  A bi-level optimization framework (BiOPT) was proposed in [3] for convex
composite optimization, which is a generalization of bi-level unconstrained
minimization framework (BLUM) given in [20].
  In this continuation paper, we introduce a $p$th-order proximal-point segment
search operator which is used to develop novel accelerated methods. Our first
algorithm combines the exact element of this operator with the estimating
sequence technique to derive new iteration points, which are shown to attain
the convergence rate $\mathcal{O}(k^{-(3p+1)/2})$, for the iteration counter
$k$.
  We next consider inexact elements of the high-order proximal-point segment
search operator to be employed in the BiOPT framework. We particularly apply
the accelerated high-order proximal-point method at the upper level, while we
find approximate solutions of the proximal-point segment search auxiliary
problem by a combination of non-Euclidean composite gradient and bisection
methods. For $q=\lfloor p/2 \rfloor$, this amounts to a $2q$th-order method
with the convergence rate $\mathcal{O}(k^{-(6q+1)/2})$ (the same as the optimal
bound of $2q$th-order methods) for even $p$ ($p=2q$) and the superfast
convergence rate $\mathcal{O}(k^{-(3q+1)})$ for odd $p$ ($p=2q+1$).
