v2007.09.17 - Convex Optimization

684 APPENDIX G. NOTATION AND A FEW DEFINITIONS→Ydg 2∇∆△ ijkIII∅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 y ,∇ 2 is second-order gradientdistance scalar, or infinitesimal difference operator, or diagonal matrixtriangle made by vertices i , j , and kRoman numeralidentity matrixindex set, a discrete set of indicesempty set, an implicit member of every set0 real zero0 origin or vector or matrix of zerosO sort-index matrix, or order of magnitude of information required,or computational intensity: O(N) is first order, O(N 2 ) is second,and so on1 real one1 vector of onese imaxmaximizexargsup Xvector whose i th entry is 1 (otherwise 0), or i th member of the standardbasis for R nmaximum [148,0.1.1] or largest element of a totally ordered setfind the maximum of a function with respect to variable xargument of operator or function, or variable of optimizationsupremum of totally ordered set X , least upper bound, may or maynot belong to set [148,0.1.1]