Complexity of an inexact proximal-point penalty method for constrained smooth non-convex optimization
Complexity of an inexact proximal-point penalty method for constrained smooth non-convex optimization
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DOI：
10.1007/s10589-022-00358-y
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发表时间：
2022-03
影响因子：
2.2
通讯作者：
Qihang Lin;Runchao Ma;Yangyang Xu
中科院分区：
文献类型：
--
作者：
Qihang Lin;Runchao Ma;Yangyang Xu
本文研究了约束优化问题的一种非精确逼近点罚方法，其中目标函数是非凸的，约束函数也可以是非凸的。该方法近似求解一系列子问题，每个子问题都是通过向原始目标函数添加与约束函数相关联的邻近项和二次惩罚项而形成的。在弱凸性假设下，每个子问题都是强凸的，并且可以通过基于最优梯度的方法有效地求解到所需的精度。分别对凸约束和非凸约束的情况分析了该方法的计算复杂度。对于这两种情况下，复杂性的结果建立在寻找一个固定点所需的近端梯度步骤的数量。当约束函数是凸的时，在斯莱特条件下，我们给出了产生-驻点的一个复杂性结果。当约束函数为非凸函数时，如果约束满足非奇异性条件，则问题的复杂性变为存在可行初始解。
In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. This method approximately solves a sequence of subproblems, each of which is formed by adding to the original objective function a proximal term and quadratic penalty terms associated to the constraint functions. Under a weak-convexity assumption, each subproblem is made strongly convex and can be solved effectively to a required accuracy by an optimal gradient-based method. The computational complexity of this approach is analyzed separately for the cases of convex constraint and non-convex constraint. For both cases, the complexity results are established in terms of the number of proximal gradient steps needed to find an-stationary point. When the constraint functions are convex, we show a complexity result ofto produce an-stationary point under the Slater’s condition. When the constraint functions are non-convex, the complexity becomesif a non-singularity condition holds on constraints and otherwiseif a feasible initial solution is available.