A toy 2D optimization problem:
geometric view via the epigraph form

Consider the toy problem

displaystylemin_x : 0.9 x_1^2 - 0.4 x_1x_2 - 0.6x_2^2 - 6.4 x_1 - 0.8 x_2 ~:~ -1 le x_1 le 2, ;; 0 le x_2 le 3 .

We can represent the problem in epigraph form, as

$displaystylemin_{x,t} : t ~:~ t ge 0.9 x_1^2 - 0.4 x_1x_2 - 0.6x_2^2 - 6.4 x_1 - 0.8 x_2, ;; -1 le x_1 le 2, ;; 0 le x_2 le 3 .$

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 t = f_0(x) for some feasible . The plot shows the unconstrained minimum of the objective function, located at hat{x} = (4,2) .

See also:

An optimization problem in two variables.
Nomenclature for the toy problem.

A toy 2D optimization problem: geometric view via the epigraph form

A toy 2D optimization problem:
geometric view via the epigraph form