In this paper, we study the complexity of the forward–backward splitting method with Beck–Teboulle’s line search for solving convex optimization problems, where the objective function can be split into the sum of a differentiable function and a nonsmooth function. We show that the method converges weakly to an optimal solution in Hilbert spaces, under mild standing assumptions without the global Lipschitz continuity of the gradient of the differentiable function involved. Our standing assumptions is weaker than the corresponding conditions in the paper of Salzo (SIAM J Optim 27:2153–2181, 2017). The conventional complexity of sublinear convergence for the functional value is also obtained under the local Lipschitz continuity of the gradient of the differentiable function. Our main results are about the linear convergence of this method (in the quotient type), in terms of both the function value sequence and the iterative sequence, under only the quadratic growth condition. Our proof technique is direct from the quadratic growth conditions and some properties of the forward–backward splitting method without using error bounds or Kurdya-Łojasiewicz inequality as in other publications in this direction.

中文翻译：

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关于前向后向分裂法的线性收敛性：第一部分——收敛性分析

在本文中，我们研究了使用 Beck-Teboulle 线搜索解决凸优化问题的前向 - 后向分裂方法的复杂性，其中目标函数可以分为可微函数和非光滑函数的总和。我们表明，在没有所涉及的可微函数梯度的全局 Lipschitz 连续性的温和常设假设下，该方法弱收敛到希尔伯特空间中的最优解。我们的常设假设弱于 Salzo 论文中的相应条件（SIAM J Optim 27:2153–2181, 2017）。函数值的次线性收敛的常规复杂度也是在可微函数梯度的局部 Lipschitz 连续性下获得的。我们的主要结果是关于该方法（商型）的线性收敛，在函数值序列和迭代序列方面，仅在二次增长条件下。我们的证明技术直接来自二次增长条件和前向后向分裂方法的一些性质，而不像在这个方向的其他出版物中那样使用误差界限或 Kurdya-Łojasiewicz 不等式。