= Quasi-Newton secant equation
{c}
{title2=$B_{k+1}p_k=y_k$}
= Secant equation in optimization
{synonym}
For $p_k=x_{k+1}-x_k$ and $y_k=\nabla f(x_{k+1})-\nabla f(x_k)$, the <Quasi-Newton secant equation> requires $B_{k+1}p_k=y_k$. It reproduces the observed change of <gradient> along the latest step. If $B_{k+1}$ is a <positive-definite symmetric matrix> and $p_k\ne0$, necessarily $p_k^Ty_k=p_k^TB_{k+1}p_k>0$. A <line search> satisfying the weak <Wolfe conditions> ensures this: with $p_k=\alpha_ks_k$,
$$
p_k^Ty_k\geq\alpha_k(\beta_2-1)\nabla f(x_k)^Ts_k>0.
$$
This necessary curvature condition is also sufficient for the existence of some <positive-definite symmetric matrix> mapping $p_k$ to $y_k$. In an <orthonormal basis> whose first vector is $p_k/\|p_k\|$, its first column must be $(a,b)^T=y_k/\|p_k\|$, with $a=p_k^Ty_k/\|p_k\|^2>0$. The block <matrix> with first row $(a,b^T)$ and lower-right block $bb^T/a+I$ has a positive <Schur complement>, so is <positive-definite>.
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