Necessary and sufficient conditions for achieving global optimal solutions in multiobjective quadratic fractional optimization problems

If \(x^*\) is a local minimum solution, then there exists a ball of radius \(r>0\) such that \(f(x)\ge f(x^*)\) for all \(x\in B(x^*,r)\). The purpose of the current study is to identify the suitable \(B(x^*,r)\) of the local optimal solution \(x^*\) for a particular multiobjective optimization problem. We provide a way to calculate the largest radius of the ball centered at local Pareto solution in which this solution is optimal. In this process, we present the necessary and sufficient conditions for achieving a global Pareto optimal solution. The results of this investigation might be useful to determine stopping criteria in the algorithms development.