# Generalized reduced gradient method

### From Glossary

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:

where has dimension . The method supposes we can partition such that:

- has dimension (and has dimension );
- the values of are strictly within their bounds: (this is a nondegeneracy assumption);
- is nonsingular at .

As in the linear case, for any there is a unique value, , such that (c.f., Implicit Function Theorem), which implies that . The idea is to choose the direction of the independent variables to be the reduced gradient: , where . Then, the step size is chosen and a correction procedure applied to return to the surface, .

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