In this paper, we determine the sufficient Karush-Kuhn-Tucker (KKT) conditions of optimality of a set-valued fractional programming problem (in short, SVFP) $(\mathrm{F}\mathrm{P})$ under the suppositions of contingent epidifferentiation and $\sigma$-arcwisely connectivity. We additionally explore the results of duality of parametric $(\mathrm{P}\mathrm{D})$, Mond-Weir $(\mathrm{M}\mathrm{W}\mathrm{D})$, Wolfe $(\mathrm{W}\mathrm{D})$, and mixed $(\mathrm{M}\mathrm{D})$ kinds for the problem $(\mathrm{F}\mathrm{P})$.

Since not every problem in optimization theory involves convex functionals, in this study, we introduced new classes of generalized convex functionals. More precisely, under generalized hypotheses, we stated new efficiency conditions associated with a class of multiple-objective optimal control models. To this end, we first defined the $G_{\theta }$-Fritz John problem and, by considering it, we established a link between the solutions of $G_{\theta }$-Fritz John problem and efficient solutions of the considered model $(P)$. In addition, we formulated the $G_{\theta }$-necessary efficiency conditions for a feasible solution in $(P)$. After that, we established a connection between the newly defined concept of $G_{\theta }-KT$ points to $(P)$ and the efficient solutions of $(P)$. Finally, we turned our attention to the $G_{\theta }$-sufficient efficiency conditions for a feasible solution to $(P)$. More precisely, we established that any feasible solution to $(P)$ will be an efficient solution if the assumption of $G_{\theta }$-convexity (and/or $G_{\theta }$-quasiconvexity, $G_{\theta }$-strictly quasiconvexity, $G_{\theta }$-monotonic quasiconvexity) is imposed on the involved functionals.

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.

This paper investigated a new family of robust multidimensional controlled models with constant-level set constraints. More precisely, we considered the extremization of a functional, given by a controlled multiple integral involving an uncertain parameter, subject to a finite set of constraints determined by path-independent curvilinear integral functionals. Necessary and sufficient optimality criteria were established for a robust feasible point. In addition, a saddle-point-type characterization of robust optimal points was studied in the second part of the paper.