We address the problem of minimizing the sum of a nonconvex, differentiable function and composite functions by DC (Difference of Convex functions) programming and DCA (DC Algorithm), powerful tools of nonconvex optimization. The main idea of DCA relies on DC decompositions of the objective function, it consists in approximating a DC (nonconvex) program by a sequence of convex ones. We first develop a standard DCA scheme especially dealing with the very specific structure of this problem. Furthermore, we extend DCA to give rise to the so-named DCA-Like, which is based on a new and efficient way to approximate the DC objective function without knowing a DC decomposition. We further improve DCA based algorithms by incorporating the Nesterov’s acceleration technique into them. The convergence properties and the convergence rate under Kurdyka-Łojasiewicz assumption of extended DCAs are rigorously studied. We prove that DCA-Like and the accelerated versions subsequently converge from every initial point to a critical point of the considered problem. Finally, we investigate the proposed algorithms for an important problem in machine learning: the t-distributed stochastic neighbor embedding. Numerical experiments on several benchmark datasets illustrate the efficiency of our algorithms.

我们解决了通过DC（凸函数的差异）编程和DCA（DC算法）（非凸优化的强大工具）来最小化非凸，可微函数和复合函数之和的问题。DCA的主要思想依赖于目标函数的DC分解，它包括通过一系列凸程序近似DC（非凸）程序。我们首先开发一种标准的DCA方案，尤其是处理此问题的非常具体的结构。此外，我们扩展了DCA以产生所谓的DCA-Like，它基于一种新的有效方法来近似DC目标函数而无需知道DC分解。通过将Nesterov的加速技术纳入其中，我们进一步改进了基于DCA的算法。严格研究了扩展DCA的Kurdyka-Łojasiewicz假设下的收敛性和收敛速度。我们证明DCA-like和加速版随后从每个初始点收敛到所考虑问题的关键点。最后，我们研究了针对机器学习中一个重要问题的拟议算法：t分布随机邻居嵌入。在几个基准数据集上的数值实验说明了我们算法的效率。