We propose a first-order method to solve the cubic regularization subproblem (CRS) based on a novel reformulation. The reformulation is a constrained convex optimization problem whose feasible region admits an easily computable projection. Our reformulation requires computing the minimum eigenvalue of the Hessian. To avoid the expensive computation of the exact minimum eigenvalue, we develop a surrogate problem to the reformulation where the exact minimum eigenvalue is replaced with an approximate one. We then apply first-order methods such as the Nesterov’s accelerated projected gradient method (APG) and projected Barzilai-Borwein method to solve the surrogate problem. As our main theoretical contribution, we show that when an \(\epsilon\)-approximate minimum eigenvalue is computed by the Lanczos method and the surrogate problem is approximately solved by APG, our approach returns an \(\epsilon\)-approximate solution to CRS in \({\tilde{O}}(\epsilon ^{-1/2})\) matrix-vector multiplications (where \({\tilde{O}}(\cdot )\) hides the logarithmic factors). Numerical experiments show that our methods are comparable to and outperform the Krylov subspace method in the easy and hard cases, respectively. We further implement our methods as subproblem solvers of adaptive cubic regularization methods, and numerical results show that our algorithms are comparable to the state-of-the-art algorithms.

我们提出了一种基于新颖的公式来解决三次正则化子问题（CRS）的一阶方法。重构是一个约束凸优化问题，其可行区域允许容易计算的投影。我们的重新公式化需要计算Hessian的最小特征值。为了避免对精确的最小特征值进行昂贵的计算，我们为重新制定公式提出了一个替代问题，在该公式中，将精确的最小特征值替换为一个近似的最小特征值。然后，我们应用一阶方法（例如Nesterov的加速投影梯度法（APG）和投影Barzilai-Borwein方法）来解决代理问题。作为我们的主要理论贡献，我们证明了当\（\ epsilon \）-approximate最小特征值是由Lanczos法和替代问题计算近似地由APG解决，我们的方法返回一个\（\小量\） -approximate溶液到CRS在\（{\代字号{Ó}}（\小量^ { -1/2}）\）矩阵向量乘法运算（其中\（{\ tilde {O}}（\ cdot）\）隐藏了对数因子）。数值实验表明，在简单和困难情况下，我们的方法分别可与Krylov子空间方法相比并胜过Krylov子空间方法。我们进一步将我们的方法实现为自适应三次正则化方法的子问题求解器，数值结果表明，我们的算法可与最新算法相媲美。