Duality for Smooth Vector Continuous-Time Programming Problems

This paper considers smooth vector continuous-time programming problems (VCTP) with phase inequality constraints defined almost everywhere on a compact interval, and develops two duality models of Mond–Weir and Wolfe type. Since many earlier results in continuous-time programming relied on an infinite-dimensional version of Gordan’s transposition theorem [20], whose validity has been questioned in [1], we instead work with a new regularity condition, (generalized) concavity assumptions, and a constraint qualification (MFCQ) to obtain weak and strong duality by using alternative tools in infinite-dimensional spaces. For the primal problem, with efficiency understood in the integral sense, we formulate a Mond–Weir continuous-time dual (MWCTD) using a primal-like trajectory, a constant weight vector, and time-dependent nonnegative multipliers. Under concavity/generalized concavity of the phase inequality constraints and pseudoconcavity (or concavity) of a scalarization functional, we derive a weak duality theorem, and under the (MFCQ) together with a suitable regularity condition we obtain a strong duality theorem. An analogous construction yields a Wolfe-type continuous-time dual (WCTD), for which weak duality follows from joint concavity and strong duality from the (MFCQ) or from a continuous-time Slater condition. Several illustrative examples are provided to demonstrate how the theoretical results can be verified in practice.