By Saul I. Gass

ISBN-10: 0486262588

ISBN-13: 9780486262581

Enjoyable, nontechnical creation covers easy ideas of linear programming and its dating to operations learn; geometric interpretation and challenge fixing, resolution thoughts, community difficulties, even more. Appendix deals exact statements of definitions, theorems, and methods, extra computational approaches. simply high-school algebra wanted. Bibliography.

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Proof. If :i and y satisfy AX = 0, :i ~ 0 and yrA < OT, we have (yT A):i = yT (AX) = 0, and hence :i = 0 because all components of yT A are strictly negative. So (I) and (II) are mutually exclusive. COROLLARY Assume now that (II) does not hold. Hence ATy ::'S b is infeasible for the particular choice b = -1. , Ax = 0, and xTb < 0 (and hence x f. 0) are satisfied, which implies that statement (I) is true. REMARK. The results of Gordan [35] actually pre-date and imply the results of Farkas [20]. As we have seen, both are consequences of the Fourier-Motzkin algorithm that is essentially due to Fourier [26] even earlier (see also Motzkin [60]).

6). Since the pivots will leave the determinants detA i and detA unchanged (see Ex. 8), the validity of Cramer's rule follows. REMARK. Cramer's rule is only of theoretical value. Gaussian Elimination will compute a solution faster. 1). 3. Symmetric and Positive Semidefinite Matrices. Recall that the matrix A E ]R"xn is said to be symmetric if A = AT. We denote the set of (real) symmetric n x n matrices by §nxn. We want to apply Gaussian Elimination to the rows and to the columns of the symmetric matrix A = (aij) with the goal of retaining symmetry after each elimination step.

Fourier-Motzkin Elimination can be viewed as Gaussian Elimination with respect to the set of non-negative scalars. In contrast to Gaussian Elimination for linear equations, however, Fourier-Motzkin Elimination may increase the number of inequalities considerably in every elimination step. This is the reason why the Fourier-Motzkin algorithm is computationally not very efficient in general. Ex. 19. 33). 36). The Satisfiability Problem. A fundamental model in artificial intelligence is concerned with boolean functions cp : {a, l}n -+ {a, I}.

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Categories: Linear Programming