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3.5 Integer Pivoting using none toreceive none for web,windows applicationbarcodes generator in The LH algor none for none ithm follows the edges of a polyhedron, which is implemented algebraically by pivoting as used by the simplex algorithm for solving a linear program. We describe an ef cient implementation that has no numerical errors by storing integers of arbitrary precision. The constraints de ning the polyhedron are thereby represented as linear equations with nonnegative slack variables.

For the polytopes P and Q in (3.6), these slack variables are nonnegative vectors s RN and r RM so that x P and y Q if and only if B x + s = 1, and x 0, s 0, r 0, y 0. (3.

9) r + Ay = 1, (3.8). VS 2010 A binding in equality corresponds to a zero slack variable. The pair (x, y) is completely labeled if and only if xi ri = 0 for all i M and yj sj = 0 for all j N, which by (3.9) can be written as the orthogonality condition x r = 0, y s = 0.

(3.10). A basic solu none none tion to (3.8) is given by n basic (linearly independent) columns of B x + s = 1 and m basic columns of r + Ay = 1, where the nonbasic variables that correspond to the m respectively n other (nonbasic) columns are set to zero, so that the basic variables are uniquely determined. A basic feasible solution also ful lls (3.

9), and de nes a vertex x of P and y of Q. The labels of such a vertex are given by the respective nonbasic columns. Pivoting is a change of the basis where a nonbasic variable enters and a basic variable leaves the set of basic variables, while preserving feasibility (3.

9). We illustrate this for the edges of the polytope P in Figure 3.2 shown as arrows, which are the edges that connect 0 to vertex c, and c to d.

The system B x + s = 1 is here =1 3x1 + 2x2 + 3x3 + s4 2x1 + 6 x2 + x3 + s5 = 1 (3.11). and the basi c variables in (3.11) are s4 and s5 , de ning the basic feasible solution s4 = 1 and s5 = 1, which is simply the right-hand side of (3.11) because the basic columns form the identity matrix.

Dropping label 2 means that x2 is no longer a nonbasic variable, so x2 enters the basis. Increasing x2 while maintaining (3.11) changes the current basic variables as s4 = 1 2x2 , s5 = 1 6x2 , and these stay nonnegative as long as x2 1/6.

The term 1/6 is the minimum ratio, over all rows in (3.11) with. equilibrium computation for two-player games positive coe f cients of the entering variable x2 , of the right-hand side divided by the coef cient. (Only positive coef cients bound the increase of x2 , which applies to at least one row since the polyhedron P is bounded.) The minimum ratio test determines uniquely s5 as the variable that leaves the basis, giving the label 5 that is picked up in that step.

The respective coef cient 6 of x2 is indicated by a box in (3.11), and is called the pivot element; its row is the pivot row and its column is the pivot column. Algebraically, pivoting is done by applying row operations to (3.

11) so that the new basic variable x2 has a unit column, so that the basic solution is again given by the right-hand side. Integer pivoting is a way to achieve this while keeping all coef cients of the system as integers; the basic columns then form an identity matrix multiplied by an integer. To that end, all rows (which in (3.

11) is only the rst row) except for the pivot row are multiplied with the pivot element, giving the intermediate system =6 18x1 + 12x2 + 18x3 + 6s4 2x1 + 6x2 + x3 + s5 = 1 (3.12). Then, suitab none none le multiples of the pivot row are subtracted from the other rows to obtain zero entries in the pivot column, giving the new system 14x1 + 16 x3 + 6s4 2s5 = 4 x3 + s5 = 1. 2x1 + 6x2 + (3.13).

In (3.13), t he basic columns for the basic variables s4 and x2 form the identity matrix, multiplied by 6 (which is pivot element that has just been used). Clearly, all matrix entries are integers.

The next step of the LH algorithm in the example is to let y5 be the entering variable in the system r + Ay = 1, which we do not show. There, the leaving variable is r3 (giving the duplicate label 3) so that the next entering variable in (3.13) is x3 .

The minimum ratio test (which can be performed using only multiplications, not divisions) shows that among the nonnegativity constraints 6s4 = 4 16x3 0 and 6x2 = 1 x3 0, the former is tighter so that s4 is the leaving variable. The pivot element, shown by a box in (3.13), is 16, with the rst row as pivot row.

The integer pivoting step is to multiply the other rows with the pivot element, giving + 16x3 + 6s4 2s5 = 4 14x1 + 16s5 = 16. 32x1 + 96x2 + 16x3 (3.14).

Subsequently none for none , a suitable multiple of the pivot row is subtracted from each other row, giving the new system + 16x3 + 6s4 2s5 = 4 14x1 6s4 + 18s5 = 12 18x1 + 96x2 (3.15). with x3 and x2 as basic variables. However, except for the pivot row, the unchanged basic variables have larger coef cients than before, because they have been multiplied with the new pivot element 16. The second row in (3.

15) can now be divided by the previous pivot element 6, and this division is integral for all coef cients in that row; this is the key feature of integer pivoting, explained shortly. The new system is 14x1 + 16x3 + 6s4 2s5 = 4 3x1 + 16x2 s4 + 3s5 = 2. (3.

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