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.
Citation: Savin Treanţă, Cristina-Florentina Marghescu, Laura-Gabriela Matei. Efficiency conditions in multiple-objective optimal control models under generalized hypotheses[J]. AIMS Mathematics, 2024, 9(9): 25184-25204. doi: 10.3934/math.20241228
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.
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Savin Treanţă, Cristina-Florentina Marghescu, Laura-Gabriela Matei. Efficiency conditions in multiple-objective optimal control models under generalized hypotheses[J]. AIMS Mathematics, 2024, 9(9): 25184-25204. doi: 10.3934/math.20241228
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