Task 6
Find minimum of the function
f = 0,001036x12 + 0,001091x22 + 0,000978x32 + 0,006604x42 + 0,003532x1x2 + 0,00309x1x3 + 0,004922x1x4 + 0,002988x2x3 + 0,006818x2x4 + 0,003766x3x4 - 0,00077x1 - 0,00091x2 - 0,00062x3 - 0,00283x4
subject to
x1 + x2 + x3 + x4 < 1
-x1 < 0
-x2 < 0
-x3 < 0
-x4 < 0

If the function is given in matrix form f(X) = (X,QX) + (L,X), then
Q =   0,001036   0,001766   0,001545   0,002461
  0,001766   0,001091   0,001494   0,003409
  0,001545   0,001494   0,000978   0,001883
  0,002461   0,003409   0,001883   0,006604

L =   -0,00077  
x1
  -0,00091
x2
  -0,00062
x3
  -0,00283
x4



Outcome

fmin = f(X0) = -0,000303183676559661

    X0   grad f   sum CkAk
x1
  0   0,000284607813446396   0,000284607813446396
x2
  0   0,000550852513628104   0,000550852513628104
x3
  0   0,000186918534221684   0,000186918534221684
x4
  0,214264082374319   0   0
Ck - Lagrange multipliers at the Kuhn-Tucker,
Ak - vector coefficients of left side of constraints number "k",

Execution constraints and Lagrange multipliers
  AkX0     B     Ck
  0,214264082374319
<
  1     0
  0
<
  0
-
  -0,000284607813446396
  0
<
  0
-
  -0,000550852513628104
  0
<
  0
-
  -0,000186918534221684
  -0,214264082374319
<
  0     0
Yellow highlighted lines that match the active inequalities at X.


At specified area the function has also one local minimum:

fmin = f(X0) = -0,000189757103574702

    X0   grad f   sum CkAk
x1
  0   0,000703015582034831   0,000703015582034831
x2
  0,41704857928506   0   0
x3
  0   0,000626141154903758   0,000626141154903758
x4
  0   1,34372135655366E-5   1,34372135655364E-5
Ck - Lagrange multipliers at the Kuhn-Tucker,
Ak - vector coefficients of left side of constraints number "k",

Execution constraints and Lagrange multipliers
  AkX0     B     Ck
  0,41704857928506
<
  1     0
  0
<
  0
-
  -0,000703015582034831
  -0,41704857928506
<
  0     0
  0
<
  0
-
  -0,000626141154903758
  0
<
  0
-
  -1,34372135655364E-5
Yellow highlighted lines that match the active inequalities at X.

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