Asymptotic Lagrange Regularity for Pseudoconcave Programming with Weak Constraint Qualification

In a previous paper we obtained characterizations of constraint sets determined by a finite number of differentiate pseudoconcave constraint functions possessing an interior point, and showed that they are convex Lagrange regular. In this paper we introduce a weaker type of constraint qualification and obtain asymptotic Lagrange regularity conditions from the constructs of semi-infinite programming. We impose the constraint qualification that each constraint function possess an interior point, possibly different for different constraint functions. Computationally speaking, it is relatively easy to verify since each constraint function is examined individually for interior points in consistent problems. Artificial vectors are adjoined to the dual problem and within the constructs of this problem involving no new data, convergence is assured asymptotically to Lagrangian type conditions. In this case the dual objective value tends to the optimal primal value in the limit. In case the constraint functions are differentiable concave, then asymptotic Lagrange regularity is obtained without any constraint qualification whatever.