Demystifying the characterization of SDP matrices in mathematical programming

TL;DR

This paper provides an accessible, empathetic introduction to semidefinite programming (SDP), emphasizing understanding eigen-decomposition of symmetric matrices through detailed proofs and insights tailored for beginners.

Contribution

It offers a beginner-friendly, detailed explanation of SDP concepts, especially eigen-decomposition, avoiding common pitfalls of overly concise proofs and fostering deeper understanding.

Findings

Two detailed proofs of eigen-decomposition are provided.

The approach enhances understanding for beginners.

Focus on capturing the 'spirit' of proofs rather than brevity.

Abstract

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.