A Generic Benders Decomposition Algorithm for the AIMMS Modeling Language Informs 2012, Phoenix

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1 A Generic Benders Decomposition Algorithm for the AIMMS Modeling Language Informs 2012, Phoenix Marcel Hunting Paragon Decision Technology Copyright by Paragon Decision Technology BV Julianastraat 30, 2012 ES Haarlem, The Netherlands Tel: Fax: No part of this Paragon publication Decision may be reproduced, Technology stored BV in Haarlem a retrieval -system, The Netherlands or transmitted in any form info@aimms.com or by any means - electronic, Tel: mechanical, photocopying, Fax: recording, or otherwise - without the permission of Paragon Decision Technology BV This document provides an outline of a presentation and is incomplete without the accompanying oral commentary and discussion.

2 Overview Introduction Benders decomposition GMP library Benders implementation in AIMMS Example: network design Future work & conclusions

3 Benders: problem formulation Minimize Subject to c T x + d T y Ax + Qy b x X y 0 η = d T y X: Tx f x can be integer, continuous or mixed

4 Benders: problem formulation Minimize Subject to c T x + η Ax + Qy b x X y 0 η = d T y X: Tx f x can be integer, continuous or mixed

5 Benders decomposition algorithm (classic) Solve master problem optimal solution (x k,η k ) Solve dual subproblem fixing x k If unbounded: get unbounded extreme ray to add feasibility cut T (b Ax) 0 to master If feasible: use dual solution to add optimality cut η T (b Ax) to master Repeat until optimality criterion satisfied

6 Benders: primal subproblem Primal subproblem Minimize Subject to d T y Qy b Ax* y 0 Feasibility problem Minimize z w i 0 Subject to Q it y w i z b i A it x* i y, z 0 Fischetti, Salvagnin, Zanette (2010): w i = 0 if A i = 0

7 Model generation Variables: JobSchedule(j,s) StartTime(s,m) TimeSpan Constraints: OneJobPerSchedule(s) OneSchedulePerJob(j) MachineStartTime(s,m) ScheduleStartTime(s,m) Solve Columns: c0 c48 c49 c118 c119 Rows: r0 r6 r7 r76 r14 r76 r77 r136

8 Generated Math Program (GMP) Symbolic MP symbolic variables symbolic constraints Generated MP Matrix generated columns generated rows generated matrix coefficients mappings from/to variables and constraints Solution Repository 1 solution status level values [basis information] 2 solution status level values [basis information]... Pool of Solver Sessions 1 solver option settings 2 solver option settings...

9 Using GMP library Normally: solve MathProgram; GMP: mygmp := GMP::Instance::Generate(MathProgram); GMP::Instance::Solve(myGMP); Modify: GMP::Column::SetUpperBound(myGMP,StartTime(s1,m1),5); GMP::Instance::FixColumns(myGMP,1,AllIntegerVariables);

10 Selection of GMP functions GMP::Instance:: Generate, CreateFeasibilityProblem, CreateDual, Solve GMP::Column:: Add, Delete, Freeze, SetLowerBound GMP::Row:: Add, Delete, SetRightHandSide GMP::Coefficient:: Set, Get, SetQuadratic GMP::Solution:: Copy, SendToModel, GetColumnValue GMP::Benders:: CreateMasterProblem, CreateSubProblem, UpdateSubProblem, AddOptimalityCut, AddFeasibilityCut

11 Using Benders (easy!) mygmp := GMP::Instance::Generate(MathProgram); GMPBenders::BendersOptimalityTolerance := 1e-6; GMPBenders::UseDual := 1; GMPBenders::FeasibilityOnly := 0; GMPBenders::AddTighteningConstraints := 1; GMPBenders::DoBendersDecomposition( /* GMP */ mygmp, /* MasterVariables */ AllIntegerVariables, /* UseStartingPoint */ 0 );

12 DoBendersDecomposition code (under the hood) gmpm := GMP::Benders::CreateMasterProblem(myGMP, MasterVariables, FeasibilityOnly:0, AddTighteningConstraints); gmps := GMP::Benders::CreateSubProblem(myGMP,gmpM,UseDual:1); GMP::Instance::Solve(gmpM); GMP::Benders::UpdateSubProblem(gmpS, gmpm, 1); GMP::Instance::Solve(gmpS); if ( GMP::Solution::GetProgramStatus(gmpS, 1) = 'Unbounded' ) then GMP::Benders::AddFeasibilityCut(gmpM, gmps, 1, NrFeasCuts); else GMP::Benders::AddOptimalityCut(gmpM, gmps, 1, NrOptCuts); endif;

13 Using Benders: modern approach Single MIP search tree (lazy constraint callback): GMPBenders::DoBendersDecompositionOneMIP ( /* GMP */ mygmp, /* MasterVariables */ AllIntegerVariables ); Two phase 1. RMIP (classic), 2. MIP (classic or single MIP): GMPBenders::DoBendersDecompositionTwoPhase ( /* GMP */ mygmp, /* MasterVariables */ AllIntegerVariables, /* UseSingleMIP */ 0 / 1 );

14 Tightening constraints Fischetti, Salvagnin, Zanette (2010) used idea with w i = 0 if A i = 0 on Network expansion problem Minimize i x i Subject to i M- y i i M+ y i b i i y i u i x i i y i 0 i x i binary i Valid inequality: i M- u i x i b i

15 Example: network design Undirected graph (V,E); capacity C e and cost c e Each edge e E can be used in both directions > (V,A) Set of commodities Q with origin s(q) and destination t(q) Demand value d q for every q Q Goal: find cheapest capacities on e E so that the resulting network can satisfy each demand 1,1 B 1,3 D A 2 3,1 2 3,1 C 3,2 1,2 E 1,1

16 Setup Instances from SNDlib ( Benders: Solve RMIP (classic) followed by MIP (single MIP search tree) Primal subproblem Feasibilty cuts only Normalize feasibility problem according to Fischetti et al (2010) Tightening constraints CPLEX 12.4 using one thread Time limit: 3 hours

17 Results SNDlib NCANN models Problem Benders CPLEX cost266uue 8.00% 6485 cost266uum 7.10% 1.46% dfn-bwindbe 27.87% 26.86% dfn-gwindbm di-yuandbe di-yuanuue francedbm germany50-dbm 7.68% 2.70% janos-usddm % janos-us-caddm 0.01% 0.18% Problem MIPLIB 2010: Recently Benders solved CPLEX newyorkuue % newyorkuum % nobel-euuue 0.39% 0.98% nobel-germadbe MIPLIB : 422 norwaydbe Open % norwayuum % ta1uue 6 87 ta2uue 7 18 zib54dbe % zib54uue

18 Future work Multiple subproblems > multi-cut versus single-cut Combinatorial Benders cuts (Codato & Fischetti, 2006) Nonlinear problems Other tightening constraints Cut bundling

19 Conclusions Benders module in AIMMS offers easy way to try feasibility of Benders algorithm for specific MIP problems Benders module uses modern developments (e.g., single MIP search tree) White box algorithm that you can modify Beta version (almost) available

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