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Introduction to BCP MCF Example Laszlo Ladanyi 1 cois Margot 2 Fran July 18, 2006 1: IBM T.J. Watson Research Center 2: Tepper School of Business, Carnegie Mellon University BCP: Branch-Cut-Price Software for branch-and-cut-and-price

  1. Introduction to BCP – MCF Example Laszlo Ladanyi 1 cois Margot 2 Fran¸ July 18, 2006 1: IBM T.J. Watson Research Center 2: Tepper School of Business, Carnegie Mellon University

  2. BCP: Branch-Cut-Price • Software for branch-and-cut-and-price • Parallel code • LP solver : Clp, Cplex, Xpress, . . . • Most flexible in COIN-OR • Research code (no stand-alone executable)

  3. BCP: Branch-Cut-Price • Software for branch-and-cut-and-price • Parallel code • LP solver : Clp, Cplex, Xpress, . . . • Most flexible in COIN-OR • Research code (no stand-alone executable) BCP code split into four directories: (see coin-Bcp/Bcp/src ) • include : all header files • Tree Manager ( TM ): Maintain the LP associated with each node, manage cuts and variables • Node level operations ( LP ): cutting, branching, heuristics, fixing, column generation • Utilities ( Member ): code for interface between TM and LP, initialization

  4. Solver Initialization Tree Manager Solver • read data

  5. Solver Initialization Tree Manager Solver • read data • pack module data

  6. Solver Initialization Tree Manager Solver • read data → • pack module data • unpack module data

  7. Solver Initialization Tree Manager Solver • read data → • pack module data • unpack module data • setup the LP solver

  8. Processing a node Tree Manager Solver • select node

  9. Processing a node Tree Manager Solver • select node • pack node LP data

  10. Processing a node Tree Manager Solver • select node → • pack node LP data • unpack node LP data

  11. Processing a node Tree Manager Solver • select node → • pack node LP data • unpack node LP data • solve • generate cuts/vars • branch • create LP data for sons

  12. Processing a node Tree Manager Solver • select node → • pack node LP data • unpack node LP data • solve • generate cuts/vars • branch • create LP data for sons • pack node LP data for sons

  13. Processing a node Tree Manager Solver • select node → • pack node LP data • unpack node LP data • solve • generate cuts/vars • branch • create LP data for sons ← • unpack node LP data • pack node LP data for for sons sons

  14. Processing a node Tree Manager Solver • select node → • pack node LP data • unpack node LP data • solve • generate cuts/vars • branch • create LP data for sons ← • unpack node LP data • pack node LP data for for sons sons • add sons to tree

  15. BCP Constraints/Variables Types of Constraints/Variables: • Core : present at all nodes • Algorithmic : separation/generation algorithm • Indexed : e.g. stored in a vector

  16. BCP Constraints/Variables Types of Constraints/Variables: • Core : present at all nodes • Algorithmic : separation/generation algorithm • Indexed : e.g. stored in a vector Algorithmic constraints and variables are local

  17. BCP Constraints/Variables Types of Constraints/Variables: • Core : present at all nodes • Algorithmic : separation/generation algorithm • Indexed : e.g. stored in a vector Algorithmic constraints and variables are local Representation: Constraints are stored as ranged constraints: lb ≤ ax ≤ ub with lb = − DBL MAX or ub = DBL MAX possible

  18. Implementing a Column Generation Application Member: • Read input • Implement variables TM: • Set up the LP at the root node • display of a solution LP: • Test feasibility of a solution • Column generation method • Computation of a lower bound • Branching decision

  19. Col. Gen. Example: Multicommodity Flow (MCF-1) • Directed graph G = ( V , E ) • N commodities • ( s i , t i ) : source-sink pair, i = 0 , . . . , N − 1 • d i : supply/demand vector for s i t i − flow , i = 0 , . . . , N − 1

  20. Col. Gen. Example: Multicommodity Flow (MCF-1) • Directed graph G = ( V , E ) • N commodities • ( s i , t i ) : source-sink pair, i = 0 , . . . , N − 1 • d i : supply/demand vector for s i t i − flow , i = 0 , . . . , N − 1 For each arc e ∈ E : • 0 : lower bound for total flow on arc • u e : finite upper bound for total flow on arc (0 ≤ u e ) • w e : unit cost (0 ≤ w e )

  21. MCF: ILP Formulation Solution: • f i : s i t i -flow with supply/demand vector d i � f i e ≤ u e for all e ∈ E • i

  22. MCF: ILP Formulation Solution: • f i : s i t i -flow with supply/demand vector d i � f i e ≤ u e for all e ∈ E • i ILP Formulation: � w T f i min i f i ≤ u � (1) i � � f i f i e = d i e − ∀ v ∈ V , ∀ i (2) v e =( v , w ) ∈ E e =( w , v ) ∈ E 0 ≤ f i ≤ u ∀ i (3) f i integral ∀ i (4)

  23. MCF: Input data Class MCF data (see Member/MCF data.hpp ): • arcs : vector of struct ( tail , head , lb , ub , weight ) • commodities : vector of struct ( source , sink , demand ) • numarcs • numnodes • numcommodities • Setup by MCF data::readDimacsFormat()

  24. MCF: Input data Class MCF data (see Member/MCF data.hpp ): • arcs : vector of struct ( tail , head , lb , ub , weight ) • commodities : vector of struct ( source , sink , demand ) • numarcs • numnodes • numcommodities • Setup by MCF data::readDimacsFormat() Parameter MCF AddDummySourceSinkArcs : Add numcommodities dummy arcs with large weight to ensure feasibility

  25. MCF: Master Problem Master Problem: • Column : s i t i -flow satisfying d i for some i • F i : matrix of all generated s i t i -flows (+ dummy flow) • λ i : multiplier for generated s i t i -flows

  26. MCF: Master Problem Master Problem: • Column : s i t i -flow satisfying d i for some i • F i : matrix of all generated s i t i -flows (+ dummy flow) • λ i : multiplier for generated s i t i -flows Example: all arcs upper capacity 2, source = 0, sink = 3, d = 2. 01 0 02 0 12 0 13 0 23 0 03 2

  27. MCF: Master Problem Master Problem: • Column : s i t i -flow satisfying d i for some i • F i : matrix of all generated s i t i -flows (+ dummy flow) • λ i : multiplier for generated s i t i -flows Example: all arcs upper capacity 2, source = 0, sink = 3, d = 2. 01 0 1 02 0 1 12 0 1 13 0 0 23 0 1 03 2 0

  28. MCF: Master Problem Master Problem: • Column : s i t i -flow satisfying d i for some i • F i : matrix of all generated s i t i -flows (+ dummy flow) • λ i : multiplier for generated s i t i -flows Example: all arcs upper capacity 2, source = 0, sink = 3, d = 2. 01 0 1 2 02 0 1 0 12 0 1 2 13 0 0 0 23 0 1 2 03 2 0 0

  29. MCF: Master Problem � w T F i λ i min i F i λ i ≤ u � (5) i e T λ i = 1 (6) ∀ i λ i ≥ 0 ∀ i (7) F i λ i integer (8) ∀ i

  30. MCF: Master Problem � w T F i λ i min i F i λ i ≤ u � (5) i e T λ i = 1 ∀ i (6) λ i ≥ 0 (7) ∀ i F i λ i integer ∀ i (8) λ 0   � F 0 F 1 F 2 � λ 1  ≤ u  λ 2 1 T λ 0     . . λ 1  = 1 1 T . .    1 T λ 2 . .

  31. MCF: Master Problem � w T F i λ i min i F i λ i ≤ u � ( π ) (5) i e T λ i = 1 ( ν i ) (6) ∀ i λ i ≥ 0 ∀ i (7) F i λ i integer (8) ∀ i Pricing of feasible s i t i -flow f : w T f weight of flow : π T f + ν i dual activity: Reduced cost of flow f = w T f − π T f − ν i = ( w T − π T ) f − ν i

  32. Class MCF vars MCF var : • int commodity : index of commodity • CoinPackedVector flow : positive flow on arcs • weight : objective coefficient See include/MCF var.hpp, Member/MCF var.cpp

  33. Class MCF vars MCF var : • int commodity : index of commodity • CoinPackedVector flow : positive flow on arcs • weight : objective coefficient See include/MCF var.hpp, Member/MCF var.cpp MCF lp::vars to cols() : generate columns of the master problem for vars

  34. MCF: Setting the Master at the Root Variables: • Dummy flow variables are algorithmic variables ( λ i 0 ∀ i ) • All generated variables are algorithmic See in TM/MCF tm.cpp : MCF tm::initialize core MCF tm::create root

  35. MCF: Setting the Master at the Root Variables: • Dummy flow variables are algorithmic variables ( λ i 0 ∀ i ) • All generated variables are algorithmic Constraints: • All constraints are core constraints • Upper bound constraints: 0 ≤ u e ∀ e ∈ E • Dummy upper bound constraints: dem ( i ) λ i 0 ≤ dem ( i ) ∀ i • Convexity constraints: λ i 0 = 1 ∀ i See in TM/MCF tm.cpp : MCF tm::initialize core MCF tm::create root

  36. Class MCF tm : Derived from BCP tm user Data: • MCF data data Methods: • pack module data() : pack data needed at the node level. Called once for each processor used as a solver. • initialize core() : Transmit core constraints/variables to BCP. • create root : set up the problem at the root node • pack var algo() : pack algorithmic vars • unpack var algo() : unpack algorithmic vars • display feasible solution() : display solution

  37. Node operations 1. Initialize new node 2. Solve node LP 3. Test feasibility of node LP solution 4. Compute lower bound for node LP 5. Fathom node (if possible) 6. Perform fixing on vars 7. Update row effectiveness records 8. Generate cuts, Generate vars 9. Generate heuristic solution 10. Fathom node (if possible) 11. Decide to branch, fathom, or repeat loop 12. Add to node LP the cuts/vars generated, if loop is repeated 13. Purge cut pool, var pool

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