Topic 14. Optimization

1. The statement of the optimization problem

Optimization takes place in any field of human activity: in economic planning, management, production processes, designing complex processes, etc. - where the search for the best option is sought. Only mathematically it is possible to give definition of optimization - when instead of words "better", "bad" are quantitative characteristics in the form of a function.

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 .

Most simply from a mathematical point of view, when the target function is given by a derivative formula. In this case, when looking for an extremum, you can use the derivative function.

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 jar through its radius: h = V / pr2. Then we get: S = 2pr2 + (2V) / h, L = 4pr + V / pr2, where ¥ <r <¥. Thus, one should find r, at which min S i L.

To do this, find the derivative of these functions by:

                                                           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 elements from a given set X such x'ÎX, 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 the dependence of the minimized index on the initial parameters.

In practice there are three classes of optimization tasks:

• unconditional optimization (without limitations) - no restrictions will be imposed on the value;
• conditional optimization (with restrictions) - the value is in the given interval;
• optimization for incomplete data - values ​​are not defined.