introduction to scipscip.zib.de/download/slides/scip-introduction.pdfintroduction to scip marc...

Introduction to SCIP

Marc PfetschZuse Institute Berlin

DFG Research Center MATHEONMathematics for key technologies

Berlin, 10/11/2007

Outline

1 SCIP Facts2 Branch-And-Cut and Constraint Programming3 SCIP Basics

SCIP PluginsProblem SpecificationPresolvingSolving

4 Further Information

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What is SCIP?

SCIP (Solving Constraint Integer Programs) . . .. is a branch-and-cut-and-price framework,. incorporates a full-scale mixed integer programming (MIP) solver,. is also a constraint programming (CP) solver,. incorporates SAT-solving features (conflict analysis, restarts),. is free for academic purposes,. and is available in source-code!

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Comparison of MIP-Solvers

0

1000

2000

3000

4000

Tim

ein

seco

nds

non-

com

mer

cial

com

mer

cial

SymphonylpsolveGLPKMintoCBC 1.2SCIP–CLPSCIP-SoPlexSCIP-CPLEXCPLEX 10.2

7.05x 6.99x

6.12x

4.10x

2.12x

1.06x 1.00x0.37x 0.13x

not solved 73% 81% 72% 55% 40% 16% 16% 1% 4%

. MIP-solver as fast as CPLEX 9.0

. Underlying LP-solver: treated as black box

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SCIP History

1998 SIP – Solving Integer Programs (Alexander Martin)

10/2002 start of SCIP development (Tobias Achterberg)02/2003 first version to solve MIPs05/2003 Gomory cuts08/2003 conflict analysis03/2004 knapsack cover separator (Kati Wolter)04/2004 restarts, c-MIR cuts01/2005 feasibility pump (Timo Berthold)09/2005 first public version 0.8009/2006 version 0.9009/2007 version 1.00

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SCIP Facts

. Approx. 220.000 lines of C code

. Main SCIP developer: Tobias Achterberg → ILOGTimo Berthold (heuristics), Kati Wolter (cuts)

. Current development: Timo Berthold, Kati Wolter, . . .

. Development together with TU Darmstadt.

. Free for academic use (as is SoPlex)

. Available at http://scip.zib.de

. ZIB Optimization Suite = SCIP + SoPlex + ZIMPLhttp://zibopt.zib.de

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http://scip.zib.de

http://zibopt.zib.de

Outline




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Definition: Mixed Integer Program (MIP)

min cTxs.t. Ax ≤ b

l ≤ x ≤ uxj ∈ Z j ∈ J

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min cTx

s.t. Ax ≤ bl ≤ x ≤ u

xj ∈ Z j ∈ J

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l ≤ x ≤ uxj ∈ Z j ∈ J

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l ≤ x ≤ u

xj ∈ Z j ∈ J

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l ≤ x ≤ uxj ∈ Z j ∈ J

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Branch-And-Cut

Combination of

. Branch-and-Bound

. Cutting planes

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Constraint Programming

Constraint Program (CP)

. Domains of variables = (finite) set

. Constraints = subsets of domain space

Easy example:. variables x and y. domains: Dx = Dy = {0, 1}. constraint: {(0, 1), (1, 0)} ⊂ Dx × Dy ⇔ x = 1− y .

Constraint Integer Program (CIP)

Fixing integer variables leaves LP.

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Solving CPs

. Branching:I Divide into subproblems, solve recursively

. Domain Propagation:I Reductions in variables’ domains “propagate”I E.g. x1 + 2x2 ≥ 5, x1 ≤ 2 ⇒ x2 ≥ 1.5

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Propagation Example

Alldifferent constraint:{x1, x2, x3, x4} ∈ {1, 2, 3, 4} pairwise different.

x1

x2

x3

x4

alldiff

⇒

x1

x2

x3

x4

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Outline




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Important Aspects

. SCIP is constraint basedI Advantage: flexibilityI Disadvantage: no dual representation

. A constraint knows its variables, but a variable does not know theconstraints it appears in.

. A constraint usually implements a type of inequalities, not a singleinequality.

. From CP-perspective: LP-relaxation is only an add-on! Manyconstraints do not separate inequalities at all.

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Example: TSPTraveling Salesman problem (TSP)

Given graph G = (V , E ) and distances de , find shortest tour throughall nodes.

IP-formulation:

min∑e∈E

de xe

s.t.∑

e∈δ(v)

xe = 2 ∀ v ∈ V

∑e∈δ(S)

xe ≥ 2 ∀ S ⊂ V , S 6= ∅

xe ∈ {0, 1} ∀ e ∈ E

Example available with SCIP, including visualization.15 / 60

SCIP Stages

Init

Problem Transforming

Free Transform

Presolving

Init Solve

Solving

Free Solve

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Outline




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SCIP Interface

Main SCIP interface: plugins.

. Perform all problem specific actions.

. Each step calls user defined plugins.

. SCIP knows of plugins through “include” functions.

. Plugins may have private data.

. User defined callback functions(virtual functions in C++-interface).

. Yields modular structure.

. The MIP solver is realized through plugins.

Everything SCIP knows, it knows through plugins.

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Types of Plugins

. Constraint handler: assures feasibility, strengthens formulation

. Separator: adds cuts, improves dual bound

. Pricer: allows dynamic generation of variables

. Heuristic: searches solutions, improves primal bound

. Branching rule: how to divide the problem?

. Node selection: which subproblem should be regarded next?

. Presolver: simplifies the problem in advance, strengthens structure

. Propagator: simplifies problem, improves dual bound locally

. Reader: reads problems from different formats

. Event handler: catches events (e.g., bound changes, new solutions)

. Display: allows modification of output

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Default Plugins

5 presolvers5 node selection rules

14 constraint handlers8 separators8 branching rules2 propagators

23 primal heuristics

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TSP Plugins

MIP default plugins 2-Opt Heuristic

No-Subtour Constraint Farthest Insert Heuristic

File Reader New Solution Event Handler

Search Tree LP Relaxation Presolve Management

ImplicationGraph

SolutionPool

CutPool

ConflictAnalysis

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Outline




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Problem Specification

Init


Free Transform

Presolving

Init Solve

Solving

Free Solve

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Define Variables (TSP Example)

SCIP_VAR* var;SCIP_CALL( // return value macro

SCIPcreateVar(scip , // SCIP pointer&var , // save in variable"varname", // pass variable name0.0, // lower bound1.0, // upper boundlength , // obj. valueSCIP_VARTYPE_BINARY , // typeTRUE , // initialFALSE , // removableNULL , NULL , NULL , // no callback functionsNULL // no variable data

));SCIP_CALL( SCIPaddVar(scip , var) ); // add var.

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TSP: Define Degree ConstraintsSCIP_CALL( SCIPcreateConsLinear(

scip , // SCIP pointer&cons , // save in cons"consname", // namenvar , // number of variablesvars , // array of variablesvals , // array of values2.0, // left hand side2.0, // right hand side (equation)TRUE , // initial?FALSE , // separate?TRUE , // enforce?TRUE , // check?TRUE , // propagate?FALSE , // local?FALSE , // modifable?FALSE , // dynamic?FALSE , // removable?FALSE // stick at node?

));SCIP_CALL( SCIPaddCons(scip , cons) ); // add constraintSCIP_CALL( SCIPreleaseCons(scip , &cons) ); // free cons. space

MIPs are specified using linear constraints only (may be “upgraded”).27 / 60

Transformation

Init


Free Transform

Presolving

Init Solve

Solving

Free Solve

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Transformation

Original problem Transformed problem

original variables

original constraints

transformed variables

transformed constraints

All variables and constraints are transformed:

. data are copied into separate memory area

. presolving and solving operate on transformed problem

. original data can only be modified in problem modification stage

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Outline




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Presolving

Init


Free Transform

Presolving

Init Solve

Solving

Free Solve

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Presolving

init exec. presolvers exec. constraint presolving

found?yes no

. Presolvers provide global presolving (e.g. dual fixing)

. Constraint Handlers provide constraint specific presolving, e.g.:I domain tightening,I coefficient modification,I deletion of redundant constraints,I constraint upgrading.

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Probing

Probing:. Tentatively fix binary variables to 0 and 1.. Execute domain propagation of constraints.. Try to deduce global fixings and implications.

Example: x − z ≤ 0,−y + z ≤ 0,

x + y ≤ 1

⇒ x = 0

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Outline




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Solving

Init


Free Transform

Presolving

Init Solve

Solving

Free Solve

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Flow Chart SCIP

Start Init Presolving

Stop

Node selection

Processing

Branching

Conflict analysis

Primal heuristics

LP inf.

LP feas.IP inf.

IP feas.

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Node Selection

. Node Selectors called in order of priority

. Choose:I child nodeI sibling nodeI leaf node stored in the tree

. two operation modes:I standard mode (default: best estimate)I memory-saving mode (default: depth first search)switching between modes is automatic

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Branching,Conflict Analysis, and Heuristics


Stop

Node selection

Processing

Branching

Conflict analysis

Primal heuristics

LP inf.

LP feas.IP inf.

IP feas.

→ separate talks

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Node Solving


Stop

Node selection

Processing

Branching

Conflict analysis

Primal heuristics

LP inf.

LP feas.IP inf.

IP feas.

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Node Solving

Domain Propagation Solve LP

Pricing

Separation

Domain Propagation

LP Solving

Constraint Enforcement

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TSP Example

Recall TSP IP-formulation:

min∑e∈E

de xe

s.t.∑

e∈δ(v)

xe = 2 ∀ v ∈ V

∑e∈δ(S)

xe ≥ 2 ∀ S ⊂ V , S 6= ∅

xe ∈ {0, 1} ∀ e ∈ E

Introduce no-subtour constraint: nosub(x)ensures that x does not contain a subtour.

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TSP CP-Formulation

min∑e∈E

de xe

s.t.∑

e∈δ(v)

xe = 2 ∀ v ∈ V

nosub(x)

xe ∈ {0, 1} ∀ e ∈ E

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TSP No-Subtour Constraint

∑e∈δ(S)

xe ≥ 2 ∀ S ⊂ V , S 6= ∅

No-Subtour Constraint Handler callback functions:. Separate: separate inequalities → added to LP

Compute Gomory-Hu tree (max-flow techniques) and check cuts.. Enforcing: separate subtour inequalities. Checking: check whether an integer solution (produced by a

heuristic) is feasible. Propagation: –. Other callbacks: (de)initialization, “locking”, transforming, printing

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Linear Constraint Handler

linear constraints: α ≤ aTx ≤ β

. Main constraint for MIP solving

. MIP is constructed with linear constraints.

. May be upgraded to knapsack constraints, etc.

. The linear constraint handler providesI presolving methodsI separates the corresponding inequalitiesI propagates variable bounds

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Specially Ordered Sets

Definition (Specially Ordered Set of Type 1)

In given set of variables, at most one is nonzero.

. Nonlinear constraint

. In general: no (good) inequalities (unless bounds are tight)

. Enforcing: branch to remove infeasibility:fix all variables except one to 0.

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Domain Propagation


Pricing

Separation

Domain Propagation

LP Solving


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Domain Propagation

. Propagators provide global domain reduction methods.

. Constraint handlers provide constraint specific domain reduction:Example: knapsack 4x1 + 6x2 + 7x3 ≤ 11, x2 = 1 ⇒ x3 = 0.

. With propagation frequency f :Propagation called every f th level of the branch-and-bound tree

Similar frequencies for separation, enforcing, . . .

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LP Solving


Pricing

Separation

Domain Propagation

LP Solving


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LP Solving

. LP-Interface LPI to various LP-solvers:SoPlex, CPLEX, XPress, CLP, . . .

. Primal or dual simplex

. Barrier with or without crossover (if supported)

. LP stability checked:resolution by changing parameters:scaling, tolerances, solving from scratch, other simplex method.

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Pricing


Pricing

Separation

Domain Propagation

LP Solving


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Pricing

execute pricers new columns?

resolve LPyes

no

. Pricers search for new variables with negative reduced costs

. inform constraint handlers about the new variables

. LP resolved after few new columns

. Currently: partial pricing is not allowed(LP objective value has to be a lower bound.)

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Separation


Pricing

Separation

Domain Propagation

LP Solving


→ separate talk. SCIP has a built-in cut selection methodology.

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Pricing

Separation

Domain Propagation

LP Solving


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If no separating cutting plane was found, solution can still beinfeasible for a constraint.

Enforcing is necessary for a correct implementation!

Constraint handler resolves the infeasibility by. reducing a variable’s domain,. separating a cutting plane (may use integrality),. adding a (local) constraint,. creating a branching, or. concluding that current subproblem is infeasible.

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Outline




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Implementation Hints

. SCIP has own memory handling.I Standard memory: SCIPallocMemoryArray(scip,&p,10),

SCIPfreeMemoryArray(scip,&p)I Block memory: SCIPallocBlockMemoryArray(scip,&p,10),

SCIPfreeBlockMemoryArray(scip,&p,10)I Fast buffer: SCIPallocBufferArray(scip,&p,10),

SCIPfreeBufferArray(scip,&p)

. Functions calling SCIP_RETCODE <function>() should use theSCIP_CALL() macro and should return SCIP_RETCODE.

. There are template files, e.g. cons_xxx.{c,h}.

. Only include scip/scip.h or scip/scipdefplugins.h.

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Documentation. Constraint Integer Programming

Ph.D. dissertation of Tobias Achterberg, 2007.. SCIP – a framework to integrate Constraint and Mixed Integer

ProgrammingTobias Achterberg, ZIB-Report 04-19, 2004.

. Primal Heuristics for Mixed Integer ProgramsTimo Berthold, Diploma Thesis, TU Berlin, 2006

. Implementation of Cutting Plane Separators for Mixed IntegerProgramsKati Wolter, Diploma Thesis, TU Berlin, 2006

. http://scip.zib.deDoxygen documentation, HowTos, FAQ

. See: scip.h, pub_*.h, <plugin>.h (e.g. cons_linear.h)

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http://scip.zib.de

How to Contribute?

You can/should contribute by . . .. using SCIP and report publications/projects,. sending bug-reports – better: bug-fixes,. giving feedback on documentation,. implementing plugins.

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Future Plans

. nonlinear components/constraintsI SOS1, SOS2I indicatorsI MINLP

. exact IP (without rounding errors)

. symmetries

. counting solutions

. improved column generation support

. new heuristics/cuts

. performance improvement

. arbitrary constraints in ZIMPL → SCIP

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Introduction to SCIP

Marc PfetschZuse Institute Berlin

DFG Research Center MATHEONMathematics for key technologies

Berlin, 10/11/2007

introduction to scipscip.zib.de/download/slides/scip-introduction.pdfintroduction to scip marc...

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