Mathematical optimisation problem (basic concept)

The form is as follows:

minimise $$f_0(x)$$

subject to $$f_1(x)\le b_i,i=1,...,m$$

 

The optimisation variable of the problem is the following vector:

$$x=(x_1,...,x_n)$$

The objective function is

$$f_0:R^n\to R$$

The constraint functions are

$$f_i:R^n\to R$$

The constrants are

$$b_1,...,b_m$$

Note that the constrants are the limits or bounds for the constraints.

 

The smallest objective value among all vectors that satisfy the constraints is the optimal or solution of the problem:

$$x^{\ast }$$

 

For example,

The following convex function (red coloured quadratic function) is

$$f(x)=2^2/5$$

The constraint function (black coloured linear function) is

$$f(x)=0.5x+7$$

The feasible area is coloured in yellow:

$$f_1(z)\le b_1,...,f_m(z)\le b_m$$

for any z, we have

$$f_1(z)\le f_0(x^{\ast })$$