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The EPLEX Library

The EPLEX Library. Introduction. lib(eplex) An interface between ECLiPSe and an external LP/MIP solver COIN-OR Open Solver Interface: currently to COIN’s CLP/CBC and SYMPHONY/CLP (Open Source) XPRESS-MP, a product by Dash Associates CPLEX, a product by ILOG SA Motivation to have this link:

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The EPLEX Library

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  1. The EPLEX Library

  2. Introduction • lib(eplex) • An interface between ECLiPSe and an external LP/MIP solver • COIN-OR Open Solver Interface: currently to COIN’s CLP/CBC and SYMPHONY/CLP (Open Source) • XPRESS-MP, a product by Dash Associates • CPLEX, a product by ILOG SA • Motivation to have this link: • Use state-of-the-art linear programming tools • Use ECLiPSe for modelling • Build hybrid solvers • Implementation • Tight coupling, using subroutine libraries

  3. Mathematical Programming • Making optimal decisions with limited resources • ‘Programming’ in the sense of planning • Model problem to optimise as an ‘objective’ function, subject to constraints • Classic example: transportation problem

  4. Objective Function Constraints Mathematical Programming Plant capacity Transportation cost Client demand minimize 10 A1 + 7 A2 + 11 A3 + 8 B1 + 5 B2 + 10 B3 + 5 C1 + 5 C2 + 8 C3 + 9 D1 + 3 D2 + 7 D3 subject to A1 + A2 + A3 = 200 B1 + B2 + B3 = 400 C1 + C2 + C3 = 300 D1 + D2 + D3 = 100 A1 + B1 + C1 + D1  500 A2 + B2 + C2 + D2  300 A3 + B3 + C3 + D3  400 A 200 10 1  500 8 9 5 B 7 400 5 2  300 5 3 C 300 10 11 3 8  400 7 D 100 • Leads to a matrix: • Each row is one constraint • Each column is one variable

  5. LP/MIP – A brief overview • Linear Programming: • Optimisation problem formulated using linear constraints and a linear objective function • An efficient algebraic method developed to solve such problems (Simplex, 1947) • Interior point (barrier) method • Mixed Integer Programming: • LP problems where some of the variables are constrained to be integer • No known ‘optimal’ method: generally use simplex/barrier with branch and bound search for optimal • Quadratic Programming • Quadratic objective function, linear constraints • Barrier method

  6. Possible Uses From Within ECLiPSe • Black-box • If problem is suitable for LP/QP/MIP as a whole • But: Only one solution, non-incremental, rounding errors • Incrementality and multiple instances • Results explicitly retrieved • Multiple independent subproblems • Hybrid • Solvers work on (possibly overlapping) subproblems • Programmed cooperation strategy • Data-driven triggering • Low-level interface

  7. General Usage • Loading: • :- lib(eplex). • Loads interface with default solver (OSI CLP/CBC on most machines) • Differences between the underlying solvers are largely hidden • Documentation • ECLiPSe Library Manual • Tutorial chapter on Eplex • Reference Manual

  8. Transportation Problem (black box solving) :- lib(eplex). main(Cost, Vars) :- Vars = [A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, D3], Vars :: 0.0..1.0Inf, A1 + A2 + A3 $= 200, B1 + B2 + B3 $= 400, C1 + C2 + C3 $= 300, D1 + D2 + D3 $= 100, A1 + B1 + C1 + D1 $=< 500, A2 + B2 + C2 + D2 $=< 300, A3 + B3 + C3 + D3 $=< 400, optimize(min( 10*A1 + 7*A2 + 11*A3 + 8*B1 + 5*B2 + 10*B3 + 5*C1 + 5*C2 + 8*C3 + 9*D1 + 3*D2 + 7*D3), Cost).

  9. Eplex Constraints • Constraints • Vars :: Min..Max • Expr1 $= Epxr2 eplex:(Expr1 =:= Epxr2) • Expr1 $>= Epxr2 eplex:(Expr1 >= Epxr2) • Expr1 $=< Expr2 eplex:(Expr1 =< Epxr2) • integers(Vars) • Linear expressions • X • 123 3.4 • +Expr -Expr • E1+E2 E1-E2 E1*E2 • sum( ListOfExpr ) • ListOfExpr1 * ListOfExpr2

  10. Be Aware of Conventions • In mathematical programming, non-negative variables are usually assumed as default • In ECLiPSe, the default is • -1.0Inf .. 1.0Inf • Non-negative variables need to be declared, e.g. • X :: 0.0 .. 1.0Inf

  11. Common Arithmetic Solver Interface

  12. Black-box Optimizer • optimize(+Objective, -Cost) • Objective is min(Expr) or max(Expr) • Runs the external solver as a black box • Considers all $=, $>=, $=< constraints, their variables, the variable bounds and the integers/1 constraints • Computes an optimal solution forObjective • Returns the costCost • Instantiates the variables to the computed solution values • Selects LP/QP/MIP depending on integersand Objective

  13. Y Cost X Black-box use of LP and MIP solver • A linear problem • ?- X+Y $>= 3, X-Y $= 0, optimize(min(X),Cost). • Y=1.5 • X=1.5 • Cost=1.5 • A mixed integer problem • ?- X+Y $>=3, X-Y $= 0, integers([X]), optimize(min(X),Cost). • Y=2.0 • X=2 • Cost=2.0

  14. LP / MIP search Summary: Eplex as black-box solver • CLP modelling / LP or MIP solving • 1. Do modelling and transformations in ECLiPSe • 2. Load model into CPLEX/XPRESS/MP and solve using LP or MIP solver • 3. Pass cost and solution values back to ECLiPSe External Solver ECLiPSe variables & bounds X1 X2 ... Xm solution values = c1 =< c2 constraints >= cn = Cost Obj cost

  15. Problems with Black-Box Approach • Solvable constraint class severely limited • Pure LP/QP/MIP • Numerical problems • The external solvers use floating-point numbers, subject to rounding errors • Other ECLiPSe solvers are usually precise (intervals or rational numbers) • Instantiating the ECLiPSe variables to inexact values may cause other implementations of the constraints to fail! • Lack of incrementality • Cannot find alternative optima • Cannot add constraints or variables • MIP search is a straightjacket • A complex search process inside the black box • Hard to control (via dozens of parameters) • Hard to add problem-specific heuristics • Unsuitable for solver cooperation!

  16. Opening the Black Box • Code for optimize/2: • optimize(Objective, Cost) :- • eplex_solver_setup(Objective), • eplex_solve(Cost), • eplex_get(vars, VarVector), • eplex_get(typed_solution, SolutionVector), • VarVector = SolutionVector.

  17. Incrementality • Solver setup • eplex_solver_setup(+Objective) • Run solver once • eplex_solve(-Cost) • Computes a solution with Cost but does not instantiate variables • Access solution information • Status, solutions, reduced costs, dual values, slack, statistics, etc. • eplex_get(+What, -Value) • eplex_var_get(+Var, +What, -Value) • New constraints and variables can now be added and the problem re-solved!

  18. Multiple problems (eplex instances) • Declare an eplex instance, e.g. • :- eplex_instance(prob). • (an instance calledeplexexists by default) • Constraints • prob:(Vars :: Min..Max) • prob:(Expr1 $= Epxr2) prob:(Expr1 =:= Epxr2) • prob:(Expr1 $>= Epxr2) prob:(Expr1 >= Epxr2) • prob:(Expr1 $=< Expr2) prob:(Expr1 =< Epxr2) • prob:(integers(Vars)) • Solver setup • prob:eplex_solver_setup(+Objective) • Run solver once • prob:eplex_solve(-Cost) • Computes a solution with Cost but does not instantiate variables • Access solution information • prob:eplex_get(+What, -Value) • prob:epelx_var_get(+Var, +What, -Value)

  19. Transportation Problem (with Instance API) :- lib(eplex). :- eplex_instance(prob). main(Cost, Vars) :- Vars = [A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, D3], prob:(Vars :: 0.0..1.0Inf), prob:(A1 + A2 + A3 $= 200), prob:(B1 + B2 + B3 $= 400), prob:(C1 + C2 + C3 $= 300), prob:(D1 + D2 + D3 $= 100), prob:(A1 + B1 + C1 + D1 $=< 500), prob:(A2 + B2 + C2 + D2 $=< 300), prob:(A3 + B3 + C3 + D3 $=< 400), prob:eplex_solver_setup(min( 10*A1 + 7*A2 + 11*A3 + 8*B1 + 5*B2 + 10*B3 + 5*C1 + 5*C2 + 8*C3 + 9*D1 + 3*D2 + 7*D3)), prob:eplex_solve(Cost), ( foreach(Var,Vars) do prob:eplex_var_get(Var, typed_solution, Var) ).

  20. Traditional MIP branching simplex solution: X = 4.2 • At each node: • Solve the continuous relaxation (ignoring integrality) with simplex • Select an integer variable with fractional simplex solution • Try two alternatives, with bounds forced to integers • Eventually, all variables will be narrowed to integers • (if a solution exists) X >= 5 X =< 4

  21. Doing MIP in ECLiPSe :- lib(eplex), lib(branch_and_bound). main :- ... <setup constraints> IntVars = <variables that should take integral values>, Objective = <objective function>, ... Objective $= CostVar, eplex_solver_setup(min(Objective)), eplex_solve(RelaxedCost), % initial simplex solving CostVar $>= RelaxedCost) % apply resulting cost bound ... minimize(mip_search(IntVars, CostVar), CostVar). mip_search(IntVars, CostVar) :- ... % for each X violating the integrality condition eplex_var_get(X, solution, RelaxedSol), Split is floor(RelaxedSol), ( X $=< Split ; X $>= Split+1 ), % choice eplex_solve(RelaxedCost), % simplex re-solving CostVar $>= RelaxedCost, % apply new cost bound ... no integrality constraints!

  22. Ok, but ... • The search procedure became messy • extra argument (CostVar) • explicit simplex calls • modularity lost (choices/propagation) • Do we really need to run simplex every time? • there might be no significant change • Solution: data-driven triggering, e.g. on • instantiation • instantiation far from relaxed solution • bounds change • bounds exclude relaxed solution

  23. Triggering the solver automatically eplex_solver_setup(+Objective, ?Cost, +Options, +TriggerModes) • Objective • min(Expr) or max(Expr) • Cost • variable - it does not get instantiated, but only bounded by the solution cost. • TriggerModes inst - if a variable was instantiated deviating_inst - if a variable was instantiated to a value that differs more than a tolerance from its LP-solution bounds - if a variable bound was changed deviating_bounds - if a variable bound was changed such that its LP-solution was excluded by more than a tolerance. new_constraint - when a new constraint appears <module>:<cond> - waking condition defined by other solver, e.g. ic:min trigger(Atom) - explicit triggering

  24. Using trigger conditions Example • ?- X+Y $>= 3, X-Y $= 0, eplex_solver_setup(min(X), X, [], [inst]). • Y = Y{1.5} • X = X{1.5} • Delayed goals: • lp_demon(…) • Yes. • ?- X+Y $>= 3, X-Y $= 0, eplex_solver_setup(min(X), X, [], [inst]), X=2.0. • Y = Y{2.0} • X = 2.0 • Delayed goals: • lp_demon(…) • Yes.

  25. Eplex solver as compound constraint setup solver • Solver triggered by events, then: • new bounds and constraints are sent to the external solver • external solver is run • cost bound (or failure) is exported • solution values are exported and ECLiPSe variables annotated (optional) X1 X2 ... Xm c1 = External Solver c2 =< cn >= Obj = Cost

  26. More Elegant Implementation of MIP :- lib(eplex), lib(branch_and_bound). main :- ... <setup constraints> IntVars = <variables that should take integral values>, Objective = <objective function>, ... Objective $= CostVar, eplex_solver_setup(min(Objective), CostVar, [], [bounds]), ... minimize(mip_search(IntVars), CostVar). mip_search(IntVars) :- ... % for each X violating the integrality condition eplex_var_get(X, solution, RelaxedSol), Split is floor(RelaxedSol), ( X $=< Split) ; X $>= Split+1 ), % choice ... % simplex re-solving triggered by bounds change % cost bound automatically applied ... • No explicit simplex calls, no auxiliary parameters to search procedure • Standard CLP search scheme - other choices/constraints can be added now

  27. Control Flow in Classical MIP Solver Choice Simplex step Solve continuous relaxation Choice

  28. Control Flow with Constraint Propagation Search/Choice constraints Propagation phase Search/Choice

  29. Control Flow with Constraint Propagation Eplex instance demon can be integrated similar to a global constraint Search/Choice Propagation phase constraints eplex instances Search/Choice

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