On multiple-cost optimization and extended controlled vector inequalities

In this study, we established several relations between generalized (weak) vector controlled inequalities of Minty and Stampacchia type and the associated multi-cost models. To this end, we introduced the updated concepts of preconvexity and (strictly) strong convexity for functionals governed by controlled simple integrals and a mean-value-type result. Also, we introduced the corresponding multiobjective extremization models. The theoretical notions and the main results were justified by suitable numerical examples that were non-trivial.