This page has been robot translated, sorry for typos if any. Original content here.

Mathematical programming - Nakonechny S.І.

8.6. Kuhn – Tucker Theorem

The Rozglanyuti method of Lagrange multipliers is able to find more than the local local points of the Lagrange function.

The Kuhn – Tucker theorem allows me to establish the following tasks: for many admissible solutions there is one more global extremum of the type of the type. Wonderfully tied up with the necessary and sufficient minds to identify the point.

Let’s take a look at the task of non-linear programming, a yak, not unchanging zagalnosti, a gift from a person:

, (8.22)

, (8.23)

. (8.24)

(Obviously, the sign of irregularity is possible for the duration of the plurality of the left and right parts of the square (- 1)).

Theorem 8.1. (Kuhn – Tucker theorem). The vector X * є is the optimal solution to the problems (8.22) - (8.24) Todi i Tilki Todi, as long as such a vector at for all dot є with the point of the Lagrange function


Functionality Meti for all oppressed, but function - bulges.

Reported . Neobkhіdnіst . Nekhay X * - the optimal plan of problems (8.22) - (8.24), so that є the point of the global maximum of tasks_. From now on, for all of the most recent plans, there are x x many possible permissions to resolve the issue:


Let us take a look now , scho vіdpovіdaє points of global maximum , and the value of the Lagrange functions at points , , de - a complete plan of tasks with a plurality of permissible rozv'yazkіv, - the vector of Lagrange multipliers, scho v_dpovіdaє X.

Wash (8.21) maєmo: , Todi

. (8.25)

For point with coordinates deyaki dodanki mind can be but vіdmіnni vіd zero. Oskіlki for mental tasks , then think less, , matimemo nerve:


Functionality - lіnіyna vidnosno , so that the rest of the haste will come to be for one . Otzhe point - the point of global minimum Functions of Lagrange.

For installation of nerves, which is necessary to understand (8.13), and the very same: Skoristaєmosia takozh Rivnyannyam (8.21) . Behind Brain Theorems - oppressed functions і , that will come to mind also:

From now on, at the point X * the Lagrange function has a global maximum in X , which is necessary to bring up the necessary theorems.

Adequacy . To make the welfare of the minds theorem, you need to enter , - saddle point function (tobto for to come to a halt (8.13)), and it is necessary to bring it up to the point X * є with the optimal plan for the task of convex programming.

Ideally, (8.13), the Lagrange function (8.12) for problems (8.22) - (8.23):


at all values .

Part of the law is part of the rule of law (8.26).


Remaining the hassle of the maikon for all . Besides that , so that the nerves can be less than


Toddly from the part of irregularities (8.26):


Through , comes to the nerves , yak cope for all value .

From now on, point X * is satisfied with the need for maximum value of functional tasks, especially for all function nabuva smaller values, lower at point X *, so that there is an optimal plan for the task of non-linear programming. The dignity of the minds of the tower is brought.

Understand the Kuhn - Tucker theorems to look less for tasks, which is to take revenge on functions.