Conic programming has well-documented merits in a gamut of signal processing and machine learning tasks. This contribution revisits a recently developed first-order conic descent (CD) solver, and advances it in three aspects: intuition, theory, and algorithmic implementation. It is found that CD can afford an intuitive geometric derivation that originates from the dual problem. This opens the door to novel algorithmic designs, with a momentum variant of CD, momentum conic descent (MOCO) exemplified. Diving deeper into the dual behavior CD and MOCO reveals: i) an analytically justified stopping criterion; and, ii) the potential to design preconditioners to speed up dual convergence. Lastly, to scale semidefinite programming (SDP) especially for low-rank solutions, a memory efficient MOCO variant is developed and numerically validated.

通过对第一阶锥下降（CD）求解器进行直观性、理论性和算法实现方面的改进研究，发现CD可以通过对偶问题的几何推导得到直观的解释，从而为新算法设计打开了大门，其中包括动量变量MOCO。进一步深入研究了CD和MOCO的对偶行为，揭示了解析合理的停止准则和设计预处理器以加快对偶收敛的潜力。最后，为了扩展半定规划（SDP）的规模，特别是对于低秩解，开发了一个内存高效的MOCO变体，并得到了数值验证。

内存高效优化的锥下降