SIAM Journal on Optimization, Volume 31, Issue 1, Page 972-990, January 2021.
Given a piecewise-defined function, checking whether it is convex is a nontrivial task. While it may be easy to check whether the restriction of the function to each piece is convex, ensuring the entire function is convex seems to require global conditions. However, it is known that one only needs to ensure the (convex) subdifferential is nonempty on the boundary of the pieces thereby obtaining more local conditions. We specialize the results to quadratic and cubic piecewise defined functions and provide linear-time algorithms to check their convexity. We also provide a MATLAB implementation using an edge-list data structure and discuss two applications: checking the structure of piecewise quadratic functions and optimization problems involving convexity constraints.

中文翻译：

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低阶分段多项式的线性时间凸性检验

SIAM优化杂志，第31卷，第1期，第972-990页，2021年1月。
给定一个分段定义的函数，检查它是否为凸面是一项艰巨的任务。虽然可能很容易检查对每个零件的功能限制是否是凸的，但确保整个功能是凸的似乎需要全局条件。然而，已知的是，仅需要确保（凸）次微分在块的边界上是非空的，从而获得更多的局部条件。我们将结果专用于二次和三次分段定义的函数，并提供线性时间算法来检查其凸性。我们还提供了使用边列表数据结构的MATLAB实现，并讨论了两个应用程序：检查分段二次函数的结构以及涉及凸性约束的优化问题。