Nonlinear conjugate gradient method

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

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

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

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

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

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

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

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

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

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

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

• Fletcher–Reeves:^[1]

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

• Polak–Ribière:^[2]

${\displaystyle \beta _{n}^{PR}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.}${\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}^{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:^[4]

${\displaystyle \beta _{n}^{DY}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{-s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.}${\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 ${\displaystyle \displaystyle \beta =\max\{0,\beta ^{PR}\}}${\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 ${\displaystyle O(N^{2})}${\displaystyle 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.^[6] In this accelerated optimization theory, the conjugate gradient method falls out as a nonlinear optimal feedback controller,

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

for the double integrator system,

${\displaystyle {\ddot {x}}=u}${\displaystyle {\ddot {x}}=u}

The quantities ${\displaystyle \gamma _{a}>0}${\displaystyle \gamma _{a}>0} and ${\displaystyle \gamma _{b}>0}${\displaystyle \gamma _{b}>0} are variable feedback gains.^[6]

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

• Optimization algorithms and methods
• Gradient methods