A two-dimensional toy optimization problem

As a toy example of an optimization problem in two variables, consider the 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 .

(Note that the term ‘‘subject to’’ has been replaced with the shorthand colon notation.)

The problem can be put in standard form

p^ast := displaystylemin_x : f_0(x) ~:~ f_i(x) le 0, ;; i=1,ldots, m ,

where:

the decision variable is $(x_1,x_2) in mbox{bf R}^2$ ;
the objective function $f_0 : mbox{bf R}^2 rightarrow mbox{bf R}$ , takes values

f_0(x) := 0.9 x_1^2 - 0.4 x_1x_2 - 0.6x_2^2 - 6.4 x_1 - 0.8 x_2;

the constraint functions $f_i : mbox{bf R}^n rightarrow mbox{bf R}$ , take values

$begin{array}[t]{rcl} f_1(x) &:=& -x_1 -1 , f_2(x) &:=& x_1 -2 , f_3(x) &:=& -x_2 , f_4(x) &:=& x_2-3 . end{array}$

is the optimal value, which turns out to be .

See also: