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:

$\nabla_x f=2 A^\top(Ax-b)=0$.

Whereas linear conjugate gradient seeks a solution to the linear equation $\displaystyle A^\top Ax=A^\top b$, 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.

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 $\displaystyle \Lambda x_{n}$, where $\displaystyle \Lambda x_{0}=\Delta x_{0}$:

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: $\displaystyle \Lambda x_{n}=\Delta x_{n}+\beta_{n} \Lambda x_{n-1}$
4. Perform a line search: optimize $\displaystyle \alpha_n$, $\arg \min_{\alpha_n} f(x_n+\alpha_n \Lambda x_n)$,
5. Update the position: $\displaystyle x_{n+1}=x_{n}+\alpha_{n} \Lambda x_{n}$,

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.

Three of the best known formulas for $\displaystyle \beta_n$ are titled Fletcher-Reeves (FR), Polak-Ribière (PR), and Hestenes-Stiefel (HS) after their developers. They are given by the following formulas:

• Fletcher–Reeves:

$\beta_{n}^{FR} = \frac{\Delta x_n^\top \Delta x_n} {\Delta x_{n-1}^\top \Delta x_{n-1}}$

• Polak–Ribière:

$\beta_{n}^{PR} = \frac{\Delta x_n^\top (\Delta x_n-\Delta x_{n-1})} {\Delta x_{n-1}^\top \Delta x_{n-1}}$

• Hestenes-Stiefel:

$\beta_{n}^{HS} = \frac{\Delta x_n^\top (\Delta x_n-\Delta x_{n-1})} {\Delta x_{n-1}^\top (\Delta x_{n}-\Delta x_{n-1})}$.

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.

Newton based methods - Newton-Raphson Algorithm, Quasi-Newton methods (e.g., BFGS method) - tend to converge in fewer iterations, although each iteration typically requires more computation than a conjugate gradient iteration as Newton-like methods require computing the Hessian (matrix of second derivatives) in addition to the gradient. Quasi-Newton methods also require more memory to operate (see also the limited memory L-BFGS method).

External links

• An Introduction to the Conjugate Gradient Method Without the Agonizing Pain by Jonathan Richard Shewchuk.

• Numerical Recipes in C - The Art of Scientific Computing, chapter 10, section 6: Conjugate Gradient Methods in Multidimensions; William H. Press (Editor), Saul A. Teukolsky (Editor), William T. Vetterling (Author), Brian P. Flannery (Author), Cambridge University Press; 2nd edition (1992).