The SOC functions are indeed vector-valued functions associated with SOC, which are accompanied by Jordan product. However, unlike the matrix multiplication, the Jordan product associated with SOC is not associative which is the main source of difficulty when we do the analysis. Therefore, the ideas for proofs are usually quite different from those for matrix-valued functions. In other words, although SOC and positive semidenite cone both belong to symmetric cones, the analysis for them are different. In general, the arguments are more tedious and need subtle arrangements in the SOC setting. This is due to the feature of SOC. In fact, the SOC complementarity functions or merit functions are comprised of so-called SOC functions. In other words, studying SOC functions is crucial to dealing with SOCP and SOCCP, which is the main target of this chapter.
