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A Quasi-Newton method that maintains a symmetric curvature approximation for differentiable optimization. For a positive-definite symmetric matrix $H$ approximating inverse curvature, step $s$, gradient change $y$ and $\rho =1/(y^{\mathsf{T}}s)>0$, its inverse update isMultiplying by $y$ gives $H_{+}y=s$, the inverse form of the Quasi-Newton secant equation. The update is a symmetric matrix. For any nonzero $v$,Both terms are nonnegative, and both can vanish only if $s^{\mathsf{T}}v=0$ and then $v=0$. Thus positive definiteness is preserved when $y^{\mathsf{T}}s>0$, making $-H_{+}\nabla f$ a descent direction away from a stationary point. Full updates use quadratic storage in the dimension; the method does not require explicit Hessian matrix evaluations. Superlinear convergence requires additional local smoothness and line search assumptions.^[ref]

A mathematician associated with Broyden's method and Quasi-Newton methods in numerical analysis. An institutional portrait is catalogued by the University of Essex Library Services: www.flickr.com/photos/149590996@N07/31601668406/

Limited-memory BFGS is a Quasi-Newton method that applies an inverse-Hessian matrix approximation without storing a dense square matrix. It retains only a fixed number $\ell$ of recent secant pairs $s_k=x_{k+1}-x_k$ and $y_k=\nabla f(x_{k+1})-\nabla f(x_k)$. The standard two-loop recursion computes the search direction using $O(\ell d)$ storage and arithmetic in dimension $d$.

With a positive-definite initial metric and safeguarded curvature updates, the step is $x_{k+1}=x_k-{\alpha }_k{H}_k\nabla f(x_k)$, with ${\alpha }_k$ chosen by a line search. It is useful when full-memory BFGS would require too much storage. Fixed-memory convergence results typically give linear convergence under strong convexity, bounded metric and suitable line search assumptions; one must not automatically transfer the full-memory superlinear theorem. In a nonconvex problem, a local method does not by itself certify a global minimum.^[ref]

For fixed vectors $a_1,\dots ,a_m$ in a finite-dimensional Euclidean space, defineThe gradient and Hessian matrix of $\Phi =\log F$ areTo derive the first identity, differentiate each exponential function and divide the sum of its gradients by $F$. For the second, differentiating a normalized weight gives $\nabla w_i=2w_i(\bar{a}-a_i)$. Thus $J_{\bar{a}}=-2{\mathrm{Cov}}_w(a_i)$, where the weighted covariance matrix is the sum in the displayed Hessian matrix. Differentiating $2(x-\bar{a})$ completes the calculation.

The weighted covariance matrix is positive semidefinite, so ${\nabla }^2\Phi ⪰2I$. Also $\Phi (x)\ge \Vert x-a_1{\Vert }^2\to \infty$ as $\Vert x\Vert \to \infty$. Therefore $F$ and $\Phi$ share a unique global minimizer, characterized by $x=\bar{a}(x)$. The weights place that minimizer in the convex hull of the centres.

For stable numerical evaluation, subtract the largest exponent before computing the normalized weights. This is the log-sum-exp principle and avoids floating-point overflow without changing the gradient. A naive iteration $x_{n+1}=\bar{a}(x_n)$ need not converge: in one dimension, centres $-1,1$ give $\bar{a}(x)=-\tanh (2x)$ with derivative $-2$ at its unique fixed point. A Newton root-finding iteration or a suitable Quasi-Newton method instead uses the positive curvature of $\Phi$.

Set $p=x^{k+1}-x^k={\alpha }^k{s}^k$ and $y=\nabla f(x^{k+1})-\nabla f(x^k)$. The Quasi-Newton secant equation is $B^{k+1}p=y$. If $B^{k+1}$ is a positive-definite symmetric matrix and $p\ne 0$, it follows thatThe weak curvature part of the Wolfe conditions ensures precisely this necessary condition:The last product is positive because ${\alpha }^k>0$, ${\beta }_2-1<0$, and $s^k$ is a descent direction. This explains why a Quasi-Newton method often uses the Wolfe conditions rather than only the Armijo condition.

A practical choice is a BFGS method, using exact gradients and a line search. At dimension $d=100$, its $d$-by-$d$ inverse-Hessian matrix approximation is small, whereas forming a million rank-one contributions to the exact Hessian matrix is expensive. If $F(x)={\sum }_i{e}^{\Vert x-a_i{\Vert }^2}$, the explicit derivatives areThe positive-definite Hessian matrix and coercivity show that the objective has one global minimizer.

For numerical stability one can minimize $\Phi =\log F$ instead, which has exactly the same minimizer. Put $r_i=\Vert x-a_i{\Vert }^2$, $w_i=e^{r_i}/{\sum }_j{e}^{r_j}$, and $\bar{a}={\sum }_i{w}_i{a}_i$. Then the logarithmic gradient of an exponential squared-distance sum givesEvaluate the weights as $e^{r_i-r_{\max }}/{\sum }_j{e}^{r_j-r_{\max }}$ to avoid floating-point overflow. A Quasi-Newton method step iswhere $H_n$ is the positive-definite inverse-Hessian matrix approximation updated by BFGS, and ${\alpha }_n$ is chosen by a line search, for example one satisfying the Wolfe conditions. Under the standard smoothness and nondegeneracy assumptions at this strongly convex minimizer, full-memory BFGS has local superlinear convergence. An exact Newton's method in optimization would have local quadratic convergence, but pays for those much more expensive exact second derivatives.^[ref]

A Quasi-Newton method replaces the Hessian matrix in Newton's method in optimization by a matrix $B_k$ learned from successive gradients. Solving $B_k{s}_k=-\nabla f(x_k)$ gives a descent direction whenever $B_k$ is a positive-definite symmetric matrix and the gradient is nonzero. After choosing a step size, one updates $B_k$ to satisfy the Quasi-Newton secant equation, rather than calculating every second derivative again.

The NEOS Guide's mathematical account of Quasi-Newton methods, deriving the secant condition from an integrated Hessian matrix and describing low-rank updating, including BFGS. The related nonlinear-equation account is neos-guide.org/guide/algorithms/broyden/ and discusses Broyden's method.