v2007.09.13 - Convex Optimization

675t → 0 +t goes to 0 from above; meaning, from the positive [147, p.2]: as in f : R n → R m meaning f is a mapping, or sequence of successiveintegers specified by bounds as in i:j (if j < i then sequence isdescending)f : M → Rmeaning f is a mapping from ambient space M to ambient R , notnecessarily denoting either domain or range| as in f(x) | x∈ C means with the condition(s) or such that orevaluated for, or as in {f(x) | x∈ C} means evaluated for each andevery x belonging to set Cg| xpexpression g evaluated at x pA, B as in, for example, A, B ∈ S N means A ∈ S N and B ∈ S N(A, B) open interval between A and B in R , or variable pair perhaps ofdisparate dimension[A, B ]closed interval or line segment between A and B in R( ) hierarchal, parenthetical, optional{ } curly braces denote a set or list, e.g., {Xa | a ≽ 0} the set of all Xafor each and every a ≽ 0 where membership of a to some space isimplicit, a union〈 〉 angle brackets denote vector inner-product (26) (31)[ ] matrix or vector, or quote insertion, or citation[d ij ] matrix whose ij th entry is d ij[x i ] vector whose i th entry is x ix pparticular value of xx 0particular instance of x , or initial value of a sequence x ix 1 first entry of vector x , or first element of a set or list {x i }x εextreme point