Quadratic on a Sphere

Another problem naturally leading to a different secular equation is finding stationary values of a quadratic function $f$ defined by $$f(x)=\frac12 x'Ax-b'x + c$$ on the unit sphere $\{x\mid x'x=1\}$. This was first studied by Forsythe and Golub [1965]. Their treatment was subsequently simplified and extended by Spjøtvoll [1972] and Gander [1981]. The problem has recently received some attention because of the development of trust region methods for optimization, and, indeed, because of Nesterov majorization.

The stationary equations are $$\begin{align*} (A-\lambda I)x&=b,\\ x'x&=1. \end{align*}$$ Suppose $A=K\Phi K'$ with the $\phi_1\geq\cdots\geq\phi_n$ , change variables to $y=K'x$, and define $d\mathop{=}\limits^{\Delta}K'b$. Then we must solve $$\begin{align*} (\Phi-\lambda I)y&=d,\\ y'y&=1. \end{align*}$$ Assume for now that the elements of $d$ are non-zero. Then $\lambda$ cannot be equal to one of the $\phi_i$. Thus $$y_i=\frac{d_i}{\phi_i-\lambda}$$ and we must have $h(\lambda)=1$, where $$h(\lambda)\mathop{=}\limits^{\Delta}\sum_{i=1}^n\frac{d_i^2}{(\phi_i-\lambda)^2}.$$ Again, let's look at an example of a particular $h$. The plots in Figure 1 show both $f$ ad $h$. We see that $h(\lambda)=1$ has 12 solutions, so the remaining question is which one corresponds with the minimum of $f$.