A two-dimensional toy optimization problem
As a toy example of an optimization problem in two variables, consider the problem
![\[ \min _x 0.9 x_1^2-0.4 x_1 x_2-0.6 x_2^2-6.4 x_1-0.8 x_2: -1 \leq x_1 \leq 2, \quad 0 \leq x_2 \leq 3. \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-aaab19d5b6345ff3fda2fd76cec02bef_l3.png "Rendered by QuickLaTeX.com")
(Note that the term ‘‘subject to’’ has been replaced with the shorthand colon notation.)
The problem can be put in standard form
![\[ p^*:=\min _x f_0(x): f_i(x) \leq 0, \quad i=1, \ldots, m, \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-09af86a96079db0e6a0140ad5ec9c747_l3.png "Rendered by QuickLaTeX.com")
where:
● the decision variable is 
;
● the objective function 
, takes values
![\[ f_0(x):=0.9 x_1^2-0.4 x_1 x_2-0.6 x_2^2-6.4 x_1-0.8 x_2; \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-cc03f1cf358204f87dedbf932c334653_l3.png "Rendered by QuickLaTeX.com")
● the constraint functions 
take values
![\[ \begin{aligned} & f_1(x):=-x_1-1, \\ & f_2(x):=x_1-2, \\ & f_3(x):=-x_2, \\ & f_4(x):=x_2-3. \end{aligned} \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-5031628d71be2fb295d79da971723f3e_l3.png "Rendered by QuickLaTeX.com")
● 
is the optimal value, which turns out to be 
.
● The optimal set is the singleton 
, with
![\[ x^*=\left(\begin{array}{c} 2.00 \\ 1.33 \end{array}\right). \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-de33c57e9fb7177b52978391802a5bf2_l3.png "Rendered by QuickLaTeX.com")
Since the optimal set is not empty, the problem is attained.
We can represent the problem in epigraph form, as
![\[ \min _{x, t} t: t \geq 0.9 x_1^2-0.4 x_1 x_2-0.6 x_2^2-6.4 x_1-0.8 x_2, \quad -1 \leq x_1 \leq 2, \quad 0 \leq x_2 \leq 3. \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-9c9f3ed1f453c01fd430b666c2cfc4c9_l3.png "Rendered by QuickLaTeX.com")
 |
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 a darker color), for  . This corresponds to the set of feasible points that achieves an objective value less or equal to  . |
