Finite exhaustive active-set solve for a box-constrained positive quadratic

Let $H\in {\mathbb{R}}^{n\times n}$ be symmetric positive definite, $c\in {\mathbb{R}}^n$, and $\mathrm{\ell },u\in {\mathbb{R}}^n$ with ${\mathrm{\ell }}_i\le u_i$. For $\varphi (x)=\frac{1}{2}{x}^{T}Hx+c^{T}x$, enumerate the finite set of status vectors $\sigma \in \{L,F,U{\}}^n$ in lexicographic order. For each status, set $x_i={\mathrm{\ell }}_i$ if ${\sigma }_i=L$, set $x_i=u_i$ if ${\sigma }_i=U$, and for the free indices $F$ solve

$$H_{FF}{x}_F=-c_F-H_{FA}{x}_A,$$

where $A=L\cup U$. Skip a singular solve. Retain a candidate only if ${\mathrm{\ell }}_i<x_i<u_i$ for $i\in F$, and the gradient $g=Hx+c$ satisfies $g_i\ge 0$ on $L$, $g_i=0$ on $F$, and $g_i\le 0$ on $U$. Return the first retained candidate; if none is retained, return no-candidate. Computations in this definition are exact real arithmetic. If ${\mathrm{\ell }}_i=u_i$, that coordinate may be assigned status $L$ or $U$.