  This paper proposes an efficient adaptive variant of a quadratic penalty
accelerated inexact proximal point (QP-AIPP) method proposed earlier by the
authors. Both the QP-AIPP method and its variant solve linearly set constrained
nonconvex composite optimization problems using a quadratic penalty approach
where the generated penalized subproblems are solved by a variant of the
underlying AIPP method. The variant, in turn, solves a given penalized
subproblem by generating a sequence of proximal subproblems which are then
solved by an accelerated composite gradient algorithm. The main difference
between AIPP and its variant is that the proximal subproblems in the former are
always convex while the ones in the latter are not necessarily convex due to
the fact that their prox parameters are chosen as aggressively as possible so
as to improve efficiency. The possibly nonconvex proximal subproblems generated
by the AIPP variant are also tentatively solved by a novel adaptive accelerated
composite gradient algorithm based on the validity of some key convergence
inequalities. As a result, the variant generates a sequence of proximal
subproblems where the stepsizes are adaptively changed according to the
responses obtained from the calls to the accelerated composite gradient
algorithm. Finally, numerical results are given to demonstrate the efficiency
of the proposed AIPP and QP-AIPP variants.
