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

${\displaystyle \displaystyle f(x)=\|Ax-b\|^{2},}$

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

${\displaystyle \nabla _{x}f=2A^{T}(Ax-b)=0}$.

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.

Given a function ${\displaystyle \displaystyle f(x)}$ of ${\displaystyle N}$ variables to minimize, its gradient ${\displaystyle \nabla _{x}f}$ indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction:

${\displaystyle \Delta x_{0}=-\nabla _{x}f(x_{0})}$

with an adjustable step length ${\displaystyle \displaystyle \alpha }$ and performs a line search in this direction until it reaches the minimum of ${\displaystyle \displaystyle f}$:

${\displaystyle \displaystyle \alpha _{0}:=\arg \min _{\alpha }f(x_{0}+\alpha \Delta x_{0})}$,
${\displaystyle \displaystyle x_{1}=x_{0}+\alpha _{0}\Delta x_{0}}$

After this first iteration in the steepest direction ${\displaystyle \displaystyle \Delta x_{0}}$, the following steps constitute one iteration of moving along a subsequent conjugate direction ${\displaystyle \displaystyle s_{n}}$, where ${\displaystyle \displaystyle s_{0}=\Delta x_{0}}$:

1. Calculate the steepest direction: ${\displaystyle \Delta x_{n}=-\nabla _{x}f(x_{n})}$,
2. Compute ${\displaystyle \displaystyle \beta _{n}}$ according to one of the formulas below,
3. Update the conjugate direction: ${\displaystyle \displaystyle s_{n}=\Delta x_{n}+\beta _{n}s_{n-1}}$
4. Perform a line search: optimize ${\displaystyle \displaystyle \alpha _{n}=\arg \min _{\alpha }f(x_{n}+\alpha s_{n})}$,
5. Update the position: ${\displaystyle \displaystyle x_{n+1}=x_{n}+\alpha _{n}s_{n}}$,

With a pure quadratic function the minimum is reached within N iterations (excepting roundoff error), but a non-quadratic function will make slower progress. Subsequent search directions lose conjugacy requiring the search direction to be reset to the steepest descent direction at least every N iterations, or sooner if progress stops. However, resetting every iteration turns the method into steepest descent. The algorithm stops when it finds the minimum, determined when no progress is made after a direction reset (i.e. in the steepest descent direction), or when some tolerance criterion is reached.

Within a linear approximation, the parameters ${\displaystyle \displaystyle \alpha }$ and ${\displaystyle \displaystyle \beta }$ are the same as in the linear conjugate gradient method but have been obtained with line searches. The conjugate gradient method can follow narrow (ill-conditioned) valleys, where the steepest descent method slows down and follows a criss-cross pattern.

Four of the best known formulas for ${\displaystyle \displaystyle \beta _{n}}$ are named after their developers:

• Fletcher–Reeves:[1]
${\displaystyle \beta _{n}^{FR}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.}$
• Polak–Ribière:[2]
${\displaystyle \beta _{n}^{PR}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.}$
• Hestenes-Stiefel:[3]
${\displaystyle \beta _{n}^{HS}=-{\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.}$
${\displaystyle \beta _{n}^{DY}=-{\frac {\Delta x_{n}^{T}\Delta x_{n}}{s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.}$.

These formulas are equivalent for a quadratic function, but for nonlinear optimization the preferred formula is a matter of heuristics or taste. A popular choice is ${\displaystyle \displaystyle \beta =\max\{0,\beta ^{PR}\}}$, which provides a direction reset automatically.[5]

Algorithms based on Newton's method potentially converge much faster. There, both step direction and length are computed from the gradient as the solution of a linear system of equations, with the coefficient matrix being the exact Hessian matrix (for Newton's method proper) or an estimate thereof (in the quasi-Newton methods, where the observed change in the gradient during the iterations is used to update the Hessian estimate). For high-dimensional problems, the exact computation of the Hessian is usually prohibitively expensive, and even its storage can be problematic, requiring ${\displaystyle O(N^{2})}$ memory (but see the limited-memory L-BFGS quasi-Newton method).

## References

1. ^ R. Fletcher and C. M. Reeves, "Function minimization by conjugate gradients", Comput. J. 7 (1964), 149–154.
2. ^ E. Polak and G. Ribière, "Note sur la convergence de directions conjugu´ee", Rev. Francaise Informat Recherche Operationelle, 3e Ann´ee 16 (1969), 35–43.
3. ^ M. R. Hestenes and E. Stiefel, "Methods of conjugate gradients for solving linear systems", J. Research Nat. Bur. Standards 49 (1952), 409–436 (1953).
4. ^ Y.-H. Dai and Y. Yuan, "A nonlinear conjugate gradient method with a strong global convergence property", SIAM J. Optim. 10 (1999), no. 1, 177–182.
5. ^ J. R. Shewchuk, "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain", August 1994.