Multiobjective optimization with least constraint violation: optimality conditions and exact penalization

Although multiobjective optimization problem (MOP) is useful for solving many practical optimization problems, it is possible that the constraints are inconsistent. In this paper, we reformulate MOP with possible inconsistent constraints into MOP with least constraint violation and provide necessary optimality conditions from the perspective of M-stationary point, Fritz-John stationary point and L-stationary point. A power penalty problem is proposed by using infeasibility measure of constraints. The calmness conditions of order l of the MOP with least constraint violation and the local exact penalization of order l of the power penalty problem are respectively introduced, which do not require the feasibility of the original MOP. We obtain the equivalence between the calmness of order l of the MOP with least constraint violation and the local exact penalization of order l of the power penalty problem. Necessary and sufficient conditions for calmness of order l are also established under suitable conditions.

Automated summary

In this paper, the authors reformulate the multiobjective optimization problem with possible inconsistent constraints into a problem with least constraint violation. They provide necessary optimality conditions from different stationary point perspectives and propose a power penalty problem using infeasibility measure of constraints. The authors introduce calmness conditions and local exact penalization, and establish equivalences and necessary and sufficient conditions for these properties.