Mathematical Programming problems (1)-Maxima and contours

Optimization problems

Mathematical programming problems, or optimization problems, can generally be written in the following form:
MAXS.T.F (x) g (x) =c (1) \begin{array}{} Max & F (x) \tag{1}\\ s.t. & G (x) = c \ \ \end{array}

Let 's look at the two-dimensional problem.

For the sake of simplicity, we consider the two-dimensional case, assuming x= (x1,x2) x= (x_1,x_2), then the optimization problem becomes the following form:
MAXS.T.F (X1,X2) g (X1,X2) =c (2) \begin{array}{} Max & F (x_1,x_2) \tag{2}\\ s.t. & G (x_1,x_2) = c \ \ \end{array}

The geometrical meaning is very obvious, it is required to find a point on the curve G (x1,x2) =c g (x_1,x_2) = C, so that the function f (x1,x2) F (x_1,x_2) obtains the maximum value. Because F (x1,x2) F (x_1,x_2) is a curved surface, the problem is to look for the highest point of a mountain road in the mountains. Have a chat. Contour Line

The key to solving the optimal programming problem lies in the contour of the surface. We stopped and looked at the interesting nature of the contours. For surface F (x1,x2) f (x_1,x_2), its contours can be expressed in the following form,
F (x1,x2) =c