Mathematics programmers - Nakonechny S.I.

8.2. Geometrical Interpretation of Problems of Nonlinear Programming

Geometrically, the central function (8.1) is the action of the surface, and the alternation (8.2) - (8.3) is the admissible multiplicity of the n- dimensional Euclidean space. The knowledge of the optimal rozv'yku tasks and non-linear programming is to be developed up to the point of view from the permissible multiplicity, in the reach of the surface of the naiwich (naynizhchogo) river.

Yaksho tsil'ova function is non-perishable, and the multitude of rozvozvіv is permissible is closed, non-empty and obmene, the global maximum (minimum) of tasks is minimized.

Naiprostіshimi for rozv'yazuvannya і tasksі нелінійного програмування, що містять система лінійних обмежень нелінійну цільову функцію . In the first place, the region of permissible rozv'yakiv - is opaque, non-empty, closed, toto obmezhenoyu.

Rozglyanemo butt geometric way to rozvyazvanya tasks and non-linear programming.

Know the minimum and maximum value of the function:

For the mind:

.

Signature: Fig. 8.1 Rozvjazannya . The region of permissible rozviazyv is spoken by the AVTD chotikikutnik (Figure 8.1). Geometrically, the central function is the center of the center of M (2; 2), the square of the radius of a . Tse means, scho її znachennja bude zbіlshuvatisya (zmenshuvatisya) zі zbіlshennyam (zmenshennyam) radii kola. It is carried out from point M of the number of radial radii. The function Z is two local maxima: points B (0; 6) and C (8; 0). The value of the functional is computed at the following points:

,

.

Oskilki , Then the point C (8; 0) is the point of the global maximum.

Obviously, the same radium , Тоді:

. Tobto point M є dotkomu mіnіmumu, oskilki їй vіdpovіdaє namenshe mozhliv znachenny tsil'ovoy funktsii.

Significantly, in the given time, the point of departure is for the optimal plan of tasks (the minimum value of the functional), all the bureaucratics of permissible rods are known, but in tasks of linear programming it is inconceivable.

Know the minimum value of the function:

For the mind:

.

Rozvjazuvannya . In the given application, the multiplicity of permissible rozv'yakivs is stored in two parts, partially unchipped at the top (Figure 8.2). The Tsilova function is analogous to the alternating vipacy, a colum centered at the point M (4; 4). The function Z is two local minima: in the point A ( ), And in B ( ).

Signature: Fig. 8.2. The value of the functional in the points is however, but it is:

.

Otzhe, maemo two alternative optimal plans.

Dany pri iaustruysthe one singularity of the tasks of nonlinear programming: for the problems of linear programming, the bagatogrone of acceptable rozv'yak tasks of nonlinear programming is not obogovyazku bude I put down a multiplier.

It is suggested that the main tasks of non-linear programming are the same as the adherence to the methods of the rozviania.