v2010.10.26 - Convex Optimization

783Oorder of magnitude information required, or computational intensity:O(N) is first order, O(N 2 ) is second, and so on1 real one1 vector of ones. Variant: 1 m 1 ∈ R me i vector whose i th entry is 1 (otherwise 0),i th member of the standard basis for R n (60)maxmaximizexargsup Xarg supf(x)subject tominminimizexfindxinf Xmaximum [199,0.1.1] or largest element of a totally ordered setfind maximum of objective function w.r.t independent variables x .Subscript x ← x∈ C may hold implicit constraints if context clear; e.g.,semidefinitenessargument of operator or function, or variable of optimizationsupremum of totally ordered set X , least upper bound, may or maynot belong to set [199,0.1.1]argument x at supremum of function f ; not necessarily unique or amember of function domainspecifies constraints of an optimization problem; generally, inequalitiesand affine equalities. Subject to implies: anything not an independentvariable is constant, an assignment, or substitutionminimum [199,0.1.1] or smallest element of a totally ordered setfind objective function minimum w.r.t independent variables x .Subscript x ← x∈ C may hold implicit constraints if context clear; e.g.,semidefinitenessfind any feasible solution, specified by the (“subject to”) constraints,w.r.t independent variables x . Subscript x ← x∈ C may hold implicitconstraints if context clear; e.g., semidefiniteness. “Find” is the thirdobjective of Optimizationinfimum of totally ordered set X , greatest lower bound, may or maynot belong to set [199,0.1.1]