NomenclatureOptimization Models > Gauss’ Bet | Functions and Maps | Standard Forms | Nomenclature | Problem Classes | Complexity | History
Consider the optimization problem ![]() Feasible setThe feasible set of problem ![]() A point Example: In the toy optimization problem, the feasible set is the ‘‘box’’ in The feasible set may be empty, if the constraints cannot be satisfied simultaneously. In this case the problem is said to be infeasible. What is a solution?In an optimization problem, we are usually interested in computing the optimal value of the objective function, and also often a minimizer, which is a vector which achieves that value, if any. Feasibility problemsSometimes an objective function is not provided. This means that we are just interested in finding a feasible point, or determine that the problem is infeasible. By convention, we set Optimal valueThe optimal value of the problem is the value of the objective at optimum, and we denote it by ![]() Example: In the toy optimization problem, the optimal value is Optimal setThe optimal set (or, set of solutions) of problem ![]() We take the convention that the optimal set is empty if the problem is not feasible. A standard notation for the optimal set is via the ![]() A point Example: The problem ![]() has no optimal points, as the optimal value If the optimal set is not empty, we say that the problem is attained. SuboptimalityThe ![]() (With our notation, This set allows to characterize points which are close to being optimal (when Example: Nomenclature of the two-dimensional toy problem. Local vs. global optimal pointsA point ![]() In other words, a local minimizer The term globally optimal (or, optimal for short) is used to distinguish points in the optimal set from local minima. Example: a function with local minima. Local optima may be described as the curse of general optimization, as most algorithms tend to be trapped in local minima if they exist. |