Let \(f (x,y)\) be a continuously differentiable function on \(\mathbb R^n \times\mathbb R^m\).
The authors present their work on minimizing \(f(x,y)\) over \(y \in\mathbb R^m\) subject to a finite number of constraints of the form \(h_i(x,y) < 0\) and \(h_j(x,y) = 0 \).
Generally researchers impose regularity conditions on the restraints to solve such nonlinear programming problems. Here, the authors explore a relaxed version of the constant rank condition on the error bound property, the directional differentiability of the optimal value function, and the necessity and sufficiency second order optimal condition.