论文标题
非平滑非convex优化的信任区域方法
A Trust-Region Method For Nonsmooth Nonconvex Optimization
论文作者
论文摘要
我们为一类非平滑非convex优化问题提供了一种信任区域类型方法，其中目标函数是（可能是非convex）平滑函数和（可能是非平滑）凸函数的求和。我们的信任区域子问题的模型函数始终是二次的，模型的线性项是使用抽象下降方向生成的。因此，可以轻松构建信任区的子问题，并通过廉价和标准方法有效地解决。当模型函数在子问题解决方案上的准确性不足时，我们会在提高准确性的步骤上添加一个保障。对于可以“截断”的一类函数，定义了一个附加的截断步骤，并设计了一个台阶修改策略。整体方案在全球范围内收敛，我们在适当的假设下建立了快速的局部收敛。特别是，使用平滑的Riemannian Trust-Region方法的连接，我们证明在严格的互补条件下部分平滑功能是局部二次收敛的。报告了$ \ ell_1 $优化问题的初步数值结果，并证明了我们方法的效率。
We propose a trust-region type method for a class of nonsmooth nonconvex optimization problems where the objective function is a summation of a (probably nonconvex) smooth function and a (probably nonsmooth) convex function. The model function of our trust-region subproblem is always quadratic and the linear term of the model is generated using abstract descent directions. Therefore, the trust-region subproblems can be easily constructed as well as efficiently solved by cheap and standard methods. When the accuracy of the model function at the solution of the subproblem is not sufficient, we add a safeguard on the stepsizes for improving the accuracy. For a class of functions that can be "truncated", an additional truncation step is defined and a stepsize modification strategy is designed. The overall scheme converges globally and we establish fast local convergence under suitable assumptions. In particular, using a connection with a smooth Riemannian trust-region method, we prove local quadratic convergence for partly smooth functions under a strict complementary condition. Preliminary numerical results on a family of $\ell_1$-optimization problems are reported and demonstrate the efficiency of our approach.
