Nonlinear conjugate gradient method

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

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

the minimum of $f$ is obtained when the gradient is 0:

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

Whereas linear conjugate gradient seeks a solution to the linear equation $\displaystyle A^{T}Ax=A^{T}b$, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient $\nabla _{x}f$ 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 $\displaystyle f(x)$ of $N$ variables to minimize, its gradient $\nabla _{x}f$ indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction:

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

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

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

$\displaystyle x_{1}=x_{0}+\alpha _{0}\Delta x_{0}$

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

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

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 $\displaystyle \alpha$ and $\displaystyle \beta$ 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 $\displaystyle \beta _{n}$ are named after their developers:

• Fletcher–Reeves:

$\beta _{n}^{FR}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.$

• Polak–Ribière:

$\beta _{n}^{PR}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.$

• Hestenes–Stiefel:

$\beta _{n}^{HS}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{-s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.$

• Dai–Yuan:

$\beta _{n}^{DY}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{-s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.$.

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 $\displaystyle \beta =\max\{0,\beta ^{PR}\}$, which provides a direction reset automatically.

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 $O(N^{2})$ memory (but see the limited-memory L-BFGS quasi-Newton method).

The conjugate gradient method can also be derived using optimal control theory. In this accelerated optimization theory, the conjugate gradient method falls out as a nonlinear optimal feedback controller,

$u=k(x,{\dot {x}}):=-\gamma _{a}\nabla _{x}f(x)-\gamma _{b}{\dot {x}}$

for the double integrator system,

${\ddot {x}}=u$

The quantities $\gamma _{a}>0$ and $\gamma _{b}>0$ are variable feedback gains.

See also

• Gradient descent
• Broyden–Fletcher–Goldfarb–Shanno algorithm
• Conjugate gradient method
• L-BFGS (limited memory BFGS)
• Nelder–Mead method
• Wolfe conditions