A First Course in Combinatorial Optimization by Jon Lee

By Jon Lee

Jon Lee specializes in key mathematical rules resulting in invaluable versions and algorithms, instead of on info buildings and implementation info, during this introductory graduate-level textual content for college students of operations examine, arithmetic, and computing device technological know-how. the perspective is polyhedral, and Lee additionally makes use of matroids as a unifying suggestion. themes comprise linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and community flows. difficulties and workouts are integrated all through in addition to references for additional research.

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N; n cjh j j=1 First, we suppose that P and D are feasible. The conclusion that we seek is that I is feasible. If not, then I I has a feasible solution. We investigate two cases: Case 1: τ > 0 in the solution of I I . Then we consider the points x ∈ Rn and y ∈ Rm defined by x j := τ1 h j , for j = 1, 2, . . , n, and yi := τ1 u i , for i = 1, 2, . . , m. In this case, x and y are feasible to P and D, respectively, but they violate the conclusion of the Weak Duality Theorem. Case 2: τ = 0 in the solution to I I .

Let x ∗ and y ∗ be the basic solutions associated with β. We can see that ∗ cx ∗ = cβ xβ∗ = cβ A−1 β b = y b. Therefore, if x ∗ and y ∗ are feasible, then, by the Weak Duality Theorem, x ∗ and y ∗ are optimal. In fact, if P and D are feasible, then there is a basis β that is both primal feasible and dual feasible (hence, optimal). We prove this, in a constructive manner, by specifying an algorithm. The algorithmic framework is that of a “simplex method”. A convenient way to carry out a simplex method is by working with “simplex tables”.

Y p ) is feasible for the dual of P: p m(k) yik bik min k=1 i=1 subject to: (D) p m(k) yik aikj ≥ c j , for j = 1, 2, . . , n; k=1 i=1 yik ≥ 0, for k = 1, 2, . . , p, i = 1, 2, . . , m(k). m(k) k k y i bi Optimality of x for Pk and y k for Dk implies that nj=1 ckj x j = i=1 when the Strong Duality Theorem is applied to the pair Pk , Dk . Using the n fact that we have a weight splitting, we can conclude that j=1 c j x j = p m(k) k k y b . The result follows by application of the Weak Duality Thei=1 i i k=1 orem to the pair P, D.

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