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, that is guaranteed to achieve a linear convergence rate in the strongly convex case. 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非光滑情形的自然推广。然而，这类方法通常收敛速度缓慢，且需采用递减步长才能实现收敛。本文针对带有$\ell_1$正则化的复合凸目标函数，提出一种常数步长的次梯度方法。当光滑项为强凸时，我们可建立函数值的线性收敛结果。这一结论依赖于对更新所用次微分元素的精确选择，以及在穿越不可微区域时采取的恰当处理策略。在此基础上，我们进一步提出该算法的加速版本，该版本基于保守惯性动力学与自适应重启策略，可确保在强凸情形下实现线性收敛速率。最后，我们通过若干强凸与非强凸实例验证了所提算法的性能。