Implementing Constraint Handlers in SCIP
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1 Implementing Constraint Handlers in SCIP Gregor Hendel Ambros Gleixner, Felipe Serrano September 30
2 Outline Why use constraints? Motivation Callback: Default Constraint Handlers of SCIP Implementation hints for the Nosubtour Constraint Handler Hendel,Gleixner,Serrano Constraint Handler Implementation 1 / 43
3 Outline Why use constraints? Motivation Callback: Default Constraint Handlers of SCIP Implementation hints for the Nosubtour Constraint Handler Hendel,Gleixner,Serrano Constraint Handler Implementation 2 / 43
4 Traveling Salesman Problem (TSP) Denition Given a graph G = (V, E) with edge lengths c e. Find a Hamiltonian cycle (cycle containing all nodes, tour) of minimum length. K 8 Hendel,Gleixner,Serrano Constraint Handler Implementation 3 / 43
5 Traveling Salesman Problem (TSP) Denition Given a graph G = (V, E) with edge lengths c e. Find a Hamiltonian cycle (cycle containing all nodes, tour) of minimum length. K 8 Hendel,Gleixner,Serrano Constraint Handler Implementation 3 / 43
6 Motivation For General Constraints: MIP Formulation min s.t. c e x e e E e δ(v) e δ(s) x e = 2 v V x e 2 S V, S x e {0, 1} e E. Hendel,Gleixner,Serrano Constraint Handler Implementation 4 / 43
7 Motivation For General Constraints: MIP Formulation min s.t. c e x e e E e δ(v) e δ(s) x e = 2 v V x e 2 S V, S x e {0, 1} e E. CIP Formulation min s.t. c e x e e E x e = 2 v V e δ(v) nosubtour(g, x) x e {0, 1} e E. nosubtour(g, x) C {e E x e = 1} : C is a cycle of length C < V Hendel,Gleixner,Serrano Constraint Handler Implementation 4 / 43
8 Separation Problem Separation Problem Given graph G = (V, E) and x [0, 1] E. Decide whether x satises all subtour elimination constraints. If not, nd violated subtour elimination constraint. Hendel,Gleixner,Serrano Constraint Handler Implementation 5 / 43
9 Separation Problem Separation Problem Given graph G = (V, E) and x [0, 1] E. Decide whether x satises all subtour elimination constraints. If not, nd violated subtour elimination constraint. Trivial observation: Consider G = (V, E) with edge capacities x e. x violates at least one subtour elimination constraint cut δ(s) with capacity x (δ(s)) < 2. Hendel,Gleixner,Serrano Constraint Handler Implementation 5 / 43
10 Separation Problem Separation Problem Given graph G = (V, E) and x [0, 1] E. Decide whether x satises all subtour elimination constraints. If not, nd violated subtour elimination constraint. Idea of separation algorithm: s, t V : Find (s, t)-cut δ(s) of minimum capacity If all cut capacities 2, all subtour elimination constraints satised ( ) V 2 times MaxFlow-MinCut algo Hendel,Gleixner,Serrano Constraint Handler Implementation 5 / 43
11 Separation Problem Separation Problem Given graph G = (V, E) and x [0, 1] E. Decide whether x satises all subtour elimination constraints. If not, nd violated subtour elimination constraint. Idea of separation algorithm: s, t V : Find (s, t)-cut δ(s) of minimum capacity If all cut capacities 2, all subtour elimination constraints satised V 1 times MaxFlow-MinCut algo within Gomory-Hu-Algorithm Hendel,Gleixner,Serrano Constraint Handler Implementation 5 / 43
12 Result of Gomory-Hu-Algorithm Gomory-Hu-Tree T For all s, t V : capacity of minimum (s, t)-cut in G : minimum label f e of all edges e in unique (s, t)-path in T minimum (s, t)-cut in G : bipartition of V obtained be deleting this edge from T Hendel,Gleixner,Serrano Constraint Handler Implementation 6 / 43
13 Outline Why use constraints? Motivation Callback: Default Constraint Handlers of SCIP Implementation hints for the Nosubtour Constraint Handler Hendel,Gleixner,Serrano Constraint Handler Implementation 7 / 43
14 Default Constraint Handlers Hendel,Gleixner,Serrano Constraint Handler Implementation 8 / 43
15 Special Linear Constraints min x 1 + x 2 + x 3 + x 4 + x 5 + y 1 + y 2 s.t. y x 1 y x 2 3 x x x 3 8 x 2 + x 3 + x 4 = 1 3 x x x y y 2 4 x 1, x 2, x 3, x 4 {0, 1} x 5 Z + y 1, y 2 R + Hendel,Gleixner,Serrano Constraint Handler Implementation 9 / 43
16 Special Linear Constraints min x 1 + x 2 + x 3 + x 4 + x 5 + y 1 + y 2 s.t. y x 1 y x 2 3 x x x 3 8 x 2 + x 3 + x 4 = 1 3 x x x y y 2 4 x 1, x 2, x 3, x 4 {0, 1} x 5 Z + y 1, y 2 R + Hendel,Gleixner,Serrano Constraint Handler Implementation 9 / 43
17 Special Linear Constraints min x 1 + x 2 + x 3 + x 4 + x 5 + y 1 + y 2 s.t. y x 1 y x 2 3 x x x 3 8 x 2 + x 3 + x 4 = 1 3 x x x y y 2 4 x 1, x 2, x 3, x 4 {0, 1} x 5 Z + y 1, y 2 R + Hendel,Gleixner,Serrano Constraint Handler Implementation 9 / 43
18 Special Linear Constraints min x 1 + x 2 + x 3 + x 4 + x 5 + y 1 + y 2 s.t. y x 1 y x 2 3 x x x 3 8 x 2 + x 3 + x 4 = 1 3 x x x y y 2 4 x 1, x 2, x 3, x 4 {0, 1} x 5 Z + y 1, y 2 R + Hendel,Gleixner,Serrano Constraint Handler Implementation 9 / 43
19 Special Linear Constraints min x 1 + x 2 + x 3 + x 4 + x 5 + y 1 + y 2 s.t. y x 1 y x 2 3 x x x 3 8 x 2 + x 3 + x 4 = 1 3 x x x y y 2 4 x 1, x 2, x 3, x 4 {0, 1} x 5 Z + y 1, y 2 R + Hendel,Gleixner,Serrano Constraint Handler Implementation 9 / 43
20 Knapsack Constraints Feasible region of 0-1 knapsack problem: { x {0, 1} N : j N a j x j b } weights of the variables: a j Z + for all j N capacity of the knapsack: b Z + Hendel,Gleixner,Serrano Constraint Handler Implementation 10 / 43
21 The Integral Constraint Handler The integral constraint handler ensures the integrality of discrete problem variables in a solution does not need constraints because integrality restrictions are encoded in the type of a variable is the shortest example of a working constraint handler I know Hendel,Gleixner,Serrano Constraint Handler Implementation 11 / 43
22 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Further Ingredients Private data Interface methods Properties/Parameters Hendel,Gleixner,Serrano Constraint Handler Implementation 12 / 43
23 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Further Ingredients Private data Interface methods Properties/Parameters Hendel,Gleixner,Serrano Constraint Handler Implementation 12 / 43
24 Private Data Constraint data: information needed to dene single constraint 1 struct SCIP_ConsData 2 { 3 SCIP_VAR** vars; // vari ables in knapsack 4 int nvars; // number of v ariabl es 5 SCIP_Longint* weights; // weights of v ariab les 6 SCIP_Longint capacity; // c a p a c i t y of knapsack 7 }; Constraint handler data: information belonging to constraint handler itself 1 struct SCIP_ConshdlrData 2 { 3 int maxrounds; // max nr. of sepa rounds per node 4 int maxsepacuts; // max nr. of cuts per sepa round 5 }; Hendel,Gleixner,Serrano Constraint Handler Implementation 13 / 43
25 Interface Methods Creating a single knapsack constraint. 1 SCIP_RETCODE SCIPcreateConsBasicKnapsack( 2 SCIP* scip, / < SCIP data s t r u c t u r e / 3 SCIP_CONS** cons, / < p o i n t e r to hold the created c o n s t r a i n t 4 const char* name, / < name of c o n s t r a i n t / 5 int nvars, / < number of items i n the knapsack / 6 SCIP_VAR** vars, / < array with item vari ables / 7 SCIP_Longint* weights, / < a r r a y with item weights / 8 SCIP_Longint capacity / < c a p a c i t y of knapsack / 9 ) 10 { 11 SCIP_CONSDATA* consdata; 12 SCIP_CALL( consdatacreate(scip, &consdata, nvars, vars, 13 weights, capacity) ); 14 SCIP_CALL( SCIPcreateConsBasic(scip, cons, name, conshdlr, 15 consdata) ); 16 return SCIP_OKAY; 17 } Hendel,Gleixner,Serrano Constraint Handler Implementation 14 / 43
26 Interface Methods Including the knapsack constraint handler. 1 SCIP_RETCODE SCIPincludeConshdlrKnapsack( 2 SCIP* scip // SCIP data s t r u c t u r e 3 ) 4 { 5 SCIP_CONSHDLR* conshdlr; 6 SCIP_CONSHDLRDATA* conshdlrdata; 7 8 SCIP_CALL( conshdlrdatacreate(scip, &conshdlrdata) ); / i n c l u d e c o n s t r a i n t handler i n t o SCIP / 12 SCIP_CALL( SCIPincludeConshdlrBasic(scip, &conshdlr, 13 CONSHDLR_NAME, CONSHDLR_DESC, 14 [...] 15 consenfolpknapsack, consenfopsknapsack, 16 conscheckknapsack, conslockknapsack, 17 conshdlrdata) ); / s e t non fundamental c a l l b a c k s v i a s p e c i f i c s e t t e r f u n c t i o n s / } Hendel,Gleixner,Serrano Constraint Handler Implementation 15 / 43
27 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Further Ingredients Private data Interface methods Properties/Parameters Hendel,Gleixner,Serrano Constraint Handler Implementation 16 / 43
28 Callback Methods What is a callback method? Hendel,Gleixner,Serrano Constraint Handler Implementation 17 / 43
29 Callback Methods What is a callback method? Nice quote from someone who cited an older Wikipedia version: In computer programming, a callback is a reference to executable code, or a piece of executable code, that is passed as an argument to other code. This allows a lower-level software layer to call a subroutine (or function) dened in a higher-level layer. source: Hendel,Gleixner,Serrano Constraint Handler Implementation 17 / 43
30 Callback Methods What is a callback method? Nice quote from someone who cited an older Wikipedia version: In computer programming, a callback is a reference to executable code, or a piece of executable code, that is passed as an argument to other code. This allows a lower-level software layer to call a subroutine (or function) dened in a higher-level layer. source: All the functionality of constraint handlers (as well as all other plugin types of SCIP) is dened through callback functions. Hendel,Gleixner,Serrano Constraint Handler Implementation 17 / 43
31 CONSCHECK Most important callback... usually called by primal heuristics checks given solution for feasibility wrt all constraints of its type possible result values SCIP_FEASIBLE SCIP_INFEASIBLE Given solution: (x 1, x 2, x 3 ) = (1, 0, 1) Knapsack constraints: 3x 1 + 6x 2 + 4x 3 8 2x x 3 3 Result: SCIP_INFEASIBLE Hendel,Gleixner,Serrano Constraint Handler Implementation 18 / 43
32 CONSENFOLP and CONSENFOPS CONSENFOLP: checks LP solution for feasibility CONSENFOPS: checks Pseudo solution for feasibility LP solution solution of LP relaxation min x 1 x 2 + x 3 s.t. 3 x x x 3 4 x 1, x 2, x 3 {0, 1} LP solution: (0, 1 2, 0) Pseudo Solution solution of LP relaxation with only bound constraints used if LP solving disabled, or numerical diculties occurred min x 1 x 2 + x 3 s.t. x 1, x 2, x 3 {0, 1} Pseudo Solution: (0, 1, 0) Hendel,Gleixner,Serrano Constraint Handler Implementation 19 / 43
33 CONSENFOLP and CONSENFOPS LP solution may violate a constraint not contained in the relaxation. Enforcement callbacks are necessary for a correct implementation! In addition, they can resolve an 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 o, or just saying solution infeasible. Hendel,Gleixner,Serrano Constraint Handler Implementation 20 / 43
34 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
35 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
36 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
37 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
38 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
39 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
40 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
41 CONSENFOLP and CONSENFOPS Presolving Domain propagation Conict analysis Node selection Solve LP Pricing Primal heuristics Processing Branching Enforce constraints Cuts Enforcement result of constraint handler: reduced domain added cut added constraint branched cuto infeasible feasible Hendel,Gleixner,Serrano Constraint Handler Implementation 21 / 43
42 The Integral Constraint Handler The integral constraint handler ensures the integrality of discrete problem variables in a solution does not need constraints because integrality restrictions are encoded in the type of a variable is the shortest example of a working constraint handler I know Hendel,Gleixner,Serrano Constraint Handler Implementation 22 / 43
43 The Integral Constraint Handler The integral constraint handler ensures the integrality of discrete problem variables in a solution does not need constraints because integrality restrictions are encoded in the type of a variable is the shortest example of a working constraint handler I know Question How does the integral constraint handler enforce the LP solution? Hendel,Gleixner,Serrano Constraint Handler Implementation 22 / 43
44 The Integral Constraint Handler The integral constraint handler ensures the integrality of discrete problem variables in a solution does not need constraints because integrality restrictions are encoded in the type of a variable is the shortest example of a working constraint handler I know Question How does the integral constraint handler enforce the LP solution? Answer It calls the branching rules to perform variable branching. Hendel,Gleixner,Serrano Constraint Handler Implementation 22 / 43
45 CONSLOCK Provides dual information for single constraints (useful for presolving, primal heuristics,... ) For each variable of a constraint, returns whether... increasing its value, and/or decreasing its value may lead to a violation of the constraint 3 x 1 5 x x 3 7 increasing: x 1 and x 3 decreasing: x 2 Hendel,Gleixner,Serrano Constraint Handler Implementation 23 / 43
46 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Further Ingredients Private data Interface methods Properties/Parameters Hendel,Gleixner,Serrano Constraint Handler Implementation 24 / 43
47 CONSINIT..., CONSEXIT... Init Presolving Init Solve Problem Transforming Transformed Solving Free Transform Transformed Free Solve CONSINIT and CONSEXIT: called after problem was transformed / before transformed problem is freed initialize and free statistics in SCIP_ConshdlrData Hendel,Gleixner,Serrano Constraint Handler Implementation 25 / 43
48 CONSINIT..., CONSEXIT... Init Presolving Init Solve Problem Transforming Transformed Solving Free Transform Transformed Free Solve CONSINITPRE and CONSEXITPRE: called before presolving starts / after presolving is nished initialize and free presolving data Hendel,Gleixner,Serrano Constraint Handler Implementation 25 / 43
49 CONSINIT..., CONSEXIT... Init Presolving Init Solve Problem Transforming Transformed Solving Free Transform Transformed Free Solve CONSINITSOL and CONSEXITSOL: called before branch-and-bound process starts / before branch-and-bound process is freed initialize and release branch-and-bound specic data Hendel,Gleixner,Serrano Constraint Handler Implementation 25 / 43
50 CONSINIT..., CONSEXIT... Init Presolving Init Solve Problem Transforming Transformed Solving Free Transform Transformed Free Solve CONSINITLP: called before rst LP relaxation is solved add linear relaxation of all "initial" constraints to the LP relaxation Hendel,Gleixner,Serrano Constraint Handler Implementation 25 / 43
51 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Further Ingredients Private data Interface methods Properties/Parameters Hendel,Gleixner,Serrano Constraint Handler Implementation 26 / 43
52 CONSSEPALP, CONSSEPASOL Feasible region of 0-1 knapsack problem: { x {0, 1} N : j N a j x j b } Minimal Cover: C N j C a j > b j C\{i} a j b i C Minimal Cover Inequality x j C 1 j C 5x 1 + 6x 2 + 2x 3 + 2x 4 8 Minimal cover: C = {2, 3, 4} Minimal cover inequality: x 2 + x 3 + x 4 2 separated in knapsack constraint handler Hendel,Gleixner,Serrano Constraint Handler Implementation 27 / 43
53 CONSSEPALP, CONSSEPASOL Separation is implemented in separators and constraint handlers. 1. Separators with SEPA_PRIORITY 0 (decreasing order) 2. Constraint handlers (decreasing order of CONSHDLR_SEPAPRIORITY) 3. Separators with SEPA_PRIORITY < 0 (decreasing order) Hendel,Gleixner,Serrano Constraint Handler Implementation 28 / 43
54 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Domain propagation (during subproblem processing) Reason for domain reductions (for conict analysis) Presolving (before processing root node) Hendel,Gleixner,Serrano Constraint Handler Implementation 29 / 43
55 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Called whenever... SCIP enters/leaves subtree where local constraint exists constraint is enabled/disabled (no propagation, no separation) Hendel,Gleixner,Serrano Constraint Handler Implementation 29 / 43
56 Ingredients of a Constraint Handler Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Displaying, parsing problems Copying problems between dierent SCIP environments Hendel,Gleixner,Serrano Constraint Handler Implementation 29 / 43
57 Outline Why use constraints? Motivation Callback: Default Constraint Handlers of SCIP Implementation hints for the Nosubtour Constraint Handler Hendel,Gleixner,Serrano Constraint Handler Implementation 30 / 43
58 Exercise: TSP MIP Formulation min s.t. c e x e e E e δ(v) e δ(s) x e = 2 v V x e 2 S V, S x e {0, 1} e E. CIP Formulation min s.t. c e x e e E x e = 2 v V e δ(v) nosubtour(g, x) x e {0, 1} e E. nosubtour(g, x) C {e E x e = 1} : C is a cycle of length C < V Hendel,Gleixner,Serrano Constraint Handler Implementation 31 / 43
59 Exercise: TSP Conict Analysis Cut Pool Solution Pool Implication Graph Presolve Management LP Relaxation Search Tree Hendel,Gleixner,Serrano Constraint Handler Implementation 32 / 43
60 Exercise: TSP MIP Default Plugins Conict Analysis Cut Pool Solution Pool Implication Graph Presolve Management LP Relaxation Search Tree Hendel,Gleixner,Serrano Constraint Handler Implementation 32 / 43
61 Exercise: TSP New Solution TSP Event Handler File Reader 2-Opt MIP Heuristic Default Plugins Conict Analysis Cut Pool Solution Pool Implication Graph Presolve Management LP Relaxation Search Tree Hendel,Gleixner,Serrano Constraint Handler Implementation 32 / 43
62 Exercise: TSP New Solution Event Handler TSP File Reader Nosubtour Constraint Hdlr 2-Opt MIP Heuristic Default Plugins Conict Analysis Cut Pool Solution Pool Implication Graph Presolve Management LP Relaxation Search Tree Hendel,Gleixner,Serrano Constraint Handler Implementation 32 / 43
63 Exercise: TSP New Solution Event Handler TSP File Reader Bonus: LP Greedy Heuristic Nosubtour Constraint Hdlr 2-Opt Heuristic MIP Default Plugins Conict Analysis Cut Pool Solution Pool Implication Graph Presolve Management LP Relaxation Search Tree Hendel,Gleixner,Serrano Constraint Handler Implementation 32 / 43
64 Exercise: TSP Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSINIT..., CONSEXIT... CONSSEPALP, CONSSEPASOL CONSPROP, CONSRESPROP, CONSPRESOL CONSACTIVE, CONSDEACTIVE, CONSENABLE, CONSDISABLE CONSTRANS, CONSDELETE, CONSFREE CONSPRINT, CONSPARSE, CONSCOPY CONSDELVARS, CONSGETVARS, CONSGETNVARS CONSGETDIVEBDCHGS Further Ingredients Private data Interface methods Properties Hendel,Gleixner,Serrano Constraint Handler Implementation 33 / 43
65 Exercise: TSP Callback Methods Fundamental: CONSCHECK CONSENFOLP, CONSENFOPS CONSLOCK Additional: CONSSEPALP, CONSSEPASOL CONSTRANS, CONSDELETE Further Ingredients Private data Interface methods Properties Hendel,Gleixner,Serrano Constraint Handler Implementation 33 / 43
66 CONSCHECK Feasibility Check Given a point y Q E, decide whether y satises all subtour elimination constraints. Hendel,Gleixner,Serrano Constraint Handler Implementation 34 / 43
67 CONSCHECK Feasibility Check Given a point y Q E, decide whether y satises all subtour elimination constraints. Check Priorities make this task easier: Constraint handlers check y in decreasing order of their check priority (CONSHDLR_CHECKPRIORITY). Check priority of nosubtour(g, x) smaller than linear and integral check priorities The solution y already satises integrality and (linear) degree constraints. min s.t. c e x e e E x e = 2 v V e δ(v) nosubtour(g, x) x e {0, 1} e E. Hendel,Gleixner,Serrano Constraint Handler Implementation 34 / 43
68 CONSCHECK Feasibility Check Given a point y Q E, decide whether y satises all subtour elimination constraints. Check Priorities make this task easier: Constraint handlers check y in decreasing order of their check priority (CONSHDLR_CHECKPRIORITY). Check priority of nosubtour(g, x) smaller than linear and integral check priorities The solution y already satises integrality and (linear) degree constraints. min s.t. c e x e e E x e = 2 v V e δ(v) nosubtour(g, x) x e {0, 1} e E. Hendel,Gleixner,Serrano Constraint Handler Implementation 34 / 43
69 CONSCHECK Feasibility Check Given a point y Q E, decide whether y satises all subtour elimination constraints. Check Priorities make this task easier: Constraint handlers check y in decreasing order of their check priority (CONSHDLR_CHECKPRIORITY). Check priority of nosubtour(g, x) smaller than linear and integral check priorities The solution y already satises integrality and (linear) degree constraints. min s.t. c e x e e E x e = 2 v V e δ(v) nosubtour(g, x) x e {0, 1} e E. y is the characteristic vector of a collection of cycles when passed to the nosubtour constraint handler. Hendel,Gleixner,Serrano Constraint Handler Implementation 34 / 43
70 Understanding SCIP Callbacks SCIP callback declarations are realized as macros: What you see in cons_nosubtour.c: 1 static 2 SCIP_DECL_CONSCHECK(consCheckNosubtour) 3 { 4 // Your S o p h i s t i c a t e d Code 5 } What the compiler sees (after preprocessor and some formatting): 1 static 2 SCIP_RETCODE conschecknosubtour(scip* scip, SCIP_CONSHDLR* conshdlr, 3 SCIP_CONS** conss, int nconss, SCIP_SOL* sol, 4 SCIP_Bool checkintegrality, SCIP_Bool checklprows, 5 SCIP_Bool printreason, SCIP_RESULT* result) 6 { 7 // Your S o p h i s t i c a t e d Code 8 } SCIP_DECL_CONSCHECK is a macro dened in type_cons.h. It is responsible of expanding the callback function with the needed arguments and return value. Hendel,Gleixner,Serrano Constraint Handler Implementation 35 / 43
71 Understanding SCIP Callbacks even more A compilable check callback 1 static 2 SCIP_DECL_CONSCHECK(consCheckNosubtour) 3 { 4 / TODO: add some more code / 5 *result = SCIP_INFEASIBLE; 6 7 return SCIP_OKAY; 8 } all callbacks must return a SCIP_RETCODE! Use SCIP_OKAY if everything worked often, callbacks require to correctly assign a result-pointer, examples: *result = SCIP_[IN]FEASIBLE *result = SCIP_SEPARATED *result = SCIP_DIDNOTFIND Find the full callback denition and possible return types in type_<pluginname>.h, eg, type_cons.h for constraint handlers or type_heur.h for primal heuristics. Hendel,Gleixner,Serrano Constraint Handler Implementation 36 / 43
72 The Graph implementation The graph data structure for this exercise is dened in GomoryHuTree.h: 1 typedef struct Graph 2 { 3 int nuses; 4 int nnodes; / < number of nodes of the graph / 5 int nedges; / < number of edges / 6 int nedgesnonzero; 7 8 GRAPHNODE *nodes; / < a r r a y c o n t a i n i n g the nodes of the graph / 9 10 GRAPHEDGE *edges; / < a r r a y c o n t a i n i n g a l l h a l f e d g e s... / } GRAPH; Hendel,Gleixner,Serrano Constraint Handler Implementation 37 / 43
73 The Graph implementation Every graph edge is a half-edge in GomoryHuTree.h: 1 typedef struct GraphEdge 2 { 3 double cap; / < c a p a c i t y used i n maxflow / 4 double rcap; / < r e s i d u a l capacity used in maxflow / 5 double length; / < length of the edge measured by some f i x e d metric 6 7 struct GraphEdge *next; / < i n i n c i d e n c e l i s t of node from which edge i s em 8 struct GraphEdge *back; / < p o i n t e r to r e v e r s e edge / 9 10 GRAPHNODE *adjac; / < p o i n t e r to adjacent node / SCIP_VAR* var; } GRAPHEDGE; Every edge has a variable pointer edge->var. Hendel,Gleixner,Serrano Constraint Handler Implementation 38 / 43
74 The Graph implementation Nodes of the graph only have two relevant members (for us): 1 typedef struct GraphNode 2 { 3 int id; / < number of the node / struct GraphEdge *first_edge;/ < i n l i s t of i n c i d e n t edges / } GRAPHNODE; Hendel,Gleixner,Serrano Constraint Handler Implementation 39 / 43
75 The Graph implementation Question How to traverse all incident edges of a node? Answer: 1 GRAPHEDGE* edge = node->first_edge; 2 while( edge!= NULL ) 3 { 4 / do something with t h i s edge / 5 6 edge = edge->next; 7 } Hendel,Gleixner,Serrano Constraint Handler Implementation 40 / 43
76 The Gomory-Hu Black Box Only one method from GomoryHuTree.h is relevant for us: ghc_tree(). It has the following arguments: GRAPH* gr: the graph data structure SCIP_Bool** cuts: 2-dimensional array to return cuts, needs to be allocated by the user int* ncuts: pointer to store the number of violated cuts double minviol feasibility tolerance, use SCIPfeastol(scip) The method returns TRUE if it found at least one violated cut, and false otherwise. ghc_tree() lls cuts with all minimum (s, t)-cuts with capacity 2 minviol Hendel,Gleixner,Serrano Constraint Handler Implementation 41 / 43
77 Documentation Constraint Integer Programming Tobias Achterberg, Ph.D. dissertation, TU Berlin, Doxygen documentation, HowTos, FAQ source code scip.h, pub_*.h, type_*.h Hendel,Gleixner,Serrano Constraint Handler Implementation 42 / 43
78 Please,... build groups of 3 people laptop owner C expert IP expert raise your hand, if you get stuck. Hendel,Gleixner,Serrano Constraint Handler Implementation 43 / 43
79 Implementing Constraint Handlers in SCIP Gregor Hendel Ambros Gleixner, Felipe Serrano September 30
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