The formalized problem consists of the following elements:
– objective function ;
– domain X of the definition of the objective functional f;
– constraint: C ⊂ X.
Here,
Points of the set C are called admissible points of the problem [1.1]. If C = X, then all points of the domain of definition of the function are admissible. The problem [1.1] in this case is called a problem without constraints.
The maximization problem can always be reduced to the minimization problem by replacing the functional f with the functional g = –f. And, on the contrary, the minimization problem in the same way can be reduced to the maximization problem. If the necessary conditions for the extremum in the minimization problem and maximization problem are different, then we write these conditions only for the minimization problem. If it is necessary to investigate both problems, then we write down
An admissible point
Then we write
In addition to global extremum problems, local extremum problems are also studied. Let X be a normed space. A local minimum (maximum) of the problem is reached at a point
In other words, if
in the problem
The theory of extremum problems gives general rules for solving extremum problems. The theory of necessary conditions of the extremum is more developed. The necessary conditions of the extremum make it possible to allocate a set of points among which solutions of the problem are situated. Such a set is called a critical set, and the points themselves are called critical points. As a rule, a critical set does not contain many points and a solution of the problem can be found by one or another method.
1.2. Optimization problems with objective functions of one variable
Let f: ℝ → ℝ be a function of one real variable.
DEFINITION 1.1.– A function f is said to be lower semicontinuous (upper semicontinuous) at a point if for every ε > 0, there exists a δ > 0 such that the inequality
holds true for all x ∈ (
DEFINITION 1.2.– (Equivalent) A function f is said to be lower semicontinuous (upper semicontinuous) at a point
holds true for all x ∈ (
If the function takes values in
Here are examples of semicontinuous functions:
1 1) the function y = [x] (integer part of x) is upper semicontinuous at the points of discontinuity;
2 2) the function y = {x} (fractional part of x) is lower semicontinuous at the points of discontinuity;
3 3) the Dirichlet function, which is equal to 0 at rational points and equal to 1 at irrational points, is lower semicontinuous at each rational point and upper semicontinuous at each irrational point;
