v2010.10.26 - Convex Optimization

782 APPENDIX F. NOTATION AND A FEW DEFINITIONSpartial orderoperatortightg ′g ′′→Ydg→Ydg 2∇∇ 2relation ≼ is a partial order, on a set, if it possesses reflexivity,antisymmetry, and transitivity (2.7.2.2)mapping to a vector space (a multidimensional function)with reference to a bound means a bound that can be met,with reference to an inequality means equality is achievablefirst derivative of possibly multidimensional function with respect toreal argumentsecond derivative with respect to real argumentfirst directional derivative of possibly multidimensional function g indirection Y ∈R K×L (maintains dimensions of g)second directional derivative of g in direction Ygradient from calculus, ∇f is shorthand for ∇ x f(x). ∇f(y) means∇ y f(y) or gradient ∇ x f(y) of f(x) with respect to x evaluated at ysecond-order gradient∆ distance scalar (Figure 25), or first-order difference matrix (851),or infinitesimal difference operator (D.1.4)△ ijkIII∅triangle made by vertices i , j , and kRoman numeral oneidentity matrix; I =δ 2 (I), δ(I)=1. Variant: I m I ∈ S mindex set, a discrete set of indicesempty set, an implicit member of every set0 real zero0 origin or vector or matrix of zerosOsort-index matrix