| # A two-dimensional toy optimization problem As a toy example of an optimization problem in two variables, consider the problem (Note that the term ‘‘subject to’’ has been replaced with the shorthand colon notation.) The problem can be put in standard form where: the decision variable is ; the objective function , takes values the constraint functions , take values is the optimal value, which turns out to be . The optimal set is the singleton , with Since the optimal set is not empty, the problem is attained. We can represent the problem in epigraph form, as
| Geometric view of the toy optimization problem above. The level curves (curves of constant value) of the objective function are shown. The problem amounts to find the smallest value of such that for some feasible . The plot also shows the unconstrained minimum of the objective function, located at . An -sub-optimal set for the toy problem above is shown (in darker color), for . This corresponds to the set of feasible points that achieves an objective value less or equal than .