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...unction ω(u, v) is piecewise linear convex and not differentiable on the intersection of pieces. One of the most widely used methods for this problem is the subgradient method. See for example Fisher =-=[11]-=-. Definition 7.1. (g u , g v ) is said to be a subgradient of ω at (ū, ¯v) ≥ 0 when ω(ū, ¯v) + ⟨g u , u − ū⟩ + ⟨g v , v − ¯v⟩ ≤ ω(u, v) holds for any (u, v) ≥ 0, where ⟨·, ·⟩ means the inner product. ...

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...ll denote by ω(LR(u, v)), provides an upper bound of the optimal objective function value ω(P ) of problem (P ). 5. Optimality and Duality Gap The following theorem is well known, see e.g., Geoffrion =-=[13]-=-. Theorem 5.1. Let (ū, ¯v) := ((ūijk) (i,j,k)∈N 3 < , (¯vijk) (i,j,k)∈N 3 < ) be a Lagrangian multiplier vector corresponding to all the transitivity constraints, and let x be an optimal solution of t...

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...ytope and investigated by Grötschel et al. [17]. They introduced some facet-defining valid inequalities of the polytope, and proposed a linear-programmingrelaxation-based algorithm for the problem in =-=[16]-=-. For subsequent researches on the linear ordering polytope, see [4, 10, 20, 23]. Their approach was further extended by Mitchell and Borchers [21, 22], who proposed a cutting plane algorithm based on...

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... agreement under the assignment constraints (3.1) and (3.2) together with the binary variable constraints. This is a well-known NP-hard problem and already a challenging problem when n = 25. See Çela =-=[6]-=-. 3.2. Integer linear programming formulation For a given linear ordering π let binary variables xij for (i, j) ∈ N 2 = be defined as xij = { 1 if π(i) < π(j) 0 otherwise, then the linear ordering pr...

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... as many as 250 items. Since the problem is NP-hard, see e.g., Section 2 of [8], there have been proposed several heuristic methods, e.g., Lagrangian heuristic method in [3], scatter search method in =-=[5]-=-, linear ordering construction heuristics in [9], Goddard’s method in [14], variable neighborhood local search method in [15]. Charon and Hudry [7] made an experiment of a branch-and-bound method with...

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...the ranking aggregation problem, which is known as Kemeny’s problem, the minimum violations ranking problem, and Slater’s problem. See the survey paper by Charon and Hudry [8] and the book by Reinelt =-=[25]-=-. The problem is formulated as a linear integer programming problem. The polytope being the convex hull of binary vectors each corresponding to a linear ordering was named the linear ordering polytope...

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...r integer programming problem. The polytope being the convex hull of binary vectors each corresponding to a linear ordering was named the linear ordering polytope and investigated by Grötschel et al. =-=[17]-=-. They introduced some facet-defining valid inequalities of the polytope, and proposed a linear-programmingrelaxation-based algorithm for the problem in [16]. For subsequent researches on the linear o...

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...rammingrelaxation-based algorithm for the problem in [16]. For subsequent researches on the linear ordering polytope, see [4, 10, 20, 23]. Their approach was further extended by Mitchell and Borchers =-=[21, 22]-=-, who proposed a cutting plane algorithm based on a primal-dual interior point method, and solved problems with as many as 250 items. Since the problem is NP-hard, see e.g., Section 2 of [8], there ha...

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...ard, see e.g., Section 2 of [8], there have been proposed several heuristic methods, e.g., Lagrangian heuristic method in [3], scatter search method in [5], linear ordering construction heuristics in =-=[9]-=-, Goddard’s method in [14], variable neighborhood local search method in [15]. Charon and Hudry [7] made an experiment of a branch-and-bound method with Lagrangian relaxation and some heuristics. The ...

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...rammingrelaxation-based algorithm for the problem in [16]. For subsequent researches on the linear ordering polytope, see [4, 10, 20, 23]. Their approach was further extended by Mitchell and Borchers =-=[21, 22]-=-, who proposed a cutting plane algorithm based on a primal-dual interior point method, and solved problems with as many as 250 items. Since the problem is NP-hard, see e.g., Section 2 of [8], there ha...

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... C lets the problem embrace the ranking aggregation problem, which is known as Kemeny’s problem, the minimum violations ranking problem, and Slater’s problem. See the survey paper by Charon and Hudry =-=[8]-=- and the book by Reinelt [25]. The problem is formulated as a linear integer programming problem. The polytope being the convex hull of binary vectors each corresponding to a linear ordering was named...

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... facet-defining valid inequalities of the polytope, and proposed a linear-programmingrelaxation-based algorithm for the problem in [16]. For subsequent researches on the linear ordering polytope, see =-=[4, 10, 20, 23]-=-. Their approach was further extended by Mitchell and Borchers [21, 22], who proposed a cutting plane algorithm based on a primal-dual interior point method, and solved problems with as many as 250 it...

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...18th century. As Kemeny proposed in [18], a natural solution would be a linear ordering that is “close” to all given rankings. Let σ1, . . . , σκ, . . . , σK be given rankings of items N. Let for α ∈ =-=[0, 1]-=- c (1) ij := α ∣ { κ | σκ(i) < σκ(j) }∣ ∣{ ∣ − (1 − α) κ | σκ(i) > σκ(j) }∣ ∣ , (2.1) K∑ c (2) ij := κ=1 α [σκ(j) − σκ(i)] + − (1 − α) [σκ(i) − σκ(j)] + , (2.2) where | · | denotes the cardinality of ...

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... facet-defining valid inequalities of the polytope, and proposed a linear-programmingrelaxation-based algorithm for the problem in [16]. For subsequent researches on the linear ordering polytope, see =-=[4, 10, 20, 23]-=-. Their approach was further extended by Mitchell and Borchers [21, 22], who proposed a cutting plane algorithm based on a primal-dual interior point method, and solved problems with as many as 250 it...

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... facet-defining valid inequalities of the polytope, and proposed a linear-programmingrelaxation-based algorithm for the problem in [16]. For subsequent researches on the linear ordering polytope, see =-=[4, 10, 20, 23]-=-. Their approach was further extended by Mitchell and Borchers [21, 22], who proposed a cutting plane algorithm based on a primal-dual interior point method, and solved problems with as many as 250 it...

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...thod, and solved problems with as many as 250 items. Since the problem is NP-hard, see e.g., Section 2 of [8], there have been proposed several heuristic methods, e.g., Lagrangian heuristic method in =-=[3]-=-, scatter search method in [5], linear ordering construction heuristics in [9], Goddard’s method in [14], variable neighborhood local search method in [15]. Charon and Hudry [7] made an experiment of ...

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...an heuristic method in [3], scatter search method in [5], linear ordering construction heuristics in [9], Goddard’s method in [14], variable neighborhood local search method in [15]. Charon and Hudry =-=[7]-=- made an experiment of a branch-and-bound method with Lagrangian relaxation and some heuristics. The binary integer programming formulation of the linear ordering problem has an O(n3 ) of inequality c...

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... facet-defining valid inequalities of the polytope, and proposed a linear-programmingrelaxation-based algorithm for the problem in [16]. For subsequent researches on the linear ordering polytope, see =-=[4, 10, 20, 23]-=-. Their approach was further extended by Mitchell and Borchers [21, 22], who proposed a cutting plane algorithm based on a primal-dual interior point method, and solved problems with as many as 250 it...

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...methods, e.g., Lagrangian heuristic method in [3], scatter search method in [5], linear ordering construction heuristics in [9], Goddard’s method in [14], variable neighborhood local search method in =-=[15]-=-. Charon and Hudry [7] made an experiment of a branch-and-bound method with Lagrangian relaxation and some heuristics. The binary integer programming formulation of the linear ordering problem has an ...

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...he update formulas is replaced by the optimal value ω(P ), the sequence generated will converge to an optimal solution of the Lagrangian dual problem (LD), see e.g., Geoffrion [13] and Larsson et al. =-=[19]-=-. However the value ω(u, v) does not necessarily decrease when the multiplier vector is updated. We count the number of consecutive failures to decrease the value, and when it amounts to 5, we halve t...

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...of [8], there have been proposed several heuristic methods, e.g., Lagrangian heuristic method in [3], scatter search method in [5], linear ordering construction heuristics in [9], Goddard’s method in =-=[14]-=-, variable neighborhood local search method in [15]. Charon and Hudry [7] made an experiment of a branch-and-bound method with Lagrangian relaxation and some heuristics. The binary integer programming...

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... point is not the ratings of items but their rankings. One of the well-known method for aggregation of rankings is the Borda method which was first proposed in the 18th century. As Kemeny proposed in =-=[18]-=-, a natural solution would be a linear ordering that is “close” to all given rankings. Let σ1, . . . , σκ, . . . , σK be given rankings of items N. Let for α ∈ [0, 1] c (1) ij := α ∣ { κ | σκ(i) < σκ(...

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...mize ∣ c (i,j):π(i)<π(j) ij subject to π is a linear ordering, which we will refer to as the Linear Ordering Problem (LOP for short). 2.2. Minimum violations ranking Ali et al. [1] and Pedings et al. =-=[24]-=- proposed the minimum violations ranking. Suppose we are given a matrix D := [dij] (i,j)∈N 2 such that dij is the points by which team i beats team j in their matchup, where we take the convention tha...

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... ∪ { i ∈ N | (s, i) ∈ Ā }, T0 := {t} ∪ { j ∈ N | (j, t) ∈ Ā }. We apply the well-known algorithm for computing the transitive closure proposed by Warshall in 1962, see e.g., Section 19.3 of Sedgewick =-=[26]-=-. 7. Subgradient Method for Lagrangian Dual Problem For the sake of simplicity we abbreviate ω(LR(u, v, P0, P1)) to ω(u, v) in this section. The Lagrangian dual problem, denoted by (LD), is a problem ...