# An efficient adaptive accelerated inexact proximal point method for solving linearly constrained nonconvex composite problems

## TL;DR

This paper introduces an adaptive accelerated inexact proximal point method for efficiently solving linearly constrained nonconvex composite optimization problems, improving upon previous methods with adaptive strategies and nonconvex subproblem handling.

## Contribution

It develops a novel adaptive variant of the quadratic penalty accelerated inexact proximal point method that handles nonconvex subproblems more efficiently.

## Key findings

- The proposed methods outperform existing approaches in numerical tests.
- Adaptive stepsize adjustment improves convergence speed.
- The methods effectively solve large-scale nonconvex constrained problems.

## Abstract

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.