v2010.10.26 - Convex Optimization

781vectorentrycubixquartixfeasible setsolution setoptimalfeasiblesameequivalentobjectiveprogramnatural ordercolumn vector in R nscalar element or real variable constituting a vector or matrixmember of R M×N×Lmember of R M×N×L×Kmost simply, the set of all variable values satisfying all constraints ofan optimization problemmost simply, the set of all optimal solutions to an optimization problem;a subset of the feasible set and not necessarily a single pointas in optimal solution, means solution to an optimization problem.optimal ⇒ feasibleas in feasible solution, means satisfies the (“subject to”) constraints ofan optimization problem, may or may not be optimalas in same problem, means optimal solution set for one problem isidentical to optimal solution set of another (without transformation)as in equivalent problem, means optimal solution to one problem can bederived from optimal solution to another via suitable transformationThe three objectives of Optimization are minimize, maximize, and findSemidefinite program is any convex minimization, maximization, orfeasibility problem constraining a variable to a subset of a positivesemidefinite cone.Prototypical semidefinite program conventionally means: a semidefiniteprogram having linear objective, affine equality constraints, but noinequality constraints except for cone membership. (4.1.1)Linear program is any feasibility problem, or minimization ormaximization of a linear objective, constraining the variable to anypolyhedron. (2.13.1.0.3)with reference to stacking columns in a vectorization means a vectormade from superposing column 1 on top of column 2 then superposingthe result on column 3 and so on; as in a vector made from entries of themain diagonal δ(A) means taken from left to right and top to bottom