Usually, this is due to the j operator that consumes a value produced by the i operator.
If x i, j = 1, then the i operator is scheduled to the j stage, otherwise it is not scheduled to the stage.
The operators/input variables relation is described with the F n, m) matrix: F = f 1, 1 ⋯ f 1, m ⋮ ⋱ ⋮ f n, 1 ⋯ f n, m, where f i, j ∈ {0, 1} for i ∈ N and j ∈ M. If f i, j = 1, then the j variable is an input for the i operator, otherwise it is not.
It is represented with the H n, m) matrix: H = h 1, 1 ⋯ h 1, m ⋮ ⋱ ⋮ h n, 1 ⋯ h n, m, where h i, j ∈ {0, 1} for i ∈ N and j ∈ M. If h i, j = 1, then the j variable is an output for the i operator, otherwise it is not.
The relation is represented with the P direct(n, n) matrix: P direct = p 1, 1 ⋯ p 1, n ⋮ ⋱ ⋮ p n, 1 ⋯ p n, n, where p i, j ∈ {0, 1} for i, j ∈ N. If p i, j = 1, then the i operator is a direct predecessor for the j operator, otherwise it is not.
Since β0 y) > 0, by (3.6) and (3.7), we have lim sup α min w ∈ T y α Re w, η y α, y + h y α, y ≤ 0. Since T is an (η, h -pseudo-monotone type I operator, we h -pseudo-monotone w ∈ T y α Re w, η y α, x + h -pseudo-monotone typeRe w, η y, x + h y, x. for all x ∈ X.
