Broyden-Fletcher-Goldfarb-Shanno method

BFGS method

The unconstrained optimization problem is to minimize a real-valued function $f$ of $N$ variables. That is, to find a local minimizer, i.e. a point $x ^ { * }$ such that

\begin{equation} \tag{a1} f ( x ^ { * } ) \leq f ( x ) \text { for all } x \text{ near } x ^ { * }; \end{equation}

$f$ is typically called the cost function or objective function.

A classic approach for doing this is the method of steepest descent (cf. Steepest descent, method of). The iteration, here described in terms of the transition from a current approximation $x _ { c }$ to a local minimizer $x ^ { * }$, to an update (and hopefully better) approximation is

\begin{equation} \tag{a2} x _ { + } = x _ { c } - \lambda \nabla f ( x _ { c } ). \end{equation}

By elementary calculus, $- \nabla f ( x _ { c } )$ is the direction of most rapid decrease (steepest descent) in $f$ starting from $x _ { c }$. The step length $\lambda$ must be a part of the algorithm in order to ensure that $f ( x _ { + } ) < f ( x _ { c } )$ (which must be so for a sufficiently small $\lambda$).

There are several methods for selecting an appropriate $\lambda$, [a3], [a7], for instance the classical Armijo rule, [a1], in which $\lambda = \beta ^ { m }$ for some $\beta \in ( 0,1 )$ and where $m \geq 0$ is the least integer such that the sufficient decrease condition

\begin{equation} \tag{a3} f ( x _ { c } + \lambda d ) \leq f ( x _ { c } ) + \alpha \lambda d ^ { T } \nabla f ( x _ { c } ) \end{equation}

holds. In (a3), $- \nabla f ( x _ { c } )$ is replaced by a general descent direction $d$, a vector satisfying $d ^ { T } \nabla f ( x _ { c } ) < 0$, and a parameter $\alpha$ has been introduced (typically $10 ^ { - 4 }$). One might think that simple decrease ($f ( x _ { c } + \lambda d ) < f ( x _ { c } )$) would suffice, and it usually does in practice. However, the theory requires (a3) to eliminate the possibility of stagnation.

The steepest descent iteration with the Armijo rule will, if $f$ is smooth and bounded away from $- \infty$, which is assumed throughout below, produce a sequence $\{ x _ { n } \}$ such that

\begin{equation} \tag{a4} \operatorname { lim } _ { n \rightarrow \infty } \nabla f ( x _ { n } ) = 0. \end{equation}

Hence, any limit point $x ^ { * }$ of the steepest descent sequence satisfies the first-order necessary conditions $\nabla f ( x ^ { * } ) = 0$. This does not imply convergence to a local minimizer and even if the steepest descent iterations do converge, and they often do, the convergence is usually very slow in the terminal phase when the iterates are near $x ^ { * }$. Scaling the steepest descent direction by multiplying it with the inverse of a symmetric positive-definite scaling matrix $H _ { c }$ can preserve the convergence of $\nabla f$ to zero and dramatically improve the convergence near a minimizer.

Now, if $x ^ { * }$ is a local minimizer that satisfies the standard assumptions, i.e. the Hessian of $f$, $\nabla ^ { 2 } f$, is symmetric, positive definite, and Lipschitz continuous near $x ^ { * }$, then the Newton method,

\begin{equation} \tag{a5} x _ { + } = x _ { c } - ( \nabla ^ { 2 } f ( x _ { c } ) ) ^ { - 1 } \nabla f ( x _ { c } ), \end{equation}

will converge rapidly to the minimizer if the initial iterate $x _ { 0 }$ is sufficiently near $x ^ { * }$, [a3], [a7]. However, when far from $x ^ { * }$, the Hessian need not be positive definite and cannot be used as a scaling matrix.

Quasi-Newton methods (cf. also Quasi-Newton method) update an approximation of $\nabla ^ { 2 } f ( x ^ { * } )$ as the iteration progresses. In general, the transition from current approximations $x _ { c }$ and $H _ { c }$ of $x ^ { * }$ and $\nabla f ( x ^ { * } )$ to new approximations $x_{+}$ and $H _ { + }$ is given (using a line search paradigm) by:

1) compute a search direction $d = - H _ { c } ^ { - 1 } \nabla f ( x _ { c } )$;

2) find $x _ { + } = x _ { c } + \lambda d$ using a line search to ensure sufficient decrease;

3) use $x _ { c }$, $x_{+}$, and $H _ { c }$ to update $H _ { c }$ and obtain $H _ { + }$. The way in which $H _ { + }$ is computed determines the method.

The BFGS method, [a2], [a4], [a5], [a6], updates $H _ { c }$ with the rank-two formula

\begin{equation} \tag{a6} H _ { + } = H _ { c } + \frac { y y ^ { T } } { y ^ { T } s } - \frac { ( H _ { c } s ) ( H _ { c } s ) ^ { T } } { s ^ { T } H _ { c } s }. \end{equation}

In (a6),

\begin{equation*} s = x _ { + } - x _ { c } , \quad y = \nabla f ( x _ { + } ) - \nabla f ( x _ { c } ). \end{equation*}

Local and global convergence theory.

The local convergence result, [a8], [a3], [a7], assumes that the standard assumptions hold and that the initial data is sufficiently good (i.e., $x _ { 0 }$ is near to $x ^ { * }$ and $H _ { 0 }$ is near $\nabla ^ { 2 } f ( x ^ { * } )$). Under these assumptions the BFGS iterates exist and converge $q$-superlinearly to $x ^ { * }$, i.e.,

\begin{equation} \tag{a7} \operatorname { lim } _ { n \rightarrow \infty } \frac { \left\| x _ { n + 1} - x ^ { * } \right\| } { \left\| x _ { n } - x ^ { * } \right\| } = 0. \end{equation}

By a global convergence theory one means that one does not assume that the initial data is good and one manages the poor data by making certain that $f$ is decreased as the iteration progresses. There are several variants of global convergence theorems for BFGS and related methods, [a9], [a11], [a10], [a12]. All these results make strong assumptions on the function and some require line search methods more complicated that the simple Armijo rule discussed above.

The result from [a9] is the most general and a special case illustrating the idea now follows. One must assume that the set

\begin{equation*} D = \{ x : f ( x ) \leq f ( x _ { 0 } ) \} \end{equation*}

is convex, $f$ is Lipschitz twice continuously differentiable in $D$, and that the Hessian and its inverse are uniformly bounded and positive definite in $D$. This assumption implies that $f$ has a unique minimizer $x ^ { * }$ in $D$ and that the standard assumptions hold. If $H _ { 0 }$ is symmetric and positive definite, the BFGS iterations, using the Armijo rule to enforce (a3), converge $q$-superlinearly to $x ^ { * }$.

Implementation.

The recursive implementation from [a7], described below, stores the history of the iteration and uses that information recursively to compute the action of $H _ { k } ^{- 1}$ on a vector. This idea was suggested in [a13], [a14], [a17]. Other implementations may be found in [a15], [a3], [a7], and [a16].

One can easily show that

\begin{equation} \tag{a8} H _ { + } ^ { - 1 } = \left( I - \frac { s y ^ { T } } { y ^ { T } s } \right) H _ { c } ^ { - 1 } \left( I - \frac { y s ^ { T } } { y ^ { T } s } \right) + \frac { s s ^ { T } } { y ^ { T } s }. \end{equation}

The algorithm $\operatorname{bfgsrec}$ below overwrites a given vector $d$ with $H _ { n}^{ - 1} d$. The storage needed is one vector for $d$ and $2 n$ vectors for the sequences $\{ s _ { k } , y _ { k } \} _ { k = 0 } ^ { n - 1 }$. A method for computing the product of $H _ { 0 } ^ { - 1 }$ and a vector must also be provided.

'

1 IF $n = 0$, $d = H _ { 0 } ^ { - 1 } d$; return 2 $\alpha = s _ { n-1 } ^ { T } d / y _ { n-1 } ^ { T }s$; $d = d - \alpha y _ { n - 1}$ call $ \operatorname{bfgsrec} ( n - 1 , \{ s _ { k } \} , \{ y _ { k } \} , H _ { 0 } ^ { - 1 } , d )$; 3 $d = d + ( \alpha - ( y _ { n-1 } ^ { T } { d } / y _ { n - 1 } ^ { T } s _ { n - 1 } ) s _ { n - 1 }$.

References

[a1]	L. Armijo, "Minimization of functions having Lipschitz-continuous first partial derivatives" Pacific J. Math. , 16 (1966) pp. 1–3
[a2]	C.G. Broyden, "A new double-rank minimization algorithm" Notices Amer. Math. Soc. , 16 (1969) pp. 670
[a3]	J.E. Dennis, R.B. Schnabel, "Numerical Methods for Nonlinear Equations and Unconstrained Optimization" , Classics in Applied Math. : 16 , SIAM (Soc. Industrial Applied Math.) (1996)
[a4]	R. Fletcher, "A new approach to variable metric methods" Comput. J. , 13 (1970) pp. 317–322
[a5]	D. Goldfarb, "A family of variable metric methods derived by variational means" Math. Comp. , 24 (1970) pp. 23–26
[a6]	D.F. Shanno, "Conditioning of quasi-Newton methods for function minimization" Math. Comp. , 24 (1970) pp. 647–657
[a7]	C.T. Kelley, "Iterative methods for optimization" , Frontiers in Appl. Math. , 18 , SIAM (Soc. Industrial Applied Math.) (1999)
[a8]	C.G. Broyden, J.E. Dennis, J.J. Moré, "On the local and superlinear convergence of quasi-Newton methods" J. Inst. Math. Appl. , 12 (1973) pp. 223–246
[a9]	R.H. Byrd, J. Nocedal, "A tool for the analysis of quasi-Newton methods with application to unconstrained minimization" SIAM J. Numer. Anal. , 26 (1989) pp. 727–739
[a10]	R.H. Byrd, J. Nocedal, Y. Yuan, "Global convergence of a class of quasi-Newton methods on convex problems" SIAM J. Numer. Anal. , 24 (1987) pp. 1171–1190
[a11]	M.J.D. Powell, "Some global convergence properties of a variable metric algorithm without exact line searches" R. Cottle (ed.) C. Lemke (ed.) , Nonlinear Programming , Amer. Math. Soc. (1976) pp. 53–72
[a12]	J. Werner, "Über die globale konvergenz von Variable-Metric Verfahren mit nichtexakter Schrittweitenbestimmung" Numer. Math. , 31 (1978) pp. 321–334
[a13]	K.J. Bathe, A.P. Cimento, "Some practical procedures for the solution of nonlinear finite element equations" Comput. Meth. Appl. Mech. Eng. , 22 (1980) pp. 59–85
[a14]	H. Matthies, G. Strang, "The solution of nonlinear finite element equations" Internat. J. Numerical Methods Eng. , 14 (1979) pp. 1613–1626
[a15]	R.H. Byrd, J. Nocedal, R.B. Schnabel, "Representation of quasi-Newton matrices and their use in limited memory methods" Math. Progr. , 63 (1994) pp. 129–156
[a16]	J.L. Nazareth, "Conjugate gradient methods less dependent on conjugacy" SIAM Review , 28 (1986) pp. 501–512
[a17]	J. Nocedal, "Updating quasi-Newton matrices with limited storage" Math. Comp. , 35 (1980) pp. 773–782