Nonlinear conjugate gradient method

In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function ${\displaystyle \displaystyle f(x)}$:

The minimum of ${\displaystyle f}$ is obtained when the gradient is 0:

Whereas linear conjugate gradient seeks a solution to the linear equation ${\displaystyle \displaystyle A^{T}Ax=A^{T}b}$, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient ${\displaystyle \nabla _{x}f}$ alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum and the second derivative is non-singular there.

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