What does "reduced gradient" refer to in nonlinear models?

The concept of "reduced gradient" in nonlinear models is closely related to the idea of reduced cost in linear programming. In essence, the reduced gradient is a measure that helps to determine how much a change in the values of variables will affect the objective function, relative to the constraints of the model. It provides information on the direction and magnitude of changes required in the decision variables to optimize the solution.

Specifically, when working with nonlinear models, the reduced gradient helps identify variables that can improve the objective function, similar to how reduced costs indicate whether a non-basic variable could enter the basis in linear programming. A variable with a reduced gradient of zero suggests that it is at its optimal level in the current context, meaning changing it would not produce a better solution.

This interplay between reduced gradients and the optimization process in nonlinear programming parallels the role of reduced costs in linear contexts, as both serve to inform decision-makers about variable significance and optimality within their models.