We propose a new approach to perform the boosted difference of convex functions algorithm (BDCA) on non-smooth and non-convex problems involving the difference of convex (DC) functions. The recently proposed BDCA uses an extrapolation step from the point computed by the classical DC algorithm (DCA) via a line search procedure in a descent direction to get an additional decrease of the objective function and accelerate the convergence of DCA. However, when the first function in DC decomposition is non-smooth, the direction computed by BDCA can be ascent and a monotone line search cannot be performed. In this work, we proposed a monotone improved boosted difference of convex functions algorithm (IBDCA) for certain types of non-smooth DC programs, namely those that can be formulated as the difference of a possibly non-smooth function and a smooth one. We show that any cluster point of the sequence generated by IBDCA is a critical point of the problem under consideration and that the corresponding objective value is monotonically decreasing and convergent. We also present the global convergence and the convergent rate under the Kurdyka-Lojasiewicz property. The applications of IBDCA in image recovery show the effectiveness of our proposed method. The corresponding numerical experiments demonstrate that our IBDCA outperforms DCA and other state-of-the-art DC methods in both computational time and number of iterations.

翻译：本文针对涉及凸函数差分（DC）的非光滑非凸优化问题，提出了一种执行加速凸函数差分算法（BDCA）的新方法。近期提出的BDCA通过经典DC算法（DCA）计算得到点，沿下降方向进行线性搜索执行外推步骤，从而获得目标函数的额外下降并加速DCA的收敛。然而，当DC分解中的第一个函数为非光滑时，BDCA计算得到的方向可能为上升方向，导致无法执行单调线性搜索。本工作针对特定类型的非光滑DC规划问题——即那些可表述为可能非光滑函数与光滑函数之差的规划问题——提出了一种单调改进加速凸函数差分算法（IBDCA）。我们证明IBDCA生成序列的任何聚点均为所考虑问题的临界点，且对应的目标函数值单调递减并收敛。同时，我们在Kurdyka-Lojasiewicz性质下给出了算法的全局收敛性及收敛速率分析。IBDCA在图像恢复中的应用验证了所提方法的有效性。数值实验表明，IBDCA在计算时间和迭代次数方面均优于DCA及其他先进的DC方法。