1. www.aimms.com Solving MINLP problems with AIMMS Pittsburgh June 4, 2014 Marcel Hunting AIMMS Software Developer

2. www.aimms.com Overview Introducing AIMMS Generated Math Program (GMP) Outer Approximation AIMMS Presolver Implement Branch-and-Bound

3. www.aimms.com AIMMS Modeling Structure AIMMS, integrated & interactive modeling system – Modeling language, integrated GUI, direct access to solvers, advanced deployment options, and extensive development tools

4. www.aimms.com AIMMS Modeling Structure AIMMS, integrated & interactive modeling system – Modeling language, integrated GUI, direct access to solvers, advanced deployment options, and extensive development tools

5. www.aimms.com (Some) AIMMS Features Point & Click / Drag & Drop IDE Global & local compilation Procedural execution & definitions Advanced diagnostic & development tools: debugger, profiler, MP Inspector, data pages Data management & batch run options Modeling of time constructs Broad class of commercial and open source solvers Broad call of model types (LP, MIP, NLP, MINLP, CP, etc.)

6. www.aimms.com Model generation Variables: JobSchedule(j,s) StartTime(s,m) TimeSpan Constraints: OneJobPerSchedule(s) OneSchedulePerJob(j) MachineStartTime(s,m) ScheduleStartTime(s,m) Columns: c0 … c48 c49 … c118 c119 Rows: r0 … r6 r7 … r13 r14 … r76 r77 … r136 Solve

7. www.aimms.com 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... Symbolic MP symbolic variables symbolic constraints Generated Math Program (GMP)

8. www.aimms.com Basic GMP Normally: solve MathProgram; GMP: myGMP := GMP::Instance::Generate(MathProgram); GMP::Instance::Solve(myGMP); Modify: GMP::Column::SetUpperBound(myGMP,StartTime(s1,m1),5);

9. www.aimms.com Selection of GMP functions GMP::Instance:: Generate, Solve, Copy, FixColumns, CreateFeasibilityProblem, CreatePresolved GMP::Column:: Add, Delete, Freeze, SetLowerBound GMP::Row:: Add, Delete, SetRightHandSide GMP::Coefficient:: Set, Get, SetQuadratic, GetQuadratic GMP::Solution:: Copy, SendToModel, GetColumnValue GMP::Linearization:: Add, Delete GMP::SolverSession:: Execute, AsynchronousExecute

10. www.aimms.com Outer Approximation module Module: GMPOuterApproximation Call: myGMP := GMP::Instance::Generate( myMathProgram ) ; GMPOuterApprox::DoOuterApproximation( myGMP ); Uses AIMMS presolver by default Can be combined with Multi-start module Quesada & Grossmann (1992) version for convex MINLP Uses lazy constraints callback Uses nested solve

11. www.aimms.com Results AOA - COA ProblemAOACOA BatchS151208M176 BatchS201210M416 CLay0205H175 CLay0305H318 FLay04H332 FLay05H> 3hr172 fo7_25411 fo911615183 netmod_dol238863 netmod_kar11425 no7_ar3_1142265 ProblemAOACOA o74494629 o7_ar4_12923643 RSyn0840M04H78 RSyn0840M04M3315 SLay08H635 SLay09M485 SLay10H> 3hr505 Syn40M04H22 trimloss435617 Water0303135 Water0303R2212

12. www.aimms.com Results COA: 1 versus 4 Threads Problem1 thr4 thr CLay0305H83 FLay05H17262 fo7_2115 fo95183937 netmod_dol26322 no7_ar3_126533 o7629323 Problem1 thr4 thr o7_ar4_1643432 RSyn0840M04H88 RSyn0840M04M157 SLay09H1724 SLay10H505191 trimloss41713 Water0303R1213

13. www.aimms.com AIMMS Presolver Delete redundant constraints & fixed variables Bound Tightening - Feasibility based – Variable x: range [0,inf) - ► range [10,55] – Linear & nonlinear constraints Improve coefficients (possibly using probing) Linearize quadratic constraints

14. www.aimms.com Linearize quadratic constraints

15. www.aimms.com Branch-and-Bound MINLP problem with binary variables x(i) and y(i,j). Implement branching; choose most fractional column. gmpBB: Generated Math Program for a node in B&B tree

16. www.aimms.com Branching for (i) do xLev(i) := GMP::Column::GetColumnValue( gmpBB, 1, x(i) ); xHalfGap(i) := abs( xLev(i) - 0.5 ); endfor; xMostFractionalColumn := ArgMin( i, xHalfGap(i) ); for (i,j) do yLev(i,j) := GMP::Column::GetColumnValue( gmpBB, 1, y(i,j) ); yHalfGap(i,j) := abs( yLev(i,j) - 0.5 ); endfor; yMostFractionalColumn := ArgMin( (i,j), yHalfGap(i,j) ); MostFractionalColumn := …

17. www.aimms.com Branching - Improved Vars := { ‘x’, ‘y’ }; ColNrs := GMP::Instance::GetColumnNumbers( gmpBB, Vars ); For example: ColNrs = {3,4,…,10,20,21,…,43} index: c for (c) do Lev(c) := GMP::Column::GetColumnValue( gmpBB, 1, c ); HalfGap(c) := abs( Lev(c) - 0.5 ); endfor; MostFractionalColumn := ArgMin( c, HalfGap(c) );

18. www.aimms.com Branching - Improved Vars := AllIntegerVariables; ColNrs := GMP::Instance::GetColumnNumbers( gmpBB, Vars ); For example: ColNrs = {3,4,…,10,20,21,…,43} index: c for (c) do Lev(c) := GMP::Column::GetColumnValue( gmpBB, 1, c ); HalfGap(c) := abs( Lev(c) - 0.5 ); endfor; MostFractionalColumn := ArgMin( c, HalfGap(c) );

19. www.aimms.com