Efficient Linear Programming for Dense CRFs - Supplementary Material

i ya:i = 1 ∀ a ∈ {1 . . . n} , (15d) ya:i ≥ 0 ∀ a ∈ {1 . . . n} ∀ i ∈ L . (15e) Here, the shorthand notation ∑ a,b6=a denotes the summation over both variables a and b such that a 6= b, writing it explicitly: ∑ a ∑ b 6=a. Furthermore, the shorthand ∀ a, b 6= a denotes for all a ∈ {1 . . . n} and b ∈ {{1 . . . n}, b 6= a}. Note that, these shorthand notations are consistent with the main paper, and we mention it here to make it clearer. We introduce three blocks of dual variables. Namely, α = {α ab:i, α ab:i | a, b 6= a, i ∈ L} for the constraints in Eqs. (15b) and (15c), β = {βa | a ∈ {1 . . . n}} for Eq. (15d) and γ = {γa:i | a ∈ {1 . . . n}, i ∈ L} for Eq. (15e), respectively. Now we can write the Lagrangian associated with this primal problem [2]: