Lesson 14

Most optimization tasks are reduced to finding the smallest or largest - the extreme value of some function, which is called a target function or a quality criterion .

Criterion - an indicator that allows you to determine the quality of the solution obtained by the task.

Managed variables are task parameters whose values ​​can be changed.

Target function is a function that binds managed variables and criteria.

From the mathematical point of view, when the target function is given by the formula and has a derivative, then the search for the extremum is found through the root of the equation of the derivative equal to zero.

Example 1. Find the sizes of cans with the largest volume V and the smallest length of L joints and area.

                                              V = pr ² h, S = 2pr ² + 2prh, L = 4pr + h .

Let's express the height of the can because of its radius: h = V / pr ^ 2 , where we get:

                                     S = 2pr ^ 2 + (2V) / h, L = 4pr + V / pr ^ 2.

To calculate r, for which min S i L , we find the derivative of these functions with r:

                                                                 S '(r) = .... = 0

i

                                                                L '(r) = .... = 0.

With different optimization criteria we get different answers.

The problem of one-dimensional optimization can be formulated as follows: find among the elements from a given set X such xеX, which gives the extremum of the function f (x ').

To reduce the practical task to mathematics it is necessary:

1. choose the f (x) indicator that is optimized;
2. to construct a mathematical model of dependence minimizing the index from the initial parameters.

If there are restrictions on the values ​​of the managed variables, then the optimization problem is arbitrary , otherwise it is unconditional .

If the objective function and all restrictions are linear, then this is a linear optimization problem, and if the target function or at least one limitation is nonlinear then this is the problem of nonlinear optimization /

If the vector of controlled variables has only one coordinate, then the optimization problem is one-dimensional , otherwise it is multidimensional .

One-extremity problems are tasks in which the target function f (x) in the domain of admissible values ​​has one extremum, and in multi- extreme optimization problems, the target function f (x) in the domain of admissible values ​​has several extrema.

If controlled variables are given on a continuous admissible set, then the optimization problem is continuous , and if the discrete one is a discrete optimization.