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Introduction to Optimum Design phần 4

Các điều kiện cần thiết cho những hạn chế bình đẳng và bất bình đẳng có thể được tóm tắt trong những gì thường được gọi là Karush-Kuhn-Tucker (KKT) các điều kiện cần thiết để đầu tiên hiển thị trong Định lý 4.6: Định lý 4,6 Karush-Kuhn-Tucker (KKT) Tối ưu Điều kiện Hãy để x * là một điểm thường xuyên của các thiết lập | TABLE 6-5 Pivot Step to Example 6.5 Interchange Basic Variable x4 with Nonbasic Variable xl for Initial canonical form. Basic0 X1 X2 X3 X4 b 1 x3 -1 1 1 0 4 2 x4 1 1 0 1 6 Basic solution Nonbasic variables x1 0 x2 0 Basic variables x3 4 x4 6 To interchange x1 with x4 choose row 2 as the pivot row and column 1 as the pivot column. Perform elimination using a21 as the pivot element. Result of the pivot operation second canonical form. Basic0 X1 X2 X3 X4 b 1 x3 0 2 1 1 10 2 x1 1 1 0 1 6 Basic solution Nonbasic variables x2 0 x4 0 Basic variables x1 6 x3 10 Solution. The given canonical form can be written in a tableau as shown in Table 6-5 x1 and x2 are nonbasic and x3 and x4 are basic i.e. x1 x2 0 x3 4 x4. 6. This corresponds to point A in Fig. 6-2. In the tableau the basic variables are identified in the leftmost column and the rightmost column gives their values. Also the basic variables can be identified by examining columns of the tableau. The variables associated with the columns of the identity matrix are basic e.g. variables x3 and x4 in Table 65. Location of the positive unit element in a basic column identifies the row whose right side parameter bi is the current value of the basic variable associated with that column. For example the basic column x3 has unit element in the first row and so x3 is the basic variable associated with the first row. Similarly x4 is the basic variable associated with row 2. To make x1 basic and x4 a nonbasic variable one would like to make a21 1 and a n 0. This will replace x1 with x4 as the basic variable and a new canonical form will be obtained. The second row is treated as the pivot row i.e. a21 1 p 2 q 1 is the pivot element. Performing Gauss-Jordan elimination in the first column with a21 1 as the pivot element we obtain the second canonical form as shown in Table 6-5. For this canonical form x2 x4 0 are the nonbasic variables and x1 6 and x3 10 are the basic variables. Thus referring to Fig. 6-2 this pivot step results in