Carathéodory–john-type Sufficiency Criteria in Continuous-time Nonlinear Programming under Generalized (α, Ρ)− (η, Θ)-type I Invexity

Continuous-time linear programming was first introduced by Bellman [1] in the treatment of production and inventory “bottleneck” problems. He formulated the dual problem, established a weak duality theorem and described sufficient optimality conditions. Tyndall [12] extended Bellman’s theory and obtained existence and duality theorems for a class of continuous linear programming problem. Levinson [4] generalized and simplified some results of Tyndall by considering the time-dependence matrices in which the functions from objective and constraints are continuous on [0, T ]. Some classes of continuous-time linear and nonlinear programming problems have been studied in different settings and under various assumptions. Singh and Farr [8] established the optimality criteria of Kuhn-Tucker and Fritz John type, without assuming differentiability of functions, assuming that the functions involved are either convex or concave. Singh [7] weakened the convexity (concavity) restrictions and assumed that the objective function involved pseudoconvexity “almost everywhere” and the constraints involved quasi-convex and quasi-concave functions. He established a sufficient optimality criterion in continuous-time programming, assuming differentiability of functions.