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School on Optimization, Le Croisic, 23-24, March, 2002. 1 Hybrid Constraint Solving in ECLiPSe: Framework and Applications Farid AJILI, IC-Parc, Imperial College, London. March, 2002. (CP-AI-OR’02 School On Optimization, Le Croisic, France)

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School on Optimization, Le Croisic, 23-24, March, 2002. 2 Agenda Combinatorial Optimization Integration of CP and OR techniques CP and OR: what are they ? Why both ? ECLiPSe as a platform for hybrid solvers Ingredients Features Probe Backtrack Search for CP-OR hybridization Application: scheduling with piecewise linear optimization Conclusion.

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School on Optimization, Le Croisic, 23-24, March, 2002. 3 Combinatorial Optimization (1) Intractable No efficient general algorithm is known Search for solution requires exponential time/space, e.g., · Problem size=10 solved in 1 nanosecond · Problem size=100 solved in 4 * age of the universe Capture most of “real life” requirements No solution approach is universally successful Constraint Programming Operations Research

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School on Optimization, Le Croisic, 23-24, March, 2002. 4 Combinatorial Optimization (2) Large scale prohibitively expensive Problem specificity Every problem is unique Very similar problems are actually different (sub-problem sizes, linking constraints,…) Textbook solutions often won’t work without modification/extensions Problem structure embeds well-studied sub-problems Scope for hybrid algorithms

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School on Optimization, Le Croisic, 23-24, March, 2002. 5 Constraint Programming CLP Program = Logic + Control Logic => Modelling: Expressive power (e.g., global constraints) Clear and simple semantics Control => Solving: Fully exploit the structure of the constraints Constraints communicate via variable domains Inference based on constraint propagation/domain reduction Search driven by problem-specific heuristics

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School on Optimization, Le Croisic, 23-24, March, 2002. 6 CLP Modelling Variables Type Domain Constraints N-ary relations Built-ins or user- defined predicates Model = setup program X1{1..9} alldifferent([ _, _, _ ]) X3{1..9} X2{1..9} X4{1..9} _ #> _ _ #\= _ [X1,X2,X3,X4]::1..9, X1 #> X2, alldifferent([X2,X3,X4]), X1 #\= X4.

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School on Optimization, Le Croisic, 23-24, March, 2002. 7 CLP Solving CLP solving interleaves inference and search The inference is supported by constraint propagation Extract information from the constraint network The network is simplified: inconsistent domain values are removed Propagation is monotonic, sound and terminates Search Search disjunctive constraints through backtrackable decisions Propagation might exclude some choices “Guess” a decision using heuristics

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School on Optimization, Le Croisic, 23-24, March, 2002. 8 A Solving Example (1) X{red,green} Y{red,blue} Z{red,green,blue} \= Search control Search control

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School on Optimization, Le Croisic, 23-24, March, 2002. 9 A Solving Example (2) X{red,green} Y{red,blue} Z{red,green,blue} \= Search control Search control ?

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School on Optimization, Le Croisic, 23-24, March, 2002. 10 A Solving Example (3) X{red} Y{red,blue} Z{red,green,blue} \= Search control Search control

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School on Optimization, Le Croisic, 23-24, March, 2002. 11 A Solving Example (4) X{red} Y{blue} Z{green,blue} \= Search control Search control

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School on Optimization, Le Croisic, 23-24, March, 2002. 12 A Solving Example (5) X{red,green} Y{red,blue} Z{red,green,blue} \= Search control Search control red blue green A solution

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School on Optimization, Le Croisic, 23-24, March, 2002. 13 The CLP View Constraint setup ( = Modeling ) Xi Cj Variables & Constraints Heuristics & Search control Propagation & Domain reduction Generic CLP program: solve(Vars) :- constrain(Vars), search(Vars). constrain(Vars) :- search(Vars) :-

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School on Optimization, Le Croisic, 23-24, March, 2002. 14 Operational Research Often associated to Mathematical Programming (MP) Model and analyse decision problems Assumption: optimization function and constraints are functions of the decision variables. Restricted modelling power Numeric domains Equations/Inequalities Integrality constraints Solution methods take advantage of special structure of problems Linear Programming (LP) (e.g., Simplex), Branch and Bound (e.g., MIP) Cutting Planes.

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School on Optimization, Le Croisic, 23-24, March, 2002. 15 A MP Formulation Given |J| sites and |I| customers: Let c ij be the cost of satisfying demand i from the facility located at site j Let y ij = 1 if demand of client i is served by facility j, 0 otherwise Let x j = 1 if a facility is opened at site j, 0 otherwise. LP Relaxation Non-linearity

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School on Optimization, Le Croisic, 23-24, March, 2002. 16 CP versus OR CP l more general constraints l handle integers directly l Favourable example: Variable Bounds: [X 1,…,X 100 ] :: 1..100 Previous Constraints: X 1 < X 2, …, X 98 < X 99 Resulting Bounds: X 1 :: 1..2, …, X 99 :: 99..100 New Constraint: X 1 >= 3 Result (1 step): failure! CP l more general constraints l handle integers directly l Favourable example: Variable Bounds: [X 1,…,X 100 ] :: 1..100 Previous Constraints: X 1 < X 2, …, X 98 < X 99 Resulting Bounds: X 1 :: 1..2, …, X 99 :: 99..100 New Constraint: X 1 >= 3 Result (1 step): failure! OR l restricted class of constraints l finds optimum without search l Favourable example: Variable Bounds: [X 1,X 2 ] :: 1..100 New Constraints: X 1 > X 2, X 2 > X 1 Result (1 step): failure! OR l restricted class of constraints l finds optimum without search l Favourable example: Variable Bounds: [X 1,X 2 ] :: 1..100 New Constraints: X 1 > X 2, X 2 > X 1 Result (1 step): failure!

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School on Optimization, Le Croisic, 23-24, March, 2002. 17 CP-MP Hybridization Hybridization of traditional CP & MP Successful approach for large-scale combinatorial optimization Benefits are increasingly recognized They have complementary characteristics · Pure CP · Powerful for satisfaction of combinatorial constraints · Weak on optimization because inference is based on local consistency · Pure MP · Special-purpose optimization for a special classes of problems (e.g. linear) · Weak on satisfaction of combinatorial constraints Efficiency via CP-MP hybridization Decomposition into well-structured sub-problems Exploit problem structure and specificity

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School on Optimization, Le Croisic, 23-24, March, 2002. 18 Ingredients for Hybridization Expressive power of the “hybrid” language Built-in primitives Setup complex constraints before and during search Flexible modelling The hybrid algorithm consists of 1. backtrack search procedure, and 2. “support methods” which assist it Accommodate co-operative solvers Cooperate with the search as it progresses Selection & Scope of appropriate methods to the sub-problems Provide “glue” for heterogeneous solvers Link sub-problems through variables and channels (“glue”) Advanced control of solvers

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School on Optimization, Le Croisic, 23-24, March, 2002. 19 The ECLiPSe Platform ECLiPSe is a CLP environment Development ECRC, Munich IC-Parc AND Parc-Technologies Ltd Growing academic user community More than 400 academic licenses issued Objectives of ECLiPSe Development & Delivery of software solutions to real world problems Solutions = Hybrids Novel approach: support hybrid algorithms

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School on Optimization, Le Croisic, 23-24, March, 2002. 20 Features for Hybridization Constraint setup Common conceptual model: one program for different solvers Different solvers handle different constraint classes Programmable mapping/relaxing into solver constraints Synchronisation and integration Search decisions are communicated to all solvers Solvers are dealing with parts of the same problem Information passing between solvers How the information is integrated back to the search Incremental prototyping Incremental development of the solver Incremental extension of the problem

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School on Optimization, Le Croisic, 23-24, March, 2002. 21 Solver Cooperation in ECLiPSe “Demons” and data-driven triggering Fine grained waking conditions (narrowed bounds,...) suspend(simplex_solve(Handle), 5, [X->min, X->max]) Priority scheme Cheap agents first, slow ones later e.g., interval propagation before Simplex Attributed variables Bounds, Domain, Range Tentative value (later) Repair-based forward search Search independently sub-problems Support methods to aid the backtrack search Forward search methods

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School on Optimization, Le Croisic, 23-24, March, 2002. 22 Repair library Basic idea Start with a “good” inconsistent “solution” Increase consistency incrementally Applications Repair Problems · “good” inconsistent solution: the “previous” solution Repair-Based Constraint Satisfaction · “good” inconsistent solution: the partially consistent soln. found by heuristics Repair-Based Constraint Optimization · “good” inconsistent solution: a good soln. with respect to optimization function Hybridization · “good” inconsistent solution: a good soln. produced by a forward search method

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School on Optimization, Le Croisic, 23-24, March, 2002. 23 Basic concepts Variables can have “tentative values” X::1..9, X tent_set 5. X = X{fd:[1..9], repair:5} Their changes are propagated [X,Y,Z] tent_set [1,1,1], S tent_is X+Y+Z, Y tent_set 5. S = S{repair:7} Constraints can be monitored for violation [X,Y,Z] tent_set [1,2,3], X + Y #= Z r_conflict confset, % X + Y #= Z satisfied X tent_set 2, % X + Y #= Z violated conflict_constraints(confset,Conf). % Conf=[X+Y#=Z] X{ } Y{ } fd:3..6 fd:1..3repair:3 repair:5 #>=#>= r_conflict

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School on Optimization, Le Croisic, 23-24, March, 2002. 24 Probe Backtrack Search Partition problem into easy and hard part Easy part: tractable Hard part: remaining constraints Solve the easy part using a specialised OR solver Value suggestions (a probe) as a tentative assignment Characteristics: · Optimally, satisfy the “easy sub-problem” · “Super-optimal” and partially consistent · Discrete values Incrementally repair violations in the hard part Make a choice: add an easy constraint (on backtracking, its negation) Aim: reduce hard constraint violation in subsequent probes Easy Part Cost Function Hard Part

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School on Optimization, Le Croisic, 23-24, March, 2002. 25 Algorithm Outline search:- define_problem(Vars), probe_tent_values(c,Vars), % assigns tentative values repair_label(c, Vars). repair_label(CS,_):- conflict_constraints(CS, []), !. repair_label(CS, Vars): conflict_constraints(CS, [Constr|_]), fix_constr(Constr), probe_tent_values(CS, Vars), % re-assigns tentative values to “better” values repair_label(CS, Vars). search:- define_problem(Vars), probe_tent_values(c,Vars), % assigns tentative values repair_label(c, Vars). repair_label(CS,_):- conflict_constraints(CS, []), !. repair_label(CS, Vars): conflict_constraints(CS, [Constr|_]), fix_constr(Constr), probe_tent_values(CS, Vars), % re-assigns tentative values to “better” values repair_label(CS, Vars).

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School on Optimization, Le Croisic, 23-24, March, 2002. 26 A Toy Example (1) define_problem(Vars, rc, Handle, Trigger) :- Vars = [X1,X2,X3], fdplex:(Vars:: 1.. 4), fdplex:(X2 - X3 #=<2), fdplex:(X2 #=< X1), fdplex:(-X2 + X3 #= 3), lp_demon_setup(min(X1 - X3),Cost,[],9,[trigger(simplex)],Handle), lp_get(Handle,vars,Vs), lp_get(Handle, solution, Solution), Vs tent_set Solution, sos2([X1,X2,X3]) r_conflict rc, Trigger = simplex. sos2(Vars):- Vars tent_get Vals, sos2_is_ok(Vals). define_problem(Vars, rc, Handle, Trigger) :- Vars = [X1,X2,X3], fdplex:(Vars:: 1.. 4), fdplex:(X2 - X3 #=<2), fdplex:(X2 #=< X1), fdplex:(-X2 + X3 #= 3), lp_demon_setup(min(X1 - X3),Cost,[],9,[trigger(simplex)],Handle), lp_get(Handle,vars,Vs), lp_get(Handle, solution, Solution), Vs tent_set Solution, sos2([X1,X2,X3]) r_conflict rc, Trigger = simplex. sos2(Vars):- Vars tent_get Vals, sos2_is_ok(Vals). No Integrality Constraints

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School on Optimization, Le Croisic, 23-24, March, 2002. 27 A Toy Example (2) probe_tent_values(Handle,Trigger) :- schedule_suspension(Trigger), wake, lp_get(Handle,vars,Vars), lp_get(Handle,solution,Solution), Vars tent_set Solution. fix_constr(sos2([X1,X2,X3])) :- ( X3 = 0 ; X1 = 0 ). probe_tent_values(Handle,Trigger) :- schedule_suspension(Trigger), wake, lp_get(Handle,vars,Vars), lp_get(Handle,solution,Solution), Vars tent_set Solution. fix_constr(sos2([X1,X2,X3])) :- ( X3 = 0 ; X1 = 0 ). X3=0 X1=0

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School on Optimization, Le Croisic, 23-24, March, 2002. 28 Input schedule (fixed times) s1s1 s2s2 s3s3 e1e1 e2e2 e3e3 No. of Resources Required Time 3 2 1 3 2 1 s1s1 s2s2s3s3

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School on Optimization, Le Croisic, 23-24, March, 2002. 29 Output: retime activities New Requirements : 1) Only, two resources 2) Precedence constraints: S1 :: L … U 20 S1–E2 S2– E3=60 etc … No. of Resources Required s1s1 s2s2 s3s3 S1 S2 S3 E1 E2 E3 3 2 1

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School on Optimization, Le Croisic, 23-24, March, 2002. 30 Some use-cases Dynamic scheduling problems Existing schedule is given Constantly changing schedule environment Use cases for re-scheduling Cope with resource failures Additional activities/requirements Resource saving Criteria Minimal perturbation to existing schedule Minimal schedule cost Maximise schedule revenues Starting point: industrial transportation application

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School on Optimization, Le Croisic, 23-24, March, 2002. 31 Problem Interest Piecewise Linear (PL) optimization Captures most common objective criteria · Minimal perturbation · Tardiness · Earliness Piecewise linearizations are often a good approximation Hybrid algorithm for minimal perturbation It significantly outperforms individual CP and CPLEX MIP solvers [El-Sakkout, Wallace 2000] Time_i Cost_i T0=Preferred Time

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School on Optimization, Le Croisic, 23-24, March, 2002. 32 Problem Input Scheduling with piecewise linear (PL) optimisation Input Resources with their capacities, activities and their resource demands For each activity, a PL constraint (Time_i, Cost_i) Temporal constraints: · Let # be a relation in {=, , , , }, and U,V be temporal variables · Constraints are of the form: U # c or U # V+c where c Assumption: time discreteness Time_i Cost_i

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School on Optimization, Le Croisic, 23-24, March, 2002. 33 The Problem as a CSP Constrained variables Start/End time variables of activities Boolean variables capturing whether an activity spans with an other activity start time Cost-related variables Constraint system Resource utilisation rules · Resource demand is smaller than or equal to the available amount · Resource maintenance requirements Temporal constraints · Bounding (e.g., time windows) · Distance/precedence relations Objective function constraints · Cost definition · PL constraints · Cost bounds

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School on Optimization, Le Croisic, 23-24, March, 2002. 34 Conceptual Model CP hard set MP easy set Domain reduction Interval propagation Lookahead resource bound checking Heuristics Repair Global cost Optimal assignment Discreteness InferenceRelaxation+Probe

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School on Optimization, Le Croisic, 23-24, March, 2002. 35 Probing Issues Curves are non-convex (unlike Minimal Perturbation) Pure Linear Programming is not enough Which part handles the non-convexity of the piecewise linearity ? 1. Easy part contains a linear relaxation of the PL constraints · Linear Programming is enough · More search to fix the PL violations 2. Easy part contains the PL constraints · Need to use Mixed Integer Programming (MIP) Different options for the probers The -formulation (MIP) The -formulation (MIP) Linear relaxation (LP) MIP/LP problem CP search Probing

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School on Optimization, Le Croisic, 23-24, March, 2002. 36 The Hybrid Algorithm Probing phase: Purpose: get a tentative temporal assignment (“probe”) Scope: minimize Cost subject to Temporal & PL constraints Structure: the integrality of the temporal variables is guaranteed Resource feasibility phase: Monitor resource constraints for violation Probe guides the search towards “conflict regions” Repair incrementally the violations · Select a resource violation (e.g., the maximum infeasibility,..) · “Guess” a distance constraint reducing the contention · The constraint should preserve the tractability of the probing sub-problem X

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School on Optimization, Le Croisic, 23-24, March, 2002. 37 The role of the probe Fixed Variables X{4}4 Tentative Variables Y{1..4}3 Conflict Region of Violated Constraints 1.High quality 2.Focus the search

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School on Optimization, Le Croisic, 23-24, March, 2002. 38 Outline of the Algorithm 1. Apply propagation to all the constraints 2. Temporal && PL Optimisation Apply the Prober to the easy part 3. Determine the set of violated constraints 4. Select a violated constraint 5. Feasibility (Backtrackable decision) Impose a new temporal ordering constraint C to reduce violations consistent optimum integer values none: Exit with success

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School on Optimization, Le Croisic, 23-24, March, 2002. 39 Some Notes Discreteness of the the temporal variables Make easier the integration with FD reasoning Caution: no cost bound in the MIP formulation Inference via probing Propagation based on the “Reduced costs” Fix variables to their lower/upper bounds if them will not improve the cost Close cooperation is more effective The probing sub-problem is dynamically re-shaped during search Tighten individual PL constraints [Refalo, 99]

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School on Optimization, Le Croisic, 23-24, March, 2002. 40 The -formulation Tightening the probing sub-problem Variable Fixing [Refalo, 99] Infer further “cuts” on the s For each (Time, Cost) Time Cost

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School on Optimization, Le Croisic, 23-24, March, 2002. 41 The -formulation Tightening the probing sub-problem: Similarly, some s can be fixed [Refalo, 99] Derive further “cuts” on them For each (Time, Cost) Time Cost

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School on Optimization, Le Croisic, 23-24, March, 2002. 42 Linear Relaxation Time Cost T Time T 1+T Time Model (Time, Cost) by its convex hull Split the domain of Time and branch Incrementally updated.

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School on Optimization, Le Croisic, 23-24, March, 2002. 43 The Repair Strategy Two types of violation Resource violations PL violations Repair priority 1) Resource violations 2) PL violations Advantage Favours resource feasibility · Initially ignore accuracy of costs 1 1 2 3 2 Time |Res|

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School on Optimization, Le Croisic, 23-24, March, 2002. 44 Evaluation & Discussion Evaluation on randomly generated tests Problem tightness Amount of resource saving PL optimisation: “distance” from the convex case The strength of the pruning in H(MIP) H(relax ) outperforms H( - prober): “First, repair resource violations” versus “First, repair PL violations” Cheap probing versus probe quality Ongoing work with other applications Networking area

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School on Optimization, Le Croisic, 23-24, March, 2002. 45 Conclusion Even when problem precludes OR, hybridization is beneficial! ECLiPSe is an environment for integration of Models Solvers More hybrid schemes are available Bender’s Decomposition Column Generation CP mixed with Langrangian Relaxation CP combined with Local Search Finite Domains & Real Interval Propagation and 44 solvers/libraries.

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School on Optimization, Le Croisic, 23-24, March, 2002. 46

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