem may be particularly relevant in the case of avoidance values, which are usually defined with a native uncertainty. We propose a sensitivity analysis method based on investigating the properties of attainable criteria values minimizing the distance to perturbed reference sets. Two problems of this kind are formulated and solved. An application to selecting compromise trajectories in multicriteria optimal control with two reference multifunctions is outlined in the final part of this paper.