Abstract

We propose a coderivative-based generalized regularized Newton method with positive definite regularization term (GRNM-PD) to solve \(C^{1,1}\) optimization problems. In GRNM-PD, a general positive definite symmetric matrix is used to regularize the generalized Hessian, in contrast to the recently proposed GRNM, which uses the identity matrix. Our approach features global convergence and fast local convergence rate even for problems with nonisolated solutions. To this end, we introduce the p-order semismooth \({}^*\) property which plays the same role in our analysis as Lipschitz continuity of the Hessian does in the \(C^2\) case. Imposing only the metric q-subregularity of the gradient at a solution, we establish global convergence of the proposed algorithm as well as its local convergence rate, which can be superlinear, quadratic, or even higher than quadratic, depending on an algorithmic parameter \(\rho \) and the regularity parameters p and q. Specifically, choosing \(\rho \) to be one, we achieve quadratic local convergence rate under metric subregularity and the strong semismooth \({^*}\) property. The algorithm is applied to a class of nonsmooth convex composite minimization problems through the machinery of forward–backward envelope. The greater flexibility in the choice of regularization matrices leads to notable improvement in practical performance. Numerical experiments on box-constrained quadratic programming problems demonstrate the efficiency of our algorithm.

中文翻译：

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具有正定正则化的广义牛顿法求解非孤立解的非光滑优化问题

摘要

我们提出了一种基于编码导数的广义正则化牛顿方法，具有正定正则化项（GRNM-PD）来解决\(C^{1,1}\)优化问题。在GRNM-PD中，使用广义正定对称矩阵来正则化广义Hessian矩阵，这与最近提出的使用单位矩阵的GRNM形成鲜明对比。我们的方法具有全局收敛和快速局部收敛速度，即使对于非孤立解决方案的问题也是如此。为此，我们引入p阶半光滑\({}^*\)属性，它在我们的分析中起着与 Hessian 的 Lipschitz 连续性在\(C^2\)情况下相同的作用。仅在解中施加梯度的度量q次正则性，我们建立了所提出算法的全局收敛性及其局部收敛速率，其可以是超线性的、二次的，甚至高于二次的，具体取决于算法参数 \ ( \rho \)和正则参数p和q。具体来说，选择\(\rho \)为 1，我们在度量次正则性和强半光滑\({^*}\)性质下实现了二次局部收敛速度。该算法通过前向后包络机制应用于一类非光滑凸复合最小化问题。正则化矩阵选择的更大灵活性可以显着提高实际性能。盒约束二次规划问题的数值实验证明了我们算法的效率。