home Other disciplines Books Mathematics programmers - Nakonechny S.I.

Mathematics programmers - Nakonechny S.I.

2.5. The basic authorities of the rozv'yakiv tasks of the linear programming

Vlastivostі rozv'yakіv problemy literarnogo programvannya formulyuyutsya viglyadі chotyryh theorems (brought to the theorems of this naslidki is imposed lower).

Authority 1 . (Theorem 2.2) The multiplicity of all plan problems of linear programming is omitted.

Authority 2 . (Theorem 2.3) The problem of linear programming is an optimal plan, then the extremal value of the function of a knife is embedded in one of the vertices of a bagatohedron of roses. Yaksho z tsilova funktsiya nabuvae ekstremalnogo znachenny bilsh yak in odnіy verhinі zyogo bagatogrannika, then you can reach yogo and in the best of times, but at the top of the list of such peaks.

Authority 3 . (Theorem 2.4). Clearly, the system of vectors A 1, A 2, ..., Ak ( k ≤ n ) in the rosette A 1 x 1 + A 2 x 2 + ... + Anxn = A 0, X ≥ 0 is inconclusive and taka , ∞

A 1 x 1 + A 2 x 2 + ... + Akxk = A 0,

De xi ≥ 0, then the point X = ( x 1, x 2, ..., xk , 0, ..., 0) is the dotted point of the bagatohedron of roses.

The power 4. (Theorem 2.5) If X = ( x 1, x 2, ..., xn ) is a kut point of the bagatohedron, then the vectors in the rosette A 1 x 1 + + A 2 x 2 + ... + Anxn = A 0 , X ≥ 0, it is recommended to add xj , є linnijno nezalezhnimi.

It is possible to formulate theorems.

THEOREM 2.2. The multiplicity of all the problems in the linear programming has been omitted.

Beforehand . It is necessary to inform that if X 1 and X 2 are the schedules of problems of linear programming (2.1) - (2.3), then the combinatorial Takozh - the task plan.

So yak X 1 і X 2 - the task plan, then it's like this:

AX 1 = A 0, X 1 ≥ 0; AX 2 = A 0, X 2 ≥ 0.

Yakshto pidstavti in the system omzhenzhen znachennya X , then otrimaemo:

Otrimali, scho X zadovolnyaє system obmezhen zadachi lіnіynogo programvannya (2.2), and oskilki , Tobto to satisfy and to satisfy (2.3). Such a course is brought, but X is a plan of tasks for linear programming.

THEOREM 2.3. Yakshcho task liniiynogo programvannya mae optimal plan, then the extreme value tsilova function nabuvaє in odnіy із vertices bahatogrannika rozv'yaziv. Yaksho tsіlоva functіія набуває екстремального значення більш як в одній вершині цього багатогранника, вона reach the yogo and in the best way, that's the lion's combination of such peaks.

Beforehand . Prispodimo, scho bagatogrannik rozv'yakіv tselemi obmezheny і mae skinchennu kilkist kutovih dots. Significantly yogo through Q. In two-dimensional space Q, the type of bagatoknika is shown in Fig. 2.3. Significantly kutovi points through X1, X2, ..., Xp, and the optimal plan - X0.

The problem (2.1) - (2.3) must be expressed at the maximum, annealed, if X is Q, the value of F ( X 0) ≥ F ( X ) for the value of X0 is to be violated. If X0 is a kut point, then the first part of the theorem is completed. It is permissible, but X0 is not a chisel point, then X0 is a point, and so is the multiplication of the multiplicity (brought to the end of the theorem). Otzhe, її mozhna podati yak opuklu lіnіynu kombinatsіy kutovih dot multiplied Q , tobto

X 0 = λ 1 X 1 + λ 2 X 2 + ... + λpXp ,

.

In the summation, F ( X ) is a linear function, it is:

(2.16)

In such an average distribution, the value of F ( Xi ) Vibiraeto naybіshe (pripustimo, scho vono vidpіadaє kotііy exactіі І is understandably yogo through m , tobto . It is replaced in (2.16) by the least value of F ( Xi ) to the highest values. Oskilki , Then

.

For the priposedchenyam X0 - optimal plan, annealed, one side, F ( X 0) ≥ F ( Xk ) = m , and the other is brought to F ( X 0) ≤ m , then F ( X 0) = m = F ( Xk ), and de Xk is a kut point. Otzhe, Lіnіyna funktsіya reach the maximum value in the kutovі precision ( Xk ).

In order to inform the other part of the theorem, it is possible to accept theorems, since F ( X ) gets the maximum values ​​greater than or equal to one point, at the points X1, X2, ..., Xq, (1 ≤ q ≤ p ), then F ( X1 ) = F ( X2 ) = ... = F ( Xq ) = m . Якщо Х has lowered лиінійна комбінація цих кутових точкаок, then:

Тобто лінійна функція F набирає максимальных значені Хі, яка є опуклою лінійною комбінацією кутових Points Х1, Х2 , ..., Хq.

A respect . Yaksho battutokutnik rozv'yakіv - unmediated area (Figure 2.4), then do not cut the point can be filed in the viscount of the lower leninium combination kutovih dots. In such a case, the task of linear programming with a baggage router of rozvozkiv, which is an unprocessed area, can be made to tasks in the obsolete area, by introducing into the system dodatkov interchange x 1 + x 2 ≤ L , de L - a large number. The introduction of this obmezhennia means vidtinannya direct x 1 + x 2 = L in the bagatokutnoi neobmezhenoї region of the littered bagatoknika, for yakogo vikonuetsya imposed a theorem.

Obviously, coordinate the kutovih points, which is done in the result of introducing a new obmezhennia, lie in the L. Yaksho in one of them lyniiyna funktsiya gaining the maximum value, then vano to lie down from L. Zmіnnichi L , the meaning of the functional can be zabbiti yak curiously great, and tse znachaet, scho lіynіyna funktsii neobmechana na bahatogranniku rozv'yaziv.

THEOREM 2.4. Якщо відомо, що система векторів ( K ≤ n ) for the storage , Lionno nezalezhna i taka, scho

,

De xi ≥ 0, then the point X = ( x 1, x 2, ..., xk , 0, ..., 0) is the dotted point of the bagatohedron of roses.

Beforehand . It is permissible, but the point X is not a chutney. Тоді вона може бути виражена опукалю лінійною комбінацією діхх інших точк Х 1 та Х 2 bakatokutnika rozv'yakіv, tobto:

The components of the vectors X 1 and X 2, the value λ 1 and λ 2 of the non-existent and the remaining components of the vector X are equal to zero, then the components of the vector vectors X 1 and X 2 of this vector are zero,

,

.

Oskilki X 1 ta X 2 - the plan, then

,

.

Віднімаючи від першого рівняння друге, отримаємо:

.

For pripuschennyam vectors Лінійно незалежні, тому останнє співвідношення виконується, якщо

.

Звідси:

Otzhe, X, do not hesitate to file a tax note in the form of a combination of the two points of the bagatoknak rozv'yakiv. Means X is a kut point.

THEOREM 2.5. Yakshto - Kutov point of the bagatrone rozv'yakv, then the vectors in the warehouse , , Scho vidpovidayut dodatnim , Л lyniyno nezalezhnimi.

Beforehand . Do not zarugujuchi zagalnosti, it is possible vvazhati nerivnimi zero the first elements of the vector X , annealing,

.

Zdіysnimo brought to the opposite. We assume that the system of vectors Lonely lay. Тоді існують такі числа , Not all rivin 'zero, for yakih vikonuetsya spіvvidnoshnya:

.

For my mind

.

Ask for action number , Pomnozhimo nyogo perchu rivnist, gave the result spoletto dodamo to another, and potim vіdnimemo vіd another рівняння:

,

.

Otzhe, system rіvnyan tasksі lіnіynogo programvannya There are two rozv'yazki, yakі mozhut і not budi plans.

.

All x > 0, the number Mozhna vibrati nastilki malim, scho vsi pershy componenti That Nabuvatimut dodatnih znachen, тоді That - The plan. With ts'omu , Toto X - has omitted the combinational point X 1 and X 2, and supersede the minds of the theorems, the edges of the X point.

Припущення стосовно лінійної залежності векторів Led to superpersonality. Otzhe, vno, ugly, and the system of vectors is lainly unreliable.

Naslidok 1 . Oskilki vectors Muyut rozmіrnіst m , then the kutov point bahatokutnika rozv'yakіv moe not more, nіzh m додатних componів .

Naslidok 2 . Кожній кутовій точці багатокутника розв'язків відповідає Lionno nezalezhnih vektorіv sistemi .

With the guidance of the authorities, you can visnovuvati:

Yakshcho funktsional laziі lіnіynogo programvannya obmezheniy na bahatogranniku rozv'yakiv, then:

• Існує така кутова точка багатогранника розв'язків, в якій лінійний functціонал reach of its optimal value;
• Kozhny oborony plan vidpіdaє kutovі poditsі bataotrannika rozv'yazіv.

Tomu for rozv'yana zadachi lіynіnogo programvannya neobhіdno doslіzhuvati lishut kutov points bahatogrannika (oprichnі plany), do not include before rozglyadu internal points multiply permissible plans.