Consider the following nonlinear optimization problem: \begin{align*} &\min f(x) \\ \text{such that } &h_1(x) = 0, \\ &h_2(x) = 0, \\ & \vdots \\ & h_m(x) = 0, \end{align*} where $x \in \mathbb{R}^n$. Suppose that the linear independence constraint qualification (LICQ) holds over the entire feasible region. Then, recall that we say that $x$ is a first order critical point if there exists $\lambda \in \mathbb{R}^m$ such that \begin{equation} \nabla f(x) - \sum_{i = 1}^m \lambda_i \nabla h_i(x) = 0. \end{equation} Furthermore, if such a $\lambda$ exists it is unique.
First-order criticality (aka being a first-order critical point) is a necessary condition for $x$ to be a local minimum. I know many algorithms exist which converge to first-order critical points.
Now, let $x^*$ denote a global minimum for the nonlinear program above. Let $\lambda^*$ denote the unique multiplier such that $$ \nabla f(x^*) - \sum_{i = 1}^m \lambda_i^* \nabla h_i(x^*) = 0. $$
Suppose I magically know $\lambda^*$, and I would like to obtain $x^*$. In fact, suppose I even know that every first-order critical point whose associated multiplier is $\lambda^*$ is globally optimal, and vice versa. In other words, $y$ is globally optimal if and only if $$ \nabla f(y) - \sum_{i = 1}^m \lambda_i^* \nabla h_i(y) = 0. $$ So I basically know precisely what the multiplier associated with the first-order criticality condition should be for any the global minimum.
Is there any nonlinear programming algorithm which could effectively use this information to converge to such a $y$? Obviously one approach would be to try to find a feasible point of the following system: \begin{align} &h_1(y) = 0, \\ &h_2(y) = 0, \\ & \vdots \\ & h_m(y) = 0, \\ &\nabla f(y) - \sum_{i = 1}^m \lambda_i^* \nabla h_i(y) = 0. \end{align} But this might be very hard, even if finding a feasible $y$ for the original optimization problem is easy. Thus, I'm wondering if there is some nonlinear programming algorithm which could make use of the fact that we know what the "correct" multipliers are, without just bluntly solving the feasibility problem above, which may be very difficult to solve.
There isn't much else we can assume about the original program. Strong duality, for example, doesn't hold.
Even references/ideas are much appreciated, as I'm not an expert in this area!
