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By Hans P. Künzi, Werner Rheinboldt, H. G. Tzschach, C. A. Zehnder

ISBN-10: 0124288502

ISBN-13: 9780124288508

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Extra resources for Numerical Methods of Mathematical Optimization. With ALGOL and FORTRAN Programs

Example text

Let us denote by Z n and Z21 the variables belonging to the corner points of 5 i and ^ 2 , using {x^ = 0, ^ 2 = 0) and (JC3 = 0, = 0), respectively. 84): Xl 1 ^2 1 = 0 -8 -1 2 5 3 1 ^5 maximize Β^,Χι,χ^'^ ^^3 ^4 -i -1 = 0 3 -3 1 -1 1 0 <, 12 <. 0 ú 4 maximize B^; x^, 0. ^ 0. 85) we have γ = —20. Since 7 < 0, the optimum has not yet been reached and the vector xg = ( ^ 2 ^ 2 ' 0, 1) = (20, 0, 1) has to be introduced into the basis. Let Z 2 2 be the corresponding variable. The variable leaving the basis is calculated to be x^.

8 1 ) does not hold by any column, and if degeneracy is excluded, then the optimum has been reached (see also the simplex criterion in ( 1 . 4 3 ) ) . Suppose that a feasible basic solution with m + η variables has been given. The corresponding basic vectors in tableau ( 1 . ,0}. Suppose, furthermore, that the price vector ( π , π ) with its m + η components is known. Then, by ( 1 . 4 6 ) , this vector and the basic vectors satisfy the equation πΡ,, + 7Γ, = Γ,,. 82) One complete step of the iteration now has the following form: STEP 1.

In addition, we are also interested in the set of all corner points of the domains Si. 72) Ci,^Ci^Xi,. ,n, k =i \,, .. ,ki). 76) for all i ΣPikZik k=l z<» ^ 0 k. · • · . 2 « Zu,. Cu, · C21, Pn, P21, · • > ^2», · · • > ^2^^ c„i,.. b Ρηΐ, 1 1 1 1,. . , 1 1,... 78) 44 1 LINEAR OPTIMIZATION The row price-vector ( π , π ) corresponds to the expression ( 1 . 4 6 ) and will be discussed further in ( 1 . 8 1 ) . The connection between the two formu­ lations ( 1 . 6 7 ) - ( 1 . 6 9 ) and ( 1 . 7 4 ) - ( 1 .

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Numerical Methods of Mathematical Optimization. With ALGOL and FORTRAN Programs by Hans P. Künzi, Werner Rheinboldt, H. G. Tzschach, C. A. Zehnder


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