cgbpdw-coatwork-annotated

Column Generation, Dantzig-Wolfe, Branch-Price-and-Cut Marco L¨ubbecke · OR Group · RWTH Aachen University, Germany @mluebbecke @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 1/86

Prerequisites → you already know � modeling with integer variables � basic facts about polyhedra � how the simplex algorithm works � a bit about linear programming duality � have seen some cutting planes and know what they are good for � know the branch-and-bound algorithm

The Cutting Stock Problem image source: commons.wikimedia.org , Leeco Steel - Antonio Rosset @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 4/86

The Cutting Stock Problem: Kantorovich (1939, 1960) m rolls of length L ∈ Z + , n orders of length � i ∈ Z + , and demand d i ∈ Z + , i ∈ [ n ] := { 1 , . . . , n } a minimum number of rolls has to be cut into orders; from each order i we need d i pieces in total min m � j =1 y j // minimize number of used rolls s . t . m � j =1 x ij = d i i ∈ [ n ] // every order has to be cut suf fi ciently often n � i =1 � i x ij ≤ L j ∈ [ m ] // do not exceed rolls’ lengths x ij ≤ d i y j i ∈ [ n ] , j ∈ [ m ] // we can only cut rolls that we use x ij ∈ Z + i ∈ [ n ] , j ∈ [ m ] // how often to cut order i from roll j y j ∈ { 0 , 1 } j ∈ [ m ] // whether or not to use roll j @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 5/86

The Cutting Stock Problem: Gilmore & Gomory (1961) � for each p ∈ P , denote by a ip ∈ Z + how often order i is cut in pattern p � build a model on these observations, based on entire con fi gurations λ p ∈ Z + p ∈ P // how often to cut pattern p ? @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 7/86

The Cutting Stock Problem: Gilmore & Gomory (1961) � for each p ∈ P , denote by a ip ∈ Z + how often order i is cut in pattern p � build a model on these observations, based on entire con fi gurations s.t. � p ∈ P a ip λ p = d i i ∈ [ n ] // cover all demands λ p ∈ Z + p ∈ P // how often to cut pattern p ? @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 7/86

Why would we care about different Models? image source: twitter.com , @MurrietaPD @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 8/86

Overview 1 Column Generation 2 Dantzig-Wolfe Reformulation 3 Branch-Price-and-Cut 4 Dual View

Column Generation to solve a Linear Program � but we solve a linear program, the restricted master problem (RMP), with J � ⊆ J z ∗ RMP = min � j ∈ J � c j λ j s.t. � j ∈ J � a j λ j ≥ b λ j ≥ 0 ∀ j ∈ J � � typically, | J | super huge, | J � | small @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 10/86

Column Generation to solve a Linear Program � but we solve a linear program, the restricted master problem (RMP), with J � ⊆ J z ∗ RMP = min � j ∈ J � c j λ j s.t. � j ∈ J � a j λ j ≥ b [ π ] λ j ≥ 0 ∀ j ∈ J � � typically, | J | super huge, | J � | small � use e.g., simplex method to obtain optimal primal λ and optimal dual π for RMP @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 10/86

Column Generation to solve a Linear Program � but we solve a linear program, the restricted master problem (RMP), with J � ⊆ J z ∗ RMP = min � j ∈ J � c j λ j s.t. � j ∈ J � a j λ j ≥ b [ π ] λ j ≥ 0 ∀ j ∈ J � � typically, | J | super huge, | J � | small � use e.g., simplex method to obtain optimal primal λ and optimal dual π for RMP � is λ an optimal solution to the MP as well? // maybe we are lucky! � suf fi cient optimality condition: non-negative reduced cost ¯ c j = c j − π t a j ≥ 0 , ∀ j ∈ J @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 10/86

Na¨ ı ve Idea for an Algorithm: Explicit Pricing � checking the reduced cost (to identify a promising variable, if any) is called pricing algorithm column generation with explicit pricing input: restricted master problem RMP with an initial set J � ⊆ J of variables; output: optimal solution λ to the master problem MP; repeat solve RMP to optimality, obtain λ and π ; compute all ¯ c j = c j − π t a j , j ∈ J ; // computationally prohibitive if there is a variable λ j ∗ with ¯ c j ∗ < 0 then J � ← J � ∪ { j ∗ } ; until all variables λ j , j ∈ J , have ¯ c j ≥ 0 ; @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 11/86

The Column Generation Algorithm algorithm column generation input: restricted master problem RMP with an initial set J � ⊆ J of variables; output: optimal solution λ to the master problem MP; repeat solve RMP to optimality, obtain λ and π ; solve z = min { ¯ c j | j ∈ J } ; if z < 0 then J � ← J � ∪ { j ∗ } with ¯ c j ∗ = z ; // add variable λ j ∗ to RMP until z ≥ 0 ; @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 13/86

Example: Cutting Stock: Restricted Master Problem � solve LP relaxation of Gilmore & Gomory formulation min � p ∈ P � λ p s.t. � p ∈ P � a ip λ p = d i [ π i ] i ∈ [ n ] // one dual variable per order/demand λ p ≥ 0 p ∈ P � with P � ⊆ P = { ( a 1 , . . . , a n ) t ∈ Z n + | � n i =1 � i a i ≤ L } a subset of variables → obtain optimal primal λ and optimal dual π t = ( π 1 , . . . , π n ) // solving a linear program, we always obtain both, optimal primal and optimal dual solutions @mluebbecke · CO@Work 2020 · Column Generation, Dantzig-Wolfe Reformulation , Branch-Price-and-Cut · 14/86