Multibody system

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

$f(x)=\Vert Ax-b{\Vert }^2$

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

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

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

$\mathrm{\Delta }{x}_0=-{\mathrm{\nabla }}_{x}f(x_0)$

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

${\alpha }_0:=\arg {\min }_{\alpha }f(x_0+\alpha \mathrm{\Delta }{x}_0)$,

$x_1=x_0+{\alpha }_0\mathrm{\Delta }{x}_0$

After this first iteration in the steepest direction $\mathrm{\Delta }{x}_0$, the following steps constitute one iteration of moving along a subsequent conjugate direction $s_n$, where $s_0=\mathrm{\Delta }{x}_0$:

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

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 $\alpha$ and $\beta$ 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 ${\beta }_n$ are titled Fletcher-Reeves (FR), Polak-Ribière (PR), and Hestenes-Stiefel (HS) after their developers. They are given by the following formulas:

• Fletcher–Reeves:

${\beta }_n^{FR}=\frac{\mathrm{\Delta }{x}_n^{\mathrm{\top }}\mathrm{\Delta }{x}_n}{\mathrm{\Delta }{x}_{n-1}^{\mathrm{\top }}\mathrm{\Delta }{x}_{n-1}}$

• Polak–Ribière:

${\beta }_n^{PR}=\frac{\mathrm{\Delta }{x}_n^{\mathrm{\top }}(\mathrm{\Delta }{x}_n-\mathrm{\Delta }{x}_{n-1})}{\mathrm{\Delta }{x}_{n-1}^{\mathrm{\top }}\mathrm{\Delta }{x}_{n-1}}$

• Hestenes-Stiefel:

${\beta }_n^{HS}=-\frac{\mathrm{\Delta }{x}_n^{\mathrm{\top }}(\mathrm{\Delta }{x}_n-\mathrm{\Delta }{x}_{n-1})}{s_{n-1}^{\mathrm{\top }}(\mathrm{\Delta }{x}_n-\mathrm{\Delta }{x}_{n-1})}$.

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 $\beta =\max \{0,{\beta }^{PR}\}$ 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).