The Generalized reduced gradient method (GRG) is a generalization of the reduced gradient method by allowing nonlinear constraints and arbitrary bounds on the variables. The form is:

$LaTeX: \max f(x) : h(x) = 0, \; L \le x \le U,$

where $LaTeX: h$ has dimension $LaTeX: m$. The method supposes we can partition $LaTeX: x = (v, w)$ such that:

• $LaTeX: v$ has dimension $LaTeX: m$ (and $LaTeX: w$ has dimension $LaTeX: n-m$);
• the values of $LaTeX: v$ are strictly within their bounds: $LaTeX: L_v < v < U_v$ (this is a nondegeneracy assumption);
• $LaTeX: \nabla_v h(x)$ is nonsingular at $LaTeX: x = (v,w)$.

As in the linear case, for any $LaTeX: w$ there is a unique value, $LaTeX: v(w)$, such that $LaTeX: h(v(w),w)=0$ (c.f., Implicit Function Theorem), which implies that $LaTeX: dv/dw = (\nabla_v h(x))^{-1} \nabla_w h(x)$. The idea is to choose the direction of the independent variables to be the reduced gradient: $LaTeX: \nabla_w (f(x) - y^T h(x))$, where $LaTeX: y = dv/dw = (\nabla_v h(x))^{-1}\nabla_w h(x)$. Then, the step size is chosen and a correction procedure applied to return to the surface, $LaTeX: h(x)=0$.

The main steps (except the correction procedure) are the same as the reduced gradient method, changing the working set as appropriate.