Abstract Optimization methods for difference-of-convex programs iteratively solve convex subproblems to define iterates. Although convex, depending on the problem’s structure, these subproblems are very often challenging and require specialized solvers. This work investigates a new methodology that defines iterates as approximate critical points of significantly easier difference-of-convex subproblems approximating the original one. Since there is considerable freedom to choose such more accessible subproblems, several algorithms can be designed from the given approach. In some cases, the resulting algorithm boils down to a straightforward process with iterates given in an analytic form. In other situations, decomposable subproblems can be chosen, opening the way for parallel computing even when the original program is not decomposable. Depending on the problem’s assumptions, a possible variant of the given approach is the Josephy–Newton method applied to the system of (necessary) optimality conditions of the original difference-of-convex program. In such a setting, local convergence with superlinear and even quadratic rates can be achieved under certain conditions.

中文翻译：

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凸凸序列差分规划

摘要 凸差程序的优化方法迭代求解凸子问题以定义迭代。尽管是凸的，但取决于问题的结构，这些子问题通常具有挑战性并且需要专门的求解器。这项工作研究了一种新方法，该方法将迭代定义为近似原始问题的明显更容易的凸差子问题的近似临界点。由于选择此类更容易访问的子问题具有相当大的自由度，因此可以根据给定的方法设计多种算法。在某些情况下，结果算法归结为一个简单的过程，迭代以分析形式给出。在其他情况下，可以选择可分解的子问题，即使原始程序不可分解，也为并行计算开辟了道路。根据问题的假设，给定方法的一个可能变体是应用于原始凸差程序的（必要）最优性条件系统的 Josephy-Newton 方法。在这样的设置下，在某些条件下可以实现超线性甚至二次收敛的局部收敛。