论文标题
揭开数学编程中SDP矩阵的表征
Demystifying the characterization of SDP matrices in mathematical programming
论文作者
论文摘要
该手稿之所以写，是因为我没有发现针对同一受众的SDP编程的其他介绍。与其他现有的SDP介绍相比，第一个区别在于，这项工作来自于努力理解的思想。这似乎只是一个弱点，但是矛盾的是，这既是弱点又是力量。首先，我没有试图压倒读者，但我试图尽可能地最大程度地减少作者与读者之间的距离，甚至希望获得少量的相互同情。这使我避免了一个相当普遍的陷阱：许多长期被认可的专家往往会忘记初学者的困难。其他专家试图使所有证据尽可能短，并驳回他们在职业生涯中看到数千次的某些关键结果。我也避免了这一点，即使我在第一次写这本手稿后修改了这份手稿时确实缩短了一些证据。但是，我还保留了一些看似比必要时间更长的证据，因为我觉得它们提供了更多的见识。一个重要的目标是捕获每个验证结果的“精神”，而不是将其减少到公式流中。 掌握SDP编程的第一个关键步骤是完全了解真实对称矩阵的特征分类。可以看到许多其他SDP编程介绍的方式解决此特征分类的方式，以了解其目标受众与我的不同。他们经常在没有证据的情况下列出特征分类，而我给出了两个证据，以使读者真正熟悉它。
This manuscript was written because I found no other introduction to SDP programming that targets the same audience. A first difference compared to other existing introductions to SDP is that this work comes out of a mind that was itself struggling to understand. This may seem to be only a weakness, but, paradoxically, it is both a weakness and a strength. First, I did not try to overpower the reader, but I tried to minimize the distance between the author and the reader as much as possible, even hoping to achieve a small level of mutual empathy. This enabled me avoid a quite common pitfall: many long-acknowledged experts tend to forget the difficulties of beginners. Other experts try to make all proofs as short as possible and to dismiss as unimportant certain key results they have seen thousands of time in their career. I also avoided this, even if I did shorten a few proofs when I revised this manuscript two years after it was first written. However, I also kept certain proofs that seem longer than necessary because I feel they offer more insight; an important goal is to capture the "spirit" of each proven result instead of reducing it to a flow of formulae. The very first key step towards mastering SDP programming is to get full insight into the eigen-decomposition of real symmetric matrices. It is enough to see the way many other introductions to SDP programming address this eigen-decomposition to understand that their target audience is different from mine. They often list the eigen-decomposition without proof, while I give two proofs to really familiarize the reader with it.
