By Robert J Vanderbei
This 3rd version introduces the most recent concept and purposes in optimization. It emphasizes restricted optimization, starting with linear programming after which continuing to convex research, community flows, integer programming, quadratic programming, and convex optimization. You’ll find a host of sensible company purposes in addition to non-business purposes. With its specialize in fixing useful difficulties, the ebook positive aspects loose C courses to enforce the most important algorithms coated. The book’s accompanying web site contains the C courses, JAVA instruments, and new on-line tutorial instruments and routines.
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Extra info for Linear Programming: Foundations and Extensions, 3rd Edition
I∈B Equating these two expressions for ζ, we see that cs − c∗s + c∗i ais i∈B c∗i bi . y= i∈B Since this equation must be an identity for every y, it follows that the coefficient multiplying y must vanish (as must the right-hand side): cs − c∗s + c∗i ais = 0. i∈B Now, the fact that xs is the entering variable in D implies that cs > 0. 38 3. DEGENERACY Recall that xt is the fickle variable with the largest index. Since xs is also fickle, we see that s < t. Since xs is not the entering variable in D∗ (as xt is), we see that c∗s ≤ 0.
2) and then look ¯2 , . . , w ¯3 , which is better in the sense that it has a larger obfor a new solution x ¯1 , x jective function value: x2 + 3¯ x3 > 5x1 + 4x2 + 3x3 . 5¯ x1 + 4¯ We continue this process until we arrive at a solution that can’t be improved. This final solution is then an optimal solution. To start the iterative process, we need an initial feasible solution x1 , x2 , . . , w3 . For our example, this is easy. We simply set all the original variables to zero and use the defining equations to determine the slack variables: x1 = 0, x2 = 0, x3 = 0, w1 = 5, w2 = 11, w3 = 8.
Hence, no dictionary can be degenerate. 36 3. DEGENERACY 4. Bland’s Rule The second pivoting rule we shall consider is called Bland’s rule. It stipulates that both the entering and the leaving variable be selected from their respective sets of choices by choosing the variable xk with the smallest index k. 3. The simplex method always terminates provided that both the entering and the leaving variable are chosen according to Bland’s rule. The proof may look rather involved, but the reader who spends the time to understand it will find the underlying elegance most rewarding.