A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs

This paper makes a survey on SOC complementarity functions and related solution methods for the second-order cone programming (SOCP) and second-order cone complementarity problem (SOCCP). Specifically, we discuss the properties of four classes of popular merit functions, and study the theoretical results of associated merit function methods and numerical behaviors in the solution of convex SOCPs. Then, we present suitable nonsinguarity conditions for the B-subdifferentials of the natural residual (NR) and Fischer-Burmcister (FB) nonsmooth system reformulations at a (locally) optimal solution, and test the numerical behavior of a globally convergent FB semismooth Newton method. Finally, we survey the properties of smoothing functions of the NR and FB SOC complementarity functions, and provide numerical comparisons of the smoothing Newton methods based on them. The theoretical results and numerical experience of this paper provide a comprehensive view on the development of this field in the past ten years.