……这本似乎是对最基础的优化细节（nuts and bolts）讲得最透彻的。EDIT：如果你想进一步理解这些书之外的凸性理论，Rockafellar 的《Convex Analysis》很有帮助。
…This one seemed to be the one that has the most nuts and bolts about the basics of optimization. EDIT: If you want more understanding of convexity beyond what's presented in these books, Rockafellar's Convex Analysis is helpful.
The usual good starting point is Boyd’s Convex Optimization, for a relatively rigorous but also quite practical book. The second usual, quite a bit more rigorous case is Rockafellar’s Convex Analysis, but this one deals only with general analysis as opposed to also including applications and algorithms. I highly recommend Boyd’s book, even for a mathematician ;) since it gives a lot of great motivation…
Rockafellar 的《Convex Analysis》对初学者来说挺难的。它属于研究专著。我更推荐 Hiriart-Urruty 和 Lemaréchal 写的《Fundamentals of convex analysis》
Convex analysis by Rockafellar is pretty hard for a beginner. It's a research monograph. I would recommend "Fundamentals of convex analysis" by Hiriart-Urruty & Lemaréchal
……凸的，那么它就是连续的（证明不算太难，在 Fleming 的《Functions of Several Variables》里有）。IIRC 凸性意味着几乎处处可微（证明大概在 R. T. Rockafellar 的《Convex Analysis》里）。上方图（epigraph）就是 f 的图像及其上方的所有区域，它是一个凸集，因此存在支撑超平面，并且是其所有支撑（切线）超平面构成的闭半空间的交集……
…convex, then it is continuous (proof is not very difficult but is in Fleming, Functions of Several Variables). IIRC convex implies differentiable almost everywhere (maybe proof in R. T. Rockafellar, Convex Analysis). The epigraph is the graph of f and everything above it, and it is a convex set so has supporting hyperplanes and is the intersection of all the closed half spaces from its supporting (tangent) hyperplanes…
……这本似乎是对最基础的优化细节（nuts and bolts）讲得最透彻的。EDIT：如果你想进一步理解这些书之外的凸性理论，Rockafellar 的《Convex Analysis》很有帮助。
…This one seemed to be the one that has the most nuts and bolts about the basics of optimization. EDIT: If you want more understanding of convexity beyond what's presented in these books, Rockafellar's Convex Analysis is helpful.
通常不错的切入点是 Boyd 的《Convex Optimization》，既相对严谨又非常实用。第二个常见选择是 Rockafellar 的《Convex Analysis》，要严谨得多，但这本只讲纯分析，不涉及应用和算法。我强烈推荐 Boyd 的书，哪怕对数学家来说也是如此 ;) 因为它给出了很多非常棒的动机说明……
The usual good starting point is Boyd’s Convex Optimization, for a relatively rigorous but also quite practical book. The second usual, quite a bit more rigorous case is Rockafellar’s Convex Analysis, but this one deals only with general analysis as opposed to also including applications and algorithms. I highly recommend Boyd’s book, even for a mathematician ;) since it gives a lot of great motivation…
Rockafellar 的《Convex Analysis》对初学者来说挺难的。它属于研究专著。我更推荐 Hiriart-Urruty 和 Lemaréchal 写的《Fundamentals of convex analysis》
Convex analysis by Rockafellar is pretty hard for a beginner. It's a research monograph. I would recommend "Fundamentals of convex analysis" by Hiriart-Urruty & Lemaréchal
……凸的，那么它就是连续的（证明不算太难，在 Fleming 的《Functions of Several Variables》里有）。IIRC 凸性意味着几乎处处可微（证明大概在 R. T. Rockafellar 的《Convex Analysis》里）。上方图（epigraph）就是 f 的图像及其上方的所有区域，它是一个凸集，因此存在支撑超平面，并且是其所有支撑（切线）超平面构成的闭半空间的交集……
…convex, then it is continuous (proof is not very difficult but is in Fleming, Functions of Several Variables). IIRC convex implies differentiable almost everywhere (maybe proof in R. T. Rockafellar, Convex Analysis). The epigraph is the graph of f and everything above it, and it is a convex set so has supporting hyperplanes and is the intersection of all the closed half spaces from its supporting (tangent) hyperplanes…