AI 中文总结
该研究推广了齐次自对偶嵌入模型，用于最小化两个下半连续凸函数之和，用单一不等式表示。通过Douglas-Rachford算法求解，利用问题结构高效实现，恢复了求解二次锥规划的方法，在特定凸优化问题上验证了通用性和有效性。
AI 中文摘要
我们提出了著名的齐次自对偶嵌入模型的推广，该模型广泛应用于圆锥优化。新的嵌入适用于最小化两个适当的下半连续凸函数之和的问题，并且可以表示为一个使用这些函数及其共轭函数的透视的单一不等式。当可用时，所提出的嵌入的解对原始问题进行原对偶编码，否则编码不可行证书。然后，我们使用Douglas-Rachford算法找到嵌入的解，并通过利用问题结构讨论其有效实现。所得算法恢复了一种现有的求解二次锥规划的方法作为特殊情况。我们在一类具有非光滑目标函数和非圆锥约束的凸优化问题上展示了该算法的通用性和有效性。
英文摘要
We present a generalization of the well-known homogeneous self-dual embedding model, which is widely used in conic optimization. The new embedding applies to a problem of minimizing the sum of two proper lower-semicontinuous convex functions and can be represented as a single inequality that uses perspectives of these functions and of their conjugates. A solution to the proposed embedding encodes a primal-dual solution to the original problem when available, or an infeasibility certificate otherwise. We then use the Douglas-Rachford algorithm to find a solution to the embedding and discuss its efficient implementation by exploiting the problem structure. The resulting algorithm recovers an existing method for solving quadratic cone programs as a special case. We demonstrate the generality and effectiveness of the algorithm on a class of convex optimization problems with non-smooth objective function and non-conic constraints.
