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Solving MINLP problems with AIMMS

Solving MINLP problems with AIMMS. Pittsburgh June 4, 2014 Marcel Hunting AIMMS Software Developer. Overview. Introducing AIMMS Generated Math Program (GMP) Outer Approximation AIMMS Presolver Implement Branch-and-Bound. AIMMS Modeling Structure.

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Solving MINLP problems with AIMMS

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  1. Solving MINLP problems with AIMMS Pittsburgh June 4, 2014 Marcel HuntingAIMMS Software Developer

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

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

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

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

  6. Symbolic MP Generated MP Matrix • symbolic variables • symbolic constraints • 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 . . . Generated Math Program (GMP)

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

  8. 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

  9. 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) versionfor convex MINLP • Uses lazy constraints callback • Uses nested solve

  10. Results AOA - COA

  11. Results COA: 1 versus 4 Threads

  12. 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

  13. Linearize quadratic constraints Constraint (binary, and continuous or integer) Linearization:

  14. 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

  15. 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 := …

  16. 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) );

  17. 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) );

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