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.

中文翻译：

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一种解决线性约束非凸复合问题的有效自适应加速不精确近点方法

本文提出了作者较早提出的二次惩罚加速不精确近点（QP-AIPP）方法的有效自适应变体。QP-AIPP方法及其变体都使用二次惩罚方法来求解线性集约束的非凸复合优化问题，其中所产生的惩罚子问题由基础AIPP方法的变体解决。反过来，该变体通过生成一系列近端子问题来解决给定的惩罚子问题，然后通过加速复合梯度算法对其进行求解。AIPP及其变体之间的主要区别在于，前者中的近端子问题始终是凸的，而后者中的子问题则不一定是凸的，这是因为其代理参数应尽可能积极地选择以提高效率。基于某些关键收敛不等式的有效性，还通过新颖的自适应加速复合梯度算法尝试性地解决了AIPP变体产生的可能非凸近端子问题。结果，该变体生成一系列近端子问题，其中，根据从对加速复合梯度算法的调用中获得的响应，自适应地更改步长。最后，数值结果表明了所提出的AIPP和QP-AIPP变体的效率。该变体生成一系列近端子问题，在这些子问题中，根据从对加速复合梯度算法的调用获得的响应，自适应地更改步长。最后，数值结果表明了所提出的AIPP和QP-AIPP变体的效率。该变体生成一系列近端子问题，其中逐步大小根据从对加速复合梯度算法的调用获得的响应而自适应地更改。最后，数值结果表明了所提出的AIPP和QP-AIPP变体的效率。