SCIP Workshop 2014, Berlin, September 30, Introduction to SCIP
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1 SCIP Workshop 2014, Berlin, September 30, 2014 Introduction to SCIP
2 ZIB: Fast Algorithms Fast Computers Konrad-Zuse-Zentrum für Informationstechnik Berlin non-university research institute of the state of Berlin Research Units: numerical analysis and modeling visualization and data analysis optimization scientific information systems distributed algorithms and supercomputing more information: SCIP introduction 2 / 63
3 Today s Schedule 09:00 10:30 Introduction and Overview 10:30 11:00 Coffee Break 11:00 12:30 Installation and Testing Environment 12: Lunch Break 14:00 15:00 Parameter Tuning 15:00 15:30 Coffee Break 15:30 17:30 Programming Exercise WiFi: eduroam Gast im ZIB (no password) SCIP introduction 3 / 63
4 Outline SCIP Solving Constraint Integer Programs 1 Constraint Integer Programming 2 Solving Constraint Integer Programs 3 The Solving Process of SCIP SCIP introduction 4 / 63
5 Outline SCIP Solving Constraint Integer Programs 1 Constraint Integer Programming 2 Solving Constraint Integer Programs 3 The Solving Process of SCIP SCIP introduction: Constraint Integer Programming 5 / 63
6 An example: the Traveling Salesman Problem Definition (TSP) Given a complete graph G = (V, E) and distances d e for all e E: Find a Hamiltonian cycle (cycle containing all nodes, tour) of minimum length. K 8 SCIP introduction: Constraint Integer Programming 6 / 63
7 An example: the Traveling Salesman Problem Definition (TSP) Given a complete graph G = (V, E) and distances d e for all e E: Find a Hamiltonian cycle (cycle containing all nodes, tour) of minimum length. K 8 SCIP introduction: Constraint Integer Programming 6 / 63
8 An example: the Traveling Salesman Problem Definition (TSP) Given a complete graph G = (V, E) and distances d e for all e E: Find a Hamiltonian cycle (cycle containing all nodes, tour) of minimum length. SCIP introduction: Constraint Integer Programming 6 / 63
9 What is a Constraint Integer Program? Mixed Integer Program Constraint Program Objective function: linear function Feasible set: described by linear constraints Variable domains: real or integer values Objective function: arbitrary function Feasible set: given by arbitrary constraints Variable domains: arbitrary (usually finite) min c T x min c(x) s.t. Ax b s.t. x F (x I, x C ) Z I R C (x I, x N ) Z I X SCIP introduction: Constraint Integer Programming 7 / 63
10 TSP Integer Programming Formulation Given complete graph G = (V, E) distances d e > 0 for all e E x e Binary variables x e = 1 if edge e is used K 8 SCIP introduction: Constraint Integer Programming 8 / 63
11 TSP Integer Programming Formulation Given complete graph G = (V, E) distances d e > 0 for all e E x e Binary variables x e = 1 if edge e is used min subject to d e x e e E e δ(v) e δ(s) x e = 2 x e 2 x e {0, 1} K 8 v V S V, S e E SCIP introduction: Constraint Integer Programming 8 / 63
12 TSP Integer Programming Formulation Given complete graph G = (V, E) distances d e > 0 for all e E x e Binary variables x e = 1 if edge e is used min subject to d e x e e E e δ(v) e δ(s) x e = 2 x e 2 x e {0, 1} K 8 v V node degree S V, S e E SCIP introduction: Constraint Integer Programming 8 / 63
13 TSP Integer Programming Formulation Given complete graph G = (V, E) distances d e > 0 for all e E x e Binary variables x e = 1 if edge e is used min subject to d e x e e E e δ(v) e δ(s) x e = 2 x e 2 x e {0, 1} K 8 v V S V, S subtour elimination e E SCIP introduction: Constraint Integer Programming 8 / 63
14 TSP Integer Programming Formulation Given complete graph G = (V, E) distances d e > 0 for all e E x e Binary variables x e = 1 if edge e is used min subject to d e x e e E e δ(v) e δ(s) x e = 2 x e 2 x e {0, 1} K 8 distance v V S V, S e E SCIP introduction: Constraint Integer Programming 8 / 63
15 TSP Constraint Programming Formulation Given complete graph G = (V, E) for each e E a distance d e > 0 Integer variables x v position of v V in tour x v K 8 SCIP introduction: Constraint Integer Programming 9 / 63
16 TSP Constraint Programming Formulation Given complete graph G = (V, E) for each e E a distance d e > 0 Integer variables x v position of v V in tour min length(x 1,..., x n ) subject to alldifferent(x 1,..., x n ) x v {1,..., n} v V SCIP introduction: Constraint Integer Programming 9 / 63
17 What is a Constraint Integer Program? Constraint Integer Program Objective function: linear function Feasible set: described by arbitrary constraints Variable domains: real or integer values When no more branching possible: CIP becomes an LP/NLP min s.t. Remark: c T x x F (x I, x C ) Z I R C arbitrary objective or variables modeled by constraints SCIP introduction: Constraint Integer Programming 10 / 63
18 What is a Constraint Integer Program? Constraint Integer Program Objective function: linear function Feasible set: described by arbitrary constraints min s.t. d e x e e E x e = 2 v V e δ(v) nosubtour(x) x e {0, 1} e E (CIP formulation of TSP) Variable domains: real or integer values When no more branching possible: CIP becomes an LP/NLP Single nosubtour constraint rules out subtours (e.g. by domain propagation). It may also separate subtour elimination inequalities. SCIP introduction: Constraint Integer Programming 10 / 63
19 Constraint Integer Programming Mixed Integer Programs MIP SCIP introduction: Constraint Integer Programming 11 / 63
20 Constraint Integer Programming Mixed Integer Programs SATisfiability problems MIP SAT SCIP introduction: Constraint Integer Programming 11 / 63
21 Constraint Integer Programming Mixed Integer Programs SATisfiability problems Pseudo-Boolean Optimization MIP SAT PBO SCIP introduction: Constraint Integer Programming 11 / 63
22 Constraint Integer Programming Mixed Integer Programs SATisfiability problems MIP Pseudo-Boolean Optimization Finite Domain SAT PBO FD SCIP introduction: Constraint Integer Programming 11 / 63
23 Constraint Integer Programming Mixed Integer Programs SATisfiability problems MIP Pseudo-Boolean Optimization Finite Domain SAT PBO FD CP Constraint Programming SCIP introduction: Constraint Integer Programming 11 / 63
24 Constraint Integer Programming Mixed Integer Programs SATisfiability problems Pseudo-Boolean Optimization Finite Domain Constraint Programming Constraint Integer Programming MIP CIP SAT PBO FD CP SCIP introduction: Constraint Integer Programming 11 / 63
25 Constraint Integer Programming Mixed Integer Programs SATisfiability problems Pseudo-Boolean Optimization Finite Domain Constraint Programming Constraint Integer Programming MIP CIP SAT PBO FD CP Relation to CP and MIP Every MIP is a CIP. MIP CIP Every CP over a finite domain space is a CIP. FD CIP SCIP introduction: Constraint Integer Programming 11 / 63
26 Outline SCIP Solving Constraint Integer Programs 1 Constraint Integer Programming 2 Solving Constraint Integer Programs 3 The Solving Process of SCIP SCIP introduction: Solving Constraint Integer Programs 12 / 63
27 How do we solve CIPs? MIP LP relaxation cutting planes MIP, CP, and SAT branch-and-bound CP domain propagation SAT SCIP conflict analysis periodic restarts SCIP introduction: Solving Constraint Integer Programs 13 / 63
28 What is SCIP? SCIP (Solving Constraint Integer Programs)... provides a full-scale MIP and MINLP solver, is constraint based, incorporates CP features (domain propagation), MIP features (cutting planes, LP relaxation), and SAT-solving features (conflict analysis, restarts), is a branch-cut-and-price framework, has a modular structure via plugins, is free for academic purposes, and is available in source-code under SCIP introduction: Solving Constraint Integer Programs 14 / 63
29 Some more facts about SCIP interface and usability user-friendly interactive shell interfaces to AMPL, GAMS, ZIMPL, MATLAB, Python and Java C++ wrapper classes LP solvers: CLP, CPLEX, Gurobi, MOSEK, QSopt, SoPlex, Xpress over parameters and 15 emphasis settings documentation and guidelines more than lines of C code, 20% documentation assertions, debug messages HowTos: plugins types, debugging, automatic testing 11 examples illustrating the use of SCIP active mailing list (280 members) about 5000 downloads per year SCIP introduction: Solving Constraint Integer Programs 15 / 63
30 SCIP Optimization Suite Toolbox for generating and solving constraint integer programs free for academic use, available in source code SCIP introduction: Solving Constraint Integer Programs 16 / 63
31 SCIP Optimization Suite Toolbox for generating and solving constraint integer programs free for academic use, available in source code ZIMPL model and generate LPs, MIPs, and MINLPs SCIP MIP, MINLP and CIP solver, branch-cut-and-price framework SoPlex revised primal and dual simplex algorithm GCG see talk tomorrow 9:40 generic branch-cut-and-price solver UG see talk tomorrow 10:20 framework for parallelization of MIP and MINLP solvers SCIP introduction: Solving Constraint Integer Programs 16 / 63
32 Structure of SCIP SCIP Primal Heuristic actcons coef cross over dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen zi round Variable Event Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler and bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound xor Cutpool LP clp cpx msk none qso spx xprs Dialog Display Node selector bfs dfs estimate hybrid estim restart dfs Presolver bound shift dualfix implics intto binary probing trivial Impli cations Tree Reader ccg cip cnf fix lp mps opb ppm rlp sol sos zpl Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg zero half Propa gator pseudo obj root redcost vbound Relax SCIP introduction: Solving Constraint Integer Programs 17 / 63
33 Structure of SCIP SCIP Primal Heuristic actcons coef cross over dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen zi round Variable Event Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler and bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound xor Cutpool LP clp cpx msk none qso spx xprs Dialog Display Node selector bfs dfs estimate hybrid estim restart dfs Presolver bound shift dualfix implics intto binary probing trivial Impli cations Tree Reader ccg cip cnf fix lp mps opb ppm rlp sol sos zpl Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg zero half Propa gator pseudo obj root redcost vbound Relax SCIP introduction: Solving Constraint Integer Programs 17 / 63
34 Structure of SCIP SCIP Primal Heuristic actcons coef cross over dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen zi round Variable Event Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler and bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound xor Cutpool LP clp cpx msk none qso spx xprs Dialog Display Node selector bfs dfs estimate hybrid estim restart dfs Presolver bound shift dualfix implics intto binary probing trivial Impli cations Tree Reader ccg cip cnf fix lp mps opb ppm rlp sol sos zpl Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg zero half Propa gator pseudo obj root redcost vbound Relax SCIP introduction: Solving Constraint Integer Programs 17 / 63
35 SCIP core branching tree variables conflict analysis solution pool cut pool statistics Plugin based design clique table implication graph... SCIP introduction: Solving Constraint Integer Programs 18 / 63
36 SCIP core branching tree variables conflict analysis solution pool cut pool statistics Plugin based design clique table implication graph... Plugins external callback objects interact with the framework through a very detailed interface SCIP introduction: Solving Constraint Integer Programs 18 / 63
37 SCIP core branching tree variables conflict analysis solution pool cut pool statistics Plugin based design clique table implication graph... Plugins external callback objects interact with the framework through a very detailed interface SCIP knows for each plugin type: the number of available plugins priority defining the calling order (usually) SCIP does not know any structure behind a plugin plugins are black boxes for the SCIP core SCIP introduction: Solving Constraint Integer Programs 18 / 63
38 SCIP core branching tree variables conflict analysis solution pool cut pool statistics Plugin based design clique table implication graph... Plugins external callback objects interact with the framework through a very detailed interface SCIP knows for each plugin type: the number of available plugins priority defining the calling order (usually) SCIP does not know any structure behind a plugin plugins are black boxes for the SCIP core Very flexible branch-and-bound based search algorithm SCIP introduction: Solving Constraint Integer Programs 18 / 63
39 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... SCIP introduction: Solving Constraint Integer Programs 19 / 63
40 A closer look: branching rules SCIP Primal Heuristic actcons coef cross over dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen zi round Variable Event Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler and bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound xor Cutpool LP clp cpx msk none qso spx xprs Dialog Display Node selector bfs dfs estimate hybrid estim restart dfs Presolver bound shift dualfix implics intto binary probing trivial Impli cations Tree Reader ccg cip cnf fix lp mps opb ppm rlp sol sos zpl Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg zero half Propa gator pseudo obj root redcost vbound Relax SCIP introduction: Solving Constraint Integer Programs 20 / 63
41 ct simple A closer look: branching roundingrule shif nd unc. allfull strong full strong nt ls cumu lative Branch infer ence indi cator leastinf ap ck integral relps cost random pscost mostinf ear SCIP introduction: Solving Constraint Integer Programs 20 / 63
42 What does SCIP know about branching rules? SCIP knows the number of available branching rules each branching rule has a priority SCIP calls the branching rule in decreasing order of priority the interface defines the possible results of a call: branched reduced domains added constraints detected cutoff did not run SCIP introduction: Solving Constraint Integer Programs 21 / 63
43 How does SCIP call a branching rule? /* start timing */ SCIPclockStart ( branchrule - > branchclock, set ); /* call external method */ SCIP_CALL ( branchrule - > branchexeclp ( set - >scip, branchrule, allowaddcons, result ) ); /* stop timing */ SCIPclockStop ( branchrule - > branchclock, set ); " branching rule <%s> returned invalid result code <%d> from LP \ /* evaluate result */ if( * result!= SCIP_CUTOFF && * result!= SCIP_CONSADDED && * result!= SCIP_REDUCEDDOM && * result!= SCIP_SEPARATED && * result!= SCIP_BRANCHED && * result!= SCIP_DIDNOTRUN ) { SCIPerrorMessage ( solution branching \n", branchrule - >name, * result ); return SCIP_INVALIDRESULT ; } SCIP introduction: Solving Constraint Integer Programs 22 / 63
44 What can a plugin access? Plugins are allowed to access all global (core) information branching tree variables conflict analysis solution pool cut pool statistics clique table implication graph... SCIP introduction: Solving Constraint Integer Programs 23 / 63
45 What can a plugin access? Plugins are allowed to access all global (core) information branching tree variables conflict analysis solution pool cut pool statistics clique table implication graph... Ideally, plugins should not access data of other plugins!!! SCIP introduction: Solving Constraint Integer Programs 23 / 63
46 What can a plugin access? Plugins are allowed to access all global (core) information branching tree variables conflict analysis solution pool cut pool statistics clique table implication graph... Ideally, plugins should not access data of other plugins!!! Branching Rules LP solution variables statistics SCIP introduction: Solving Constraint Integer Programs 23 / 63
47 Constraint Handlers Constraint handlers most powerful plugins in SCIP define the feasible region a single constraint may represent a whole set of inequalities Functions check and enforce feasibility of solutions can add linear representation to LP relaxation constraint-specific presolving, domain propagation, separation Result SCIP is constraint based Advantage: flexibility Disadvantage: limited global view SCIP introduction: Solving Constraint Integer Programs 24 / 63
48 Default Plugins SCIP Primal Heuristic actcons coef cross over dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen zi round Variable Event Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler and bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound xor Cutpool LP clp cpx msk none qso spx xprs Dialog Display Node selector bfs dfs estimate hybrid estim restart dfs Presolver bound shift dualfix implics intto binary probing trivial Impli cations Tree Reader ccg cip cnf fix lp mps opb ppm rlp sol sos zpl Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg zero half Propa gator pseudo obj root redcost vbound Relax SCIP introduction: Solving Constraint Integer Programs 25 / 63
49 Number of plugins SCIP Version Plugin Branching Rules Constraint Handlers Node Selectors Presolvers Primal Heuristics Propagators Readers Separators SCIP introduction: Solving Constraint Integer Programs 26 / 63
50 Extending SCIP SCIP Primal Heuristic actcons coef cross over dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen zi round Variable Event Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler and bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound xor Cutpool LP clp cpx msk none qso spx xprs Dialog Display Node selector bfs dfs estimate hybrid estim restart dfs Presolver bound shift dualfix implics intto binary probing trivial Impli cations Tree Reader ccg cip cnf fix lp mps opb ppm rlp sol sos zpl Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg zero half Propa gator pseudo obj root redcost vbound Relax SCIP introduction: Solving Constraint Integer Programs 27 / 63
51 Extending SCIP SCIP Primal Heuristic actcons coef cross over dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen Variable Event Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler and bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound Cutpool LP clp cpx msk none qso spx Dialog Display Node selector bfs dfs estimate hybrid estim Presolver bound shift dualfix implics intto binary probing Impli cations Tree Reader ccg cip cnf fix lp mps opb ppm rlp sol sos Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg Propa gator pseudo obj root redcost Relax SCIP introduction: Solving Constraint Integer Programs 27 / 63
52 Extending SCIP: TSP SCIP Primal Heuristic farthest insert coef 2-Opt dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen Variable Event sol found Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler subtour bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound Cutpool LP clp cpx msk none qso spx Dialog Display Node selector bfs dfs estimate hybrid estim Presolver bound shift dualfix implics intto binary probing Impli cations Tree Reader tsp cip cnf fix lp mps opb ppm rlp sol sos Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg Propa gator pseudo obj root redcost Relax SCIP introduction: Solving Constraint Integer Programs 27 / 63
53 Extending SCIP: Coloring SCIP Primal Heuristic init coef 2-Opt dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen Variable Event Branch Ryan& Foster strong Ryan& Foster infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler store Graph bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound Cutpool LP clp cpx msk none qso spx Dialog Display Node selector bfs dfs estimate hybrid estim Presolver bound shift dualfix implics intto binary probing Impli cations Tree Reader col csol cnf fix lp mps opb ppm rlp sol sos Pricer coloring Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg Propa gator pseudo obj root redcost Relax SCIP introduction: Solving Constraint Integer Programs 27 / 63
54 Extending SCIP: STP SCIP Primal Heuristic TM coef local dins feaspump fixand infer frac guided int int shifting linesearch local branching mutation subnlp objpscost octane oneopt pscost rens rins rootsol rounding shifting shift& prop simple rounding trivial trysol twoopt under cover veclen Variable Event sol found Branch allfull strong full strong infer ence leastinf mostinf pscost random relps cost Conflict Constraint Handler stp bound disjunc. count sols cumu lative indi cator integral knap sack linear linking logicor or orbi tope quadr atic setppc soc sos1 sos2 var bound Cutpool LP clp cpx msk none qso spx Dialog Display Node selector bfs dfs estimate hybrid estim Presolver stp dualfix implics intto binary probing Impli cations Tree Reader stp cip cnf fix lp mps opb ppm rlp sol sos Pricer Separator clique cmir flow cover gomory implied bounds intobj mcf odd cycle rapid learn redcost strong cg Propa gator pseudo obj root redcost Relax SCIP introduction: Solving Constraint Integer Programs 27 / 63
55 Outline SCIP Solving Constraint Integer Programs 1 Constraint Integer Programming 2 Solving Constraint Integer Programs 3 The Solving Process of SCIP SCIP introduction: The Solving Process of SCIP 28 / 63
56 Operational Stages Init Init Solve Problem Transforming Presolving Solving Free Transform Free Solve SCIP introduction: The Solving Process of SCIP 29 / 63
57 Operational Stages Init Init Solve Problem Transforming Presolving Solving Free Transform Free Solve Basic data structures are allocated and initialized. User includes required plugins (or just takes plugins). SCIP introduction: The Solving Process of SCIP 29 / 63
58 Problem Specification Init Init Solve Problem Transforming Presolving Solving Free Transform Free Solve User creates and modifies the original problem instance. Problem creation is usually done in file readers. SCIP introduction: The Solving Process of SCIP 30 / 63
59 Define Variables (TSP Example) SCIP_VAR * var ; SCIP_ CALL ( // return value macro SCIPcreateVar ( scip, // SCIP pointer & var, // save in variable " varname ", // pass variable name 0.0, // lower bound 1.0, // upper bound length, // obj. value SCIP_ VARTYPE_ BINARY, // type TRUE, // initial FALSE, // removable NULL, NULL, NULL, // no callback functions NULL // no variable data ) ); SCIP_ CALL ( SCIPaddVar ( scip, var ) ); // add var. SCIP introduction: The Solving Process of SCIP 31 / 63
60 Define Variables (TSP Example) SCIP_VAR * var ; SCIP_ CALL ( // return value macro SCIPcreateVarBasic ( scip, // SCIP pointer & var, // save in variable " varname ", // pass variable name 0.0, // lower bound 1.0, // upper bound length, // obj. value SCIP_ VARTYPE_ BINARY // type ) ); SCIP_ CALL ( SCIPaddVar ( scip, var ) ); // add var. SCIP introduction: The Solving Process of SCIP 32 / 63
61 TSP: Define Degree Constraints SCIP_CALL ( SCIPcreateConsLinear ( scip, // SCIP pointer &cons, // save in cons " consname ", // name nvar, // number of variables vars, // array of variables vals, // array of values 2.0, // left hand side 2.0, // 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 constraint SCIP_CALL ( SCIPreleaseCons ( scip, & cons ) ); // free cons. space MIPs are specified using linear constraints only (may be upgraded ). SCIP introduction: The Solving Process of SCIP 33 / 63
62 TSP: Define Degree Constraints SCIP_CALL ( SCIPcreateConsBasicLinear ( scip, // SCIP pointer &cons, // save in cons " consname ", // name nvar, // number of variables vars, // array of variables vals, // array of values 2.0, // left hand side 2.0 // right hand side ( equation ) ) ); SCIP_CALL ( SCIPaddCons ( scip, cons ) ); // add constraint SCIP_CALL ( SCIPreleaseCons ( scip, & cons ) ); // free cons. space MIPs are specified using linear constraints only (may be upgraded ). SCIP introduction: The Solving Process of SCIP 34 / 63
63 Transformation Init Init Solve Problem Transforming Presolving Solving Free Transform Free Solve Creates a working copy of the original problem. SCIP introduction: The Solving Process of SCIP 35 / 63
64 Original and Transformed Problem Original CIP Transformed CIP Original variables Transformed variables Original constraints Transformed constraints data is copied into separate memory area presolving and solving operate on transformed problem original data can only be modified in problem modification stage SCIP introduction: The Solving Process of SCIP 36 / 63
65 Presolving Init Init Solve Problem Transforming Presolving Solving Free Transform Free Solve SCIP introduction: The Solving Process of SCIP 37 / 63
66 Presolving Task reduce size of model by removing irrelevant information strengthen LP relaxation by exploiting integrality information make the LP relaxation numerically more stable extract useful information Primal Reductions: based on feasibility reasoning no feasible solution is cut off Dual Reductions: consider objective function at least one optimal solution remains SCIP introduction: The Solving Process of SCIP 38 / 63
67 Solving Init Init Solve Problem Transforming Presolving Solving Free Transform Free Solve SCIP introduction: The Solving Process of SCIP 39 / 63
68 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 40 / 63
69 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 40 / 63
70 Node Selection Techniques Task improve primal bound keep comp. effort small improve global dual bound basic rules depth first search (DFS) early feasible solutions best bound search (BBS) improve dual bound best estimate search (BES) improve primal bound combinations BBS or BES with plunging hybrid BES/BBS interleaved BES/BBS SCIP introduction: The Solving Process of SCIP 41 / 63
71 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 42 / 63
72 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 42 / 63
73 Domain Propagation x 1 x 2 x 3 x 4 Task x 1 x 2 x 3 x 4 simplify model locally improve local dual bound detect infeasibility Techniques constraint specific each cons handler may provide a propagation routine reduced presolving (usually) dual propagation root reduced cost strengthening objective function special structures variable bounds SCIP introduction: The Solving Process of SCIP 43 / 63
74 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 44 / 63
75 LP Solving LP solver is a black box interface to different LP solvers: SoPlex, CPLEX, XPress, Gurobi, CLP,... primal/dual simplex barrier with/without crossover feasibility double-checked by SCIP condition number check resolution by changing parameters: scaling, tolerances, solving from scratch, other simplex SCIP introduction: The Solving Process of SCIP 45 / 63
76 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 46 / 63
77 Pricing Pricing find variable with negative reduced costs or prove that there exists none typically problem specific Branch-and-Price huge number of variables start with subset add others, when needed dynamic aging of variables problem variable pricer to add them again early branching possible lazy variable bounds SCIP introduction: The Solving Process of SCIP 47 / 63
78 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 48 / 63
79 Cutting Plane Separation Techniques Task strengthen relaxation add valid constraints generate on demand general cuts complemented MIR cuts Gomory mixed integer cuts strong Chvátal-Gomory cuts implied bound cuts reduced cost strengthening problem specific cuts 0-1 knapsack problem stable set problem 0-1 single node flow problem SCIP introduction: The Solving Process of SCIP 49 / 63
80 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 50 / 63
81 Constraint Enforcement LP solution may violate a constraint not contained in the relaxation. 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, Concluding that the subproblem is infeasible and can be cut off, or Just saying solution infeasible. SCIP introduction: The Solving Process of SCIP 51 / 63
82 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
83 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
84 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
85 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
86 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
87 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
88 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
89 Constraint Enforcement Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Reduced domain Added cut Cutoff Infeasible Added constraint Branched Feasible SCIP introduction: The Solving Process of SCIP 52 / 63
90 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 53 / 63
91 Branching Rules Techniques Task divide into (disjoint) subproblems improve local bounds branching on variables most infeasible least infeasible random branching strong branching pseudocost reliability VSIDS hybrid reliability/inference branching on constraints SOS1 SOS2 SCIP introduction: The Solving Process of SCIP 54 / 63
92 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 55 / 63
93 Primal Heuristics Techniques Task improve primal bound effective on average guide remaining search rounding possibly solve final LP least infeasible guided objective objective feasibility pump Large Neighborhood Search RINS, local branching RENS combinatorial SCIP introduction: The Solving Process of SCIP 56 / 63
94 Flow Chart SCIP Presolving Stop Domain propagation Conflict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts SCIP introduction: The Solving Process of SCIP 57 / 63
95 Conflict Analysis Techniques Task Analyze infeasibility Derive valid constraints Help to prune other nodes Analyze: Propagation conflicts Infeasible LPs Bound-exceeding LPs Strong branching conflicts Detection: Cut in conflict graph LP: Dual ray heuristic Use conflicts: Only for propagation As cutting planes SCIP introduction: The Solving Process of SCIP 58 / 63
96 Operational Stages Init Init Solve Problem Transforming Presolving Solving Free Transform Free Solve SCIP introduction: The Solving Process of SCIP 59 / 63
97 Implementation Hints SCIP has own memory handling. Standard memory: SCIPallocMemoryArray(scip,&p,10), SCIPfreeMemoryArray(scip,&p) Block memory: SCIPallocBlockMemoryArray(scip,&p,10), SCIPfreeBlockMemoryArray(scip,&p,10) Fast buffer: SCIPallocBufferArray(scip,&p,10), SCIPfreeBufferArray(scip,&p) Functions calling SCIP RETCODE <function>() should use the SCIP CALL() macro and should return SCIP RETCODE. There are template files, e.g. cons xyz.{c,h}. Only include scip/scip.h, pub *.h, or scip/scipdefplugins.h. SCIP introduction: The Solving Process of SCIP 60 / 63
98 Documentation Constraint Integer Programming Ph.D. dissertation of Tobias Achterberg, Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite ZIB Report, Doxygen documentation, HowTos, FAQ See: scip.h, pub *.h, <plugin>.h (e.g. cons linear.h) If nothing helps: SCIP mailing list SCIP introduction: The Solving Process of SCIP 61 / 63
99 Wrap-up SCIP is... a solver for MIP and MINLP a framework for branch-and-bound based algorithms (branch-and-cut, branch-and-price,... ) part of the SCIP Optimization Suite Advantages of SCIP: broad scope ( next two days) available in source code ( coming next) actively developed ( tomorrow) hundreds of parameters to play with ( afternoon) easily extendable ( afternoon) SCIP introduction: The Solving Process of SCIP 62 / 63
100 Today s Schedule 09:00 10:30 Introduction and Overview 10:30 11:00 Coffee Break 11:00 12:30 Installation and Testing Environment 12: Lunch Break 14:00 15:00 Parameter Tuning 15:00 15:30 Coffee Break 15:30 17:30 Programming Exercise WiFi: eduroam Gast im ZIB (no password) SCIP introduction: The Solving Process of SCIP 63 / 63
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