Nomenclature of a toy 2D optimization problem

Consider the toy optimization problem in two variables:

 displaystylemin_x : x_1^2 - x_1x_2 + 2x_2^2 -3x_1 - 1.5 x_2  ~:~ -1 le x_1 le 2, ;; 0 le x_2 le 3 .

For this problem:

  • The optimal value is p^ast = -10.2667.

  • The optimal set is the singleton mathbf{X}^{rm opt} = {x^{ast}}, with

 x^{ast} = left( begin{array}{c} 2.00  1.33 end{array} right) .
  • Since the optimal set is not empty, the problem is attained.

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An epsilon-sub-optimal set for the toy problem above is shown (in darker color), for epsilon = 0.9. This corresponds to the set of feasible points that achieves an objective value less or equal than p^ast+epsilon.

See also: