In this chapter, we introduce the SOC-convexity and SOC-monotonicity which are natural extensions of traditional convexity and monotonicity. These kinds of SOC-convex and SOC-monotone functions are also parallel to matrix-convex and matrix-monotone functions, see [21, 74]. We start with studying the SOC-convexity and SOC-monotonicity for some simple functions, e.g., f(t)=t 2 , t 3 ,1/t,t 1/2 , |t| and [t]+. Then, we explore characterizations of SOC-convex and SOC-monotone functions.
Chen, J. S. (2019). SOC-convexity and SOC-monotonity. In Springer Optimization and Its Applications (Vol. 143, pp. 39–99). Springer International Publishing. https://doi.org/10.1007/978-981-13-4077-2_2