Convex Optimization |
"Prior to 1984 [renaissance of interior-point methods of solution] linear and nonlinear programming, one a subset of the other, had evolved for the most part along unconnected paths, without even a common terminology. The use of programming to mean optimization serves as a persistent reminder of these differences."
The goal, then, becomes conversion of a given problem (perhaps a nonconvex problem statement) to an equivalent convex form or to an alternation of convex subproblems convergent to a solution of the original problem: A fundamental property of convex optimization problems is that any locally optimal point is also globally optimal. Given convex real objective function g and convex feasible set C subject to X in C where constraints are abstract here in the membership of variable X to feasible set C. Inequality constraint functions of a convex optimization problem are convex while equality constraint functions are conventionally affine, but not necessarily so. Affine equality constraint functions (necessarily convex), as opposed to the larger set of all convex equality constraint functions having convex level sets, make convex optimization tractable. Similarly, the problem maximize g(X) |