Generalized Lagrange multiplier method

1. Given $LaTeX: (u, v)$ with $LaTeX: u \ge 0$, find $LaTeX: x \in \arg\!\max \{f(x) - u^T g(x) - v^T h(x) : x \in X\} = L^*(u,v)$.
2. Test if 0 is in the convex hull of $LaTeX: S(x, u, v) := \{(b, c): g(x) \le b, \; h(x) = c,\; u^T g(x) =u^T b\}$. If so, stop ($LaTeX: x$ solves the original mathematical program if 0 is in $LaTeX: S(x, u, v)$; otherwise, search alternative optima; if no maximum generates 0 as a member of $LaTeX: S$, then the right-hand side (0) is in a duality gap).
3. Given $LaTeX: (x, u, v)$, choose $LaTeX: (u', v')$ to reduce the value of $LaTeX: L^*(u, v)$.