SOC-Convexity and SOC-Monotonity

Abstract

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.