Convex Optimization 2026, 10 - Optimization Problems
For some optimization problem, define an optimal point x* that is feasible and f0(x*) = p*. The optimal set is the set of all optimal points. The optimal value is achieved if the problem is solvable through such an optimal point. If Xopt is empty, then the optimal value is not achieved. A feasible point x with f0(x) ≤ p* + ϵ, ϵ > 0 is ϵ-suboptimal. A feasible point x is locally optimal if there is an R > 0 with f0(x) = inf{f0(z) | fi(z) ≤ 0, i = 1, ..., m, hi(z) = 0, i = 1, ..., p, ||z - x||2 ≤ R} with variable z. x then minimizes f0 over nearby points in the feasible set. If x is feasible and fi(x) = 0, the i-th inequality constraint is considered active at x. If fi(x) < 0, it's inactive (4.1.1.2). If the objective function is constantly 0, the optimal value is either zero (feasible set is nonempty) or ∞ (feasible set is empty). This defines the feasibility problem which can be used to determine whether the constraints are consistent (4.1.1.3). It's convention to express optimization problems as minimization problems, though maximization problems are non-problematic to formulate (4.1.2.2).