论文标题
基于大化最小化的Levenberg-摩尔夸伯特的方法，用于受约束的非线性最小二乘
Majorization-Minimization-Based Levenberg--Marquardt Method for Constrained Nonlinear Least Squares
论文作者
论文摘要
描述了一种新的Levenberg--marquardt（LM）方法，用于求解具有凸约限制的非线性最小二乘问题。已经提出了各种版本的LM方法，其主要差异是选择阻尼参数。在本文中，我们提出了一个新规则，以更新参数，以便即使在存在凸约限制集的存在下也可以实现全局和局部收敛。我们结果的关键是从多数化最小化方法的LM方法的新观点。具体而言，我们表明，如果以特定方式设置阻尼参数，则在某些标准假设下，LM方法中标准子问题的目标函数成为原始目标函数上的上限。 我们的方法使用（加速）投影梯度方法求解了一个子问题的序列。它在$ o（ε^{ - 2}）$计算后找到了一个$ε$ -Stationary点，并在本地错误绑定条件下实现了零分离问题的本地二次收敛。压缩传感和基质分解的数值结果表明，在许多情况下，我们的方法比现有方法更快。
A new Levenberg--Marquardt (LM) method for solving nonlinear least squares problems with convex constraints is described. Various versions of the LM method have been proposed, their main differences being in the choice of a damping parameter. In this paper, we propose a new rule for updating the parameter so as to achieve both global and local convergence even under the presence of a convex constraint set. The key to our results is a new perspective of the LM method from majorization-minimization methods. Specifically, we show that if the damping parameter is set in a specific way, the objective function of the standard subproblem in LM methods becomes an upper bound on the original objective function under certain standard assumptions. Our method solves a sequence of the subproblems approximately using an (accelerated) projected gradient method. It finds an $ε$-stationary point after $O(ε^{-2})$ computation and achieves local quadratic convergence for zero-residual problems under a local error bound condition. Numerical results on compressed sensing and matrix factorization show that our method converges faster in many cases than existing methods.
