Subgradient methods are the natural extension to the non-smooth case of the classical gradient descent for regular convex optimization problems. However, in general, they are characterized by slow convergence rates, and they require decreasing step-sizes to converge. In this paper we propose a subgradient method with constant step-size for composite convex objectives with $\ell_1$-regularization. If the smooth term is strongly convex, we can establish a linear convergence result for the function values. This fact relies on an accurate choice of the element of the subdifferential used for the update, and on proper actions adopted when non-differentiability regions are crossed. Then, we propose an accelerated version of the algorithm, based on conservative inertial dynamics and on an adaptive restart strategy. Finally, we test the performances of our algorithms on some strongly and non-strongly convex examples.

翻译：次梯度方法是经典梯度下降法在正则凸优化问题中针对非光滑情形的自然推广。然而，一般情况下，这类方法收敛速度较慢，且需采用递减步长方可收敛。本文针对含$\ell_1$正则化的复合凸目标函数，提出一种基于常步长的次梯度方法。当光滑项满足强凸性时，我们可建立关于函数值的线性收敛结果。该结论依赖于更新过程中次微分元素的精确选取，以及穿越不可微区域时所采用的恰当策略。在此基础上，我们结合保守惯性动力学与自适应重启策略，提出了算法的加速版本。最后，通过若干强凸与非强凸算例检验了所提算法的性能。