Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform

Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform François Vanderbeck B. Detienne, F. Clautiaux, R. Griset, T. Leite, G. Marques, V. Nesello, A. Pessoa, E. Uchoa, R. Sadykov, H. Sen, I. Tahiri, L. Wolsey ROADEF, February 2019 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 1 / 54

Formulation Combinatorial Optimization Problem (CO) min{c(s) : s S} where S is the discrete set of feasible solutions. (7) (6) (5) S (4) François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 2 / 54

Formulation Combinatorial Optimization Problem (CO) min{c(s) : s S} where S is the discrete set of feasible solutions. Integer Programming Formulation A polyhedron P = {x n : Ax a} is a formulation for (CO) iff min{c(s) : s S} min{cx : x P I = P n }. x2 x2 (7) 5 5 (6) (5) 3 (7) 3 (7) (6) (5) (6) (5) S (4) 2 (4) 2 (4) 1 1 1 2 3 4 x1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 2 / 54

Alternative formulations A formulation is typically not unique P and P can be alternative formulations for (CO) if (CO) min{cx : x P n } min{c x : x P n } x2 5 3 (7) (6) (5) 2 1 (4) 1 2 3 4 x1 More generally: can expressed in different variable-spaces. François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 3 / 54

Quality of Formulations Stronger formulation (in the same space) Formulation P n is a stronger than P n if P P. Then, min{cx : x P } min{cx : x P} x2 5 3 (7) (6) (5) 2 1 (4) 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 4 / 54

Extended Formulation Given an initial compact formulation: x2 5 3 (7) (6) (5) 2 1 (4) 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 5 / 54

Extended Formulation w 5 3 5 x2 (5) 2 3 1 2 (4) 1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 6 / 54

Extended Formulation w 5 3 5 x2 (5) 2 3 1 2 (4) 1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 7 / 54

Extended Formulation w 5 3 5 x2 (5) 2 3 1 2 (4) 1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 8 / 54

Projection The Projection of Q = {(x, w) n+e : Gx + Hw d} on the x-space is: proj x (Q) := {x n : w e such that(x, w) Q}. w 5 3 5 x2 (5) 2 3 1 2 (4) 1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 9 / 54

Take away messages Integer Programming Reformulations A powerful way to exploit the combinatorial structure. Benders & Dantzig Wolfe decomposition are generic schemes to derive & handle reformulations Size can be coped with using dynamic generation: a small % of variables and constraints are needed; hence it scales up to real-life applications. With efficiency enhancement features, these approaches can be highly competitive Implementation can be generic, with tools such as Coluna.jl François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 10 / 54

Outline 1 Sub-gradient linearization : Dantzig & Benders 2 Solution Methods 3 Efficient Implementation 4 Integer Solutions 5 Coluna.jl François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 11 / 54

Subgradient linearization of a convex function f (x) max f(x) g(x) 0 x 0 where f (.) and/or g(.) are defined by a bundle of linear pieces that are generated dynamically during the optimization process. x François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 12 / 54

Subgradient linearization of convex functions η max η η f(x) g(x) 0 x 0 where f (.) and/or g(.) are defined by a bundle of linear pieces that are generated dynamically during the optimization process. x François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 13 / 54

Subgradient linearization of convex functions η max η η f(x) g(x) 0 x 0 where f (.) and/or g(.) are defined by a bundle of linear pieces that are generated dynamically during the optimization process. Separation: Evaluate the true fonction f at the current point x yields a proof of optimality or a violated constraint. x François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 13 / 54

Dantzig-Wolfe Decomposition Assume a bounded integer integer problem with structure: [F] min c x : A x a x X = { B x b x n } François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 15 / 54

Dantzig-Wolfe Decomposition Assume a bounded integer integer problem with structure: [F] min c x : A x a x X = { B x b where B is typically block diagonal x n } [F B ] min c 1 x 1 + c 2 x 2 +... + c K x K A 1 x 1 + A 2 x 2 +... + A K x K a B 1 x 1 b 1 B 2 x 2 b 2...... B K x K x 1 p 1 x 2 p 2..., x K p K b K François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 15 / 54

Dantzig-Wolfe Decomposition Assume a bounded integer integer problem with structure: [F] min c x : A x a x X = { B x b x n } Assume that subproblem [SP] min{c x : x X} (1) is relatively easy to solve compared to problem [F]. Then, X = {x q } q Q François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 15 / 54

Dantzig-Wolfe Decomposition Assume a bounded integer integer problem with structure: [F] min c x : A x a x X = { B x b x n } Assume that subproblem [SP] min{c x : x X} (1) is relatively easy to solve compared to problem [F]. Then, X = {x q } q Q conv(x) = {x n + : x = x q λ q, λ q = 1, λ q 0 q Q} q Q q Q François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 15 / 54

Dantzig-Wolfe Decomposition Assume a bounded integer integer problem with structure: [F] min c x : A x a x X = { B x b x n } Assume that subproblem [SP] min{c x : x X} (1) is relatively easy to solve compared to problem [F]. Then, X = {x q } q Q conv(x) = {x n + : x = x q λ q, λ q = 1, λ q 0 q Q} q Q q Q [F] min{ cx q λ q : Ax q λ q a, λ q = 1, x q λ q n } q Q q Q q Q q François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 15 / 54

The Cutting Stock Problem (CSP) 7 x 656 688 waste : 0 min 432 432 480 2 x waste : 0 422 352 568 2 x waste : 2 420 572 352 2 x waste : 0 8 x 672 672 waste : 0 4 x 688 608 waste : 48 5 x 722 568 waste : 54 4 x 560 560 waste : 224 8 x 722 572 waste : 50 6 x 692 572 waste : 80 560 432 352 4 x waste : 0 s.t. K k=1 y k K x ik d i i k=1 w i x ik W y k k i x ik i, k y k {0, 1} k 8 x 576 608 waste : 160 432 400 488 3 x waste : 24 488 488 352 4 x waste : 16 576 400 352 5 x waste : 16 14 x 692 608 waste : 44 2 x 576 400 waste : 368 Number of distinct cutting patterns : 17 Number of rolls : 88 Total waste : 5090 min q λ q x q i λ q d i i = 1,, n q λ q 0 q François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 16 / 54

The Vehicle Routing Problem (VRP) min s.t. c e x r e e,r e δ(i),r x r e 2 i x r X r r 6, 4, 3 min e,r,q Q r c e x q e λ q e δ(i),r,q Q r x q e λ q 2 i q Q r λ q U r r λ q 0 r, q Q r François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 17 / 54

Lagrangian Duality [F] = min {c x : A x a; B x b; x n } L(π) = min q Q {c xq + π (a Ax q )} [LD] = max L(π) π m + η η L(π) π o π π François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 18 / 54

The Lagrangian Dual as an LP η [LD] max min {π a + (c π m + q Q πa)xq }; max{η, η cx q + π(a Ax q ) q Q, π m +, η 1 }; η L(π) π L(π) π François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 19 / 54

The Lagrangian Dual as an LP η [LD] max min {π a + (c π m + q Q πa)xq }; max{η, η cx q + π(a Ax q ) q Q, π m +, η 1 }; min{ (cx q )λ q, q Q (Ax q )λ q a, q Q λ q = 1, q Q λ q 0 q Q}; η L(π) L(π) François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 19 / 54 π π

Cut generation in the dual L(u) x2 x1 L(u) x2 x1 x3 x3 x4 x4 x5 x5 u u L(u) x2 x1 L(u) x2 x1 x3 x3 x4 x4 x5 x5 u u François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 20 / 54

Column generation in the primal Xlp Xlp Xlp Xlp François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 21 / 54

Properties Restricted Master: DUAL DM t max L t () where L t (π) = min q Q t{πa + (c πa)x q } PRIMAL PM t min{cx : Ax a, x conv({x q } q Q t)}. Valid upper bound: UB η L(π) (π t, η t ) π François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 22 / 54

Properties Restricted Master: DUAL DM t max L t () where L t (π) = min q Q t{πa + (c πa)x q } PRIMAL PM t min{cx : Ax a, x conv({x q } q Q t)}. Valid upper bound: UB η L(π) (π t, η t ) SubProb: Computing L(π t ) yields a 1 most violated constr. for DM t 2 most neg. red. cost col. for PM t 3 a correction on L t (.) at point π t 4 a sub-gradient g t = (a A x t ) of L(.) π Valid lower bound: LB François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 22 / 54

The benefits of the approach Better Dual Bound: Lagrangian duality Mastering the size of the formulation: projection w 5 3 5 x2 (5) 2 3 1 2 (4) 1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 23 / 54

The benefits of the approach Better Dual Bound: Lagrangian duality Mastering the size of the formulation: projection Divide and conquer & Allow to Exploit problem structure complexity size + +... François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 23 / 54

The benefits of the approach Better Dual Bound: Lagrangian duality Mastering the size of the formulation: projection Divide and conquer & Allow to Exploit problem structure Avoid symmetry between identical blocks & Allow a compact representation of a solution min cx 1 + c x 2 +... + c x K A x 1 + A x 2 +... + A x K a B x 1 b 1 B x 2 b 2...... B x K x 1 p 1 x 2 p 2..., x K p K If identical subproblems, solution x q is taken λ q times b K François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 23 / 54

Dual Decompositions On a subset of constraints (Dantzig-Wolfe): multi-resources min c 1 x 1 + c 2 x 2 +... + c K x K A x 1 + A x 2 +... + A x K a B 1 x 1 b 1 B 2 x 2 b 2...... B K x K b K On a subset of variables (Benders): multi-decision-levels min a x + b 1 y 1 + b 2 y 2 +... + b K y K A x + B 1 y 1 c 1 A x + B 2 y 2 c 2 A x +...... A x + B K y K c K François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 24 / 54

Capacitated Facility Location P fx + Pi i i s.t. y Pj ij i yij yij xi min François Vanderbeck (U. de Bordeaux & Atoptima) P i,j cij yij si xi i dj j 0 i, j {0, 1}, i Integer Programming Reformulations 25 / 54

Multi Commodity Network Design min i,j f ijx ij + i,j,k c ijky ijk s.t. k y ijk u ij x ij i, j j y ijk j y jik = b ik i, k y ijk 0 i, j, k x ij {0, 1} i, j François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 26 / 54

Benders Decomposition max a x + b 1 y 1 + b 2 y 2 +... + b K y K A x + B 1 y 1 c 1 A x + B 2 y 2 c 2 A x +...... A x + B K y K c K max { a x + b y : A x + B y c, x n, y p + }, François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 27 / 54

Benders Decomposition max { a x + b y : A x + B y c, φ(x) := max {b y : B y c A x, x n, y p + }, y p + } First stage: max x,η {η : η a x + φ(x), x n, η }, François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 28 / 54

Benders Decomposition max { a x + b y : A x + B y c, φ(x) := max {b y : B y c A x, x n, y p + }, y p + } First stage: max x,η {η : η a x + φ(x), x n, η }, Second stage: min ν 0? b y + ν (η a x) B y + 1 ν (A x c) (y, ν) p+1 + François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 28 / 54

Benders Decomposition max { a x + b y : A x + B y c, φ(x) := max {b y : B y c A x, x n, y p + }, y p + } First stage: max x,η {η : η a x + φ(x), x n, η }, u q 0 (η a x) + uq (A x c) 0 q Second stage: min ν 0? b y + ν (η a x) B y + 1 ν (A x c) (y, ν) p+1 + max u 0 (η a x) + u (A x c) 0? u 0 b u B 0 u 0 + 1x 1 (u 0, u) 1+m 2 + {u 1,..., u T } are its extreme dual solutions. François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 28 / 54

Benders Decomposition max { a x + b y : A x + B y c, φ(x) := max {b y : B y c A x, x n, y p + }, y p + } First stage: Relaxed Master UB max x,η {η : η a x + φ(x), x n, η }, Second stage: min ν 0? b y + ν (η a x) B y + 1 ν (A x c) (y, ν) p+1 + u q 0 (η a x) + uq (A x c) 0 q Sub-Problem Lagrangian LB max u 0 (η a x) + u (A x c) 0? u 0 b u B 0 u 0 + 1x 1 (u 0, u) 1+m 2 + {u 1,..., u T } are its extreme dual solutions. François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 28 / 54

Benders Relaxed Master PRIMAL: [M t ] UB max x,η {η : x q 0 η + (xq A x q 0 a) x cxq q Q t x n +, η }, (x t, η t ) DUAL: [DM t ] min{ (c x q ) λ q : q Q t (x q A x q 0 a) λ q 0, q Q t x q 0 λ q = 1, q λ q 0 q Q t }. (π t, σ t ) = q (uq, u q 0 )λ q SP: L(x, η) = η + min q {c u q (u q A u q 0 a) x uq 0 η} u, LB François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 29 / 54

Method s Classification 1 Ascent: max π {L(π)} Subgradient Volume Conjugate SubGradient Subgradient: π t+1 = (π t + s t η L(π) g t g t )+ + simple unconstraint optimization + low memory use - myoptic : use local info. - slow convergence tail π François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 31 / 54

Method s Classification 1 Ascent: Subgradient Volume Conjugate SubGradient f (x) 2 Polyhedral - Active Set: Kelley Bundle x + use global info + restricted formulation - combinatorics of active set - oscillation bt extreme sol. François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 31 / 54

Method s Classification 1 Ascent: Subgradient Volume Conjugate SubGradient 2 Polyhedral - Active Set: Kelley Bundle 3 Polyhedral - Interior Point: Analytic Center Cutting Plane Primal Dual Col Gen + nb iterations less sensitive to size - full size solution François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 31 / 54

Comparing Methods Best Methods? Ascent methods are computationally light but only allow a rough optimization (good for: warm start, huge problems [Nesterov-14]) Active set methods are best when combinatoric is low Extreme solutions are good to yield integer solutions Otherwise, Interior Point approaches (ACCPM, PDCGM) take over Quality of the implementation François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 32 / 54

Outline 1 Sub-gradient linearization: Dantzig & Benders 2 Solution Methods 3 Efficient Implementation 4 Integer Solutions 5 Coluna.jl François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 33 / 54

Convergence Issue {π t } t π Π Oscillations: π t (bang-bang), {L(π t )} (yo-yo) and π t π > π t 1 π Tailing-off effect: cuts co-linear with objective (inherent to cutting plane proc.) step size very small. Multiple optima 844 760 675 591 506 422 338 253 169 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 34 / 54

Stabilization - Acceleration Techniques 1 Proximality of a stability center (to avoid oscillations): Volume: π t = ( ˆπ + s t dt d t )+. Bundle: π t = argmax π {πa + σ 1 2θ π ˆπ 2 } ( ˆπ, ˆL) L(π) François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 35 / 54

Stabilization - Acceleration Techniques 1 Proximality of a stability center (to avoid oscillations): 2 Central Point Cutting (to avoid tailing and degeneracy): Optimal face center Chebyshev center Analytic center (of trust polyhedra) Interior Point LP-Solver η = η t ( ˆπ, ˆL) (π t, η t ) η ˆL η ˆL ( ˆπ, ˆL) François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 35 / 54

Stabilization - Acceleration Techniques 1 Proximality of a stability center (to avoid oscillations): 2 Central Point Cutting (to avoid tailing and degeneracy): 3 Smoothing (to avoid oscillations): Primal Smoothing Volume: d t = α d t 1 + (1 α) g t Dual Smoothing (also contributes to centralize) [Neame-99] π t = α π t 1 + (1 α) π t [Wentges-97] π t = α ˆπ + (1 α) π t G x y0 G x(a)=x a G OUT SEP d y2 y1 IN f(x)=c1 f(x)=c3<c2 f(x)=c2<c1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 35 / 54

Outline 1 Sub-gradient linearization: Dantzig & Benders 2 Solution Methods 3 Efficient Implementation 4 Integer Solutions 5 Coluna.jl François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 36 / 54

Integrality enforcement in Branch-and-Price w 5 3 5 x2 (5) 2 3 1 2 (4) 1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 37 / 54

Integrality enforcement in Branch-and-Price w 5 3 5 x2 (5) 2 3 1 2 (4) 1 1 2 3 4 x1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 38 / 54

Integrality enforcement in Branch-and-Price Assume a bounded integer integer problem with structure: Assume that subproblem [F] min c x : A x a x Z = { B x b x n } [SP] min{c x : x Z} (2) with conv(z) = {x n + : x = x q λ q, λ q = 1, λ q 0 q Q} q Q q Q The reformulation of [F] is : min{ cx q λ q : Ax q λ q a, λ q = 1, q Q q Q q Q q xq λ q n } François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 39 / 54

Integrality enforcement in Branch-and-Price Assume a bounded integer integer problem with structure: Assume that subproblem [F] min c x : A x a x Z = { B x b x n } [SP] min{c x : x Z} (2) with conv(z) = {x n + : x = x q λ q, λ q = 1, λ q 0 q Q} q Q q Q The reformulation of [F] is : min{ cx q λ q : Ax q λ q a, λ q = 1, q Q But in the case of identical sub-problems... q Q q Q q xq λ q n } François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 39 / 54

Integrality enforcement in B&P: identical SP A Mapping that partitions the aggregate SP solutions: λ 1 = 1 2 λ 2 = 1 2 λ 3 = 1 2 λ 4 = 1 2 4 1 1 1 4 1 2 2 2 2 1 1 1 1 x 1 = x 2 = 4 1 1 2 2 1 1 François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 40 / 54

Integrality enforcement in B&P: identical SP A Mapping that partitions the aggregate SP solutions: 1 2 1 1 2 1 2 λ q 1 2 1 1 x 1 1 1 1 1 0 0 0 x 2 1 1 0 0 1 1 0 x 3 0 0 1 0 1 0 1 x 4 1 0 1 0 1 1 0 X 1 = {x X : x 1 1} and X 12 = {x X : x 1 1, x 2 1} Branching: [V, MP 2009] q X 12 λ q 2 or q (X 1 \X 12 ) λ q 2 or q (X\X 1 ) λ q 3 dynamically increases the # of SP, up to K non-empty nested SP François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 41 / 54

Integer Benders Algorithm (text-book) Oracle Multi-cuts: separation for each integer solution of the pool François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 42 / 54

Benders using Lazy Constraints Oracle Oracle Oracle Oracle Oracle François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 43 / 54

Benders using Cut-and-Branch Oracle Oracle Oracle Oracle Oracle Oracle François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 44 / 54

Benders using Branch-and-Cut Oracle Oracle Oracle Oracle Oracle Oracle Oracle Oracle Oracle François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 45 / 54

Outline 1 Sub-gradient linearization: Dantzig & Benders 2 Solution Methods 3 Efficient Implementation 4 Integer Solutions 5 Coluna.jl François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 46 / 54

Software What: Uses: Coluna.jl a Column-and-Row generation code Open-Source: MPLicence in Julia 1.0 Extending JuMP / JuliaOpt and BlockDecomposition.jl Dantzig-Wolfe & Benders decomposition Robust & Stochastic Optimization Machine Learning First release: Jan. 19 Support: Math Optimization Society (MOS) & JuliaOpt URL https://github.com/atoptima/coluna.jl François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 47 / 54

Coluna Environment François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 48 / 54

Cutting Stock Problem: identical subproblems François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 49 / 54

Vehicle Routing Problem: Q-route subproblem oracle François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 50 / 54

Vehicle Routing Problem: Cut & Pricing oracles François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 51 / 54

Production Problem: combining Dantzig & Benders François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 52 / 54

Coluna s Future Present Modeling JuMP Extension Automated Reformul. Algorithms Column Generation Pricing Call-backs Branch-and-Price Preprocessing Primal Heuristics Applications RCSP Oracle VRP Application (MIT L) François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 53 / 54

Coluna s Future Present Coming Soon Modeling JuMP Extension Automated Reformul. Algorithms Column Generation Pricing Call-backs Branch-and-Price Preprocessing Primal Heuristics Applications RCSP Oracle VRP Application (MIT L) Efficiency Stabilization Strong Branching Further Primal Heuristics Pricing Strategies Parallelisation Cutomization Cutting Callback Branching Callback Functionality Ad-hoc Branching Benders Decomposition Nested Decomposition François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 53 / 54

Coluna & the community Welcome β-users Integration Algorithms: Subgradient, Interior Point, Primal Heuristics Oracles: Pricing, Separation Code review Co-development François Vanderbeck (U. de Bordeaux & Atoptima) Integer Programming Reformulations 54 / 54