On optimality and duality in nonsmooth vector fractional continuous-time programming: Strengthened conditions for mixed-affine models

We investigate a nonsmooth vector fractional continuous-time programming problem with inequality-type phase constraints, motivated by applied-mechanics contexts in which performance is quantified by ratio-type, time-accumulated indices under pointwise-in-time operational constraints, including abrasive machining and grinding, quasi-steady aircraft cruise efficiency, and energy-aware gait scheduling in legged robotics. We derive saddle-point and Karush-Kuhn-Tucker type necessary optimality conditions for properly efficient solutions by combining a continuous-time Slater-type condition with a regularity requirement for convex inequality systems, and we also establish a sufficient saddle-point optimality condition that holds in the convex framework. For models with mixed affine structure, we strengthen the theory beyond classical Slater-based frameworks by introducing two additional verifiable hypotheses, a solvability condition and a separation direction condition. These assumptions yield sharper multiplier conclusions, including nontriviality of multipliers associated with nonaffine constraints, and lead to refined optimality statements without auxiliary parameters. A key lemma is established and provides the main tool underlying these results. We then introduce a vector-valued Lagrangian and formulate a corresponding vector dual model. For the dual problem, we prove weak and strong duality results, including a strong duality theorem that guarantees the absence of a duality gap. Several examples illustrate how the assumptions can be verified and how the theoretical results apply through explicit multiplier constructions.