A central path approaches the boundary and the Hessian conditioning diverges

The closed feasible triangle $\{x_1\ge 0,x_2\ge 0,x_1+x_2\le 1\}$ is compact, and $x_1,x_2,q$ lie in $[0,1]$. On its strict interior, each barrier term is nonnegative; as any boundary point is approached, at least one of $x_1,x_2,q$ tends to zero and its negative logarithm tends to $+\mathrm{\infty }$, while $t{x}_1$ is bounded. Fix any strict-interior point. Its objective sublevel set is therefore separated from the boundary, closed in the compact triangle by continuity, and compact; the objective attains a strict-interior minimizer. Direct differentiation gives $H_t=\left(\begin{array}{cc}{x}_1^{-2}+q^{-2}& q^{-2}\\ q^{-2}& x_2^{-2}+q^{-2}\end{array}\right)$, whose quadratic form is $h_1^2\mathrm{/}{x}_1^2+h_2^2\mathrm{/}{x}_2^2+(h_1+h_2{)}^2\mathrm{/}{q}^2>0$ for every nonzero $h$, so the minimizer is unique. At the central point, the $x_2$ derivative is $-1\mathrm{/}{x}_2+1\mathrm{/}q=0$, hence $q=x_2$ and $x_1+2x_2=1$. The $x_1$ derivative is $t-1\mathrm{/}{x}_1+1\mathrm{/}q=0$, so $1\mathrm{/}{x}_1=t+1\mathrm{/}{x}_2>t$ and $0<x_1<1\mathrm{/}t$. Consequently $x_1\to 0$ and $x_2=q\to 1\mathrm{/}2$ as $t\to \mathrm{\infty }$.

For $t\ge 4$, step 1.1 gives $x_1<1\mathrm{/}4$ and $x_2=q=(1-x_1)\mathrm{/}2>3\mathrm{/}8$. By [F2] applied to the coordinate unit vectors, ${\lambda }_{\max }(H_t)\ge (H_t{)}_{11}>x_1^{-2}>t^2$ and $0<{\lambda }_{\min }(H_t)\le (H_t{)}_{22}=x_2^{-2}+q^{-2}<2(8\mathrm{/}3{)}^2=128\mathrm{/}9$. With ${\kappa }_2(H_t):={\lambda }_{\max }(H_t)\mathrm{/}{\lambda }_{\min }(H_t)$ for this positive-definite Hessian, ${\kappa }_2(H_t)>9t^2\mathrm{/}128$, exhibiting growing conditioning as the central path approaches the boundary.