Primal Heuristics in SCIP
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1 Primal Heuristics in SCIP Timo Berthold Zuse Institute Berlin DFG Research Center MATHEON Mathematics for key technologies Berlin, 10/11/2007
2 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 2 / 34
3 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 3 / 34
4 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 4 / 34
5 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics 5 / 34
6 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics 5 / 34
7 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics 5 / 34
8 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics 5 / 34
9 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics Often find good solutions 5 / 34
10 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics Often find good solutions in a reasonable time 5 / 34
11 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics Often find good solutions in a reasonable time without any warranty! 5 / 34
12 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Branch-And-Cut Heuristics Often find good solutions in a reasonable time without any warranty! Integrate into exact solver 5 / 34
13 Primal Heuristics Why use heuristics inside an exact solver? Able to prove feasibility of the model Often nearly optimal solution suffices in practice Feasible solutions guide remaining search process Characteristics 6 / 34
14 Primal Heuristics Why use heuristics inside an exact solver? Able to prove feasibility of the model Often nearly optimal solution suffices in practice Feasible solutions guide remaining search process Characteristics Highest priority to feasibility Keep control of effort! Use as much information as you can get 6 / 34
15 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 7 / 34
16 Adding A Primal Heuristic SCIPincludeHeur( scip, // scip "Christofides", // HEUR_NAME "Start heuristic for TSP", // HEUR_DESC X, // HEUR_DISPCHAR , // HEUR_PRIORITY 0, // HEUR_FREQ 0, // HEUR_FREQOFS 0, // HEUR_MAXDEPTH SCIP_HEURTIMING_BEFORENODE // HEUR_TIMING ); 9 / 34
17 Heuristic Timings LP Solving Domain Propagation Solve LP Pricing Separation Constraint Enforcement Domain Propagation Primal Heuristics 10 / 34
18 Heuristic Timings LP Solving Domain Propagation Solve LP Pricing Separation Constraint Enforcement Domain Propagation Primal Heuristics #define HEUR_TIMING SCIP_HEURTIMING_AFTERNODE 10 / 34
19 Heuristic Timings Primal Heuristics Domain Propagation LP Solving Solve LP Pricing Separation Constraint Enforcement Domain Propagation Primal Heuristics #define HEUR_TIMING SCIP_HEURTIMING_BEFORENODE 10 / 34
20 Heuristic Timings Primal Heuristics Domain Propagation LP Solving Solve LP Pricing Heurs Separation Constraint Enforcement Domain Propagation Primal Heuristics #define HEUR_TIMING SCIP_HEURTIMING_DURINGLPLOOP 10 / 34
21 Categories Two main categories Start heuristics Often already at root node Mostly start from LP optimum Improvement heuristics Require feasible solution Normally at most once for each incumbent 11 / 34
22 Categories Two main categories Start heuristics Often already at root node Mostly start from LP optimum Improvement heuristics Require feasible solution Normally at most once for each incumbent #define HEUR_FREQOFS 0 11 / 34
23 Categories Two main categories Start heuristics Often already at root node Mostly start from LP optimum Improvement heuristics Require feasible solution Normally at most once for each incumbent if( SCIPgetLPSolstat(scip)!= SCIP_LPSOLSTAT_OPTIMAL ) return SCIP_OKAY; 11 / 34
24 Categories Two main categories Start heuristics Often already at root node Mostly start from LP optimum Improvement heuristics Require feasible solution Normally at most once for each incumbent if( SCIPgetNSols(scip) <= 0 ) return SCIP_OKAY; 11 / 34
25 Categories Two main categories Start heuristics Often already at root node Mostly start from LP optimum Improvement heuristics Require feasible solution Normally at most once for each incumbent struct SCIP_HeurData { SCIP_SOL* lastsol; } 11 / 34
26 Approaches Five main approaches Rounding assign integer values to fractional variables Diving: DFS in the Branch-And-Bound-tree Objective diving: manipulate objective function LargeNeighborhoodSearch: solve some submip Pivoting: manipulate simplex algorithm 12 / 34
27 Approaches Five main approaches Rounding assign integer values to fractional variables Diving: DFS in the Branch-And-Bound-tree Objective diving: manipulate objective function LargeNeighborhoodSearch: solve some submip Pivoting: manipulate simplex algorithm 12 / 34
28 Overview Implemented into SCIP 5 Rounding heuristics 8 Diving heuristics 3 Objective divers 4 LNS improvement heuristics 3 Others 13 / 34
29 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions 14 / 34
30 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions 14 / 34
31 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions 14 / 34
32 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions int nlocks = SCIPvarGetNLocksUp(var); 14 / 34
33 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions int nlocks = SCIPvarGetNLocksDown(var); 14 / 34
34 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions int nlocks = SCIPvarGetNLocksDown(var); 14 / 34
35 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions int nlocks = SCIPvarGetNLocksUp(var); 14 / 34
36 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions 14 / 34
37 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions 14 / 34
38 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions 14 / 34
39 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Real pscost = SCIPgetVarPseudocost(scip, var, delta); 14 / 34
40 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Real pscost = SCIPgetVarPseudocost(scip, var, delta); 14 / 34
41 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Real pscost = SCIPgetVarPseudocost(scip, var, delta); 14 / 34
42 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Real pscost = SCIPgetVarPseudocost(scip, var, delta); 14 / 34
43 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions 14 / 34
44 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Real rootsolval = SCIPvarGetRootSol(var); 14 / 34
45 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Real solval = SCIPgetSolVal(scip, NULL, var); 14 / 34
46 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Sol* bestsol = SCIPgetBestSol(scip); SCIP_Real solval = SCIPgetSolVal(scip, bestsol, var); 14 / 34
47 Useful Information Statistics & points Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP solution Current best solution Other known solutions SCIP_Sol** sols = SCIPgetSols(scip); SCIP_Real solval = SCIPgetSolVal(scip, sols[i], var); 14 / 34
48 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 15 / 34
49 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 16 / 34
50 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
51 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
52 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
53 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
54 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
55 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
56 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
57 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
58 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
59 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
60 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
61 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
62 Rounding Heuristics Guideline: Stay feasible! Features Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers, Integer Shifting finally solves an LP. 17 / 34
63 Rens: An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i ; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting submip 18 / 34
64 Rens: An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i ; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting submip 18 / 34
65 Rens: An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i ; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting submip 18 / 34
66 Rens: An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i ; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting submip 18 / 34
67 Rens: An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i ; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting submip 18 / 34
68 Rens: An LNS Rounding Heuristic Observations Solutions found by Rens are roundings of x Yields best possible rounding Yields certificate, if no rounding exists Results 19 / 34
69 Rens: An LNS Rounding Heuristic Observations Solutions found by Rens are roundings of x Yields best possible rounding Yields certificate, if no rounding exists Results 82 of 129 test instances are roundable Rens finds a global optimum for 23 instances! Dominates all other rounding heuristics 19 / 34
70 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 20 / 34
71 Diving Heuristics I Idea Alternately solve the LP and round a variable Simulates DFS in Branch-And-Bound-tree Complementary target for branching Backtracking possible 21 / 34
72 Diving Heuristics II Applied branching rules Fractional Diving: lowest fractionality Coefficient Diving: lowest locking number Linesearch Diving: highest increase since root Guided Diving: lowest difference to best known solution Pseudocost Diving: highest ratio of pseudocosts Vectorlength Diving: lowest ratio of objective change and number of rows containing the variable 22 / 34
73 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; 23 / 34
74 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; 23 / 34
75 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; 23 / 34
76 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; 23 / 34
77 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; (x, x) = x j x j 23 / 34
78 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; (x, x) = x j x j 23 / 34
79 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; (x, x) = x j x j 23 / 34
80 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; 23 / 34
81 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; 23 / 34
82 The Feasibility Pump (Fischetti, Lodi et al.) Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change objective; 7. Go to 1; 23 / 34
83 Objective Feasibility Pump (Achterberg & B.) Improvements Objective c T x regarded at each step: := (1 α) (x) + αc T x, with α [0, 1] Algorithm able to resolve from cycling Quality of solutions much better Results 24 / 34
84 Objective Feasibility Pump (Achterberg & B.) Improvements Objective c T x regarded at each step: := (1 α) (x) + αc T x, with α [0, 1] Algorithm able to resolve from cycling Quality of solutions much better Results Finds a solution for 74% of the test instances On average 5.5 seconds running time Optimality gap decreased from 107% to 38% 24 / 34
85 Other Objective Divers Other objective divers Objective Pseudocost Diving analogon to Pseudocost Diving Punishment by high objective coefficients Rootsolution Diving analogon to Linesearch Diving Objective function faded out 25 / 34
86 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 26 / 34
87 LNS heuristics Idea: Create submip by fixing variables or adding constraints Approaches Rins: fix variables equal in LP optimum and incumbent Crossover: fix variables equal in different feasible solutions Mutation: fix variables randomly Local Branching: add distance constraint wrt. incumbent 27 / 34
88 Other Heuristics Combinatorial heuristics 1-Opt Shifts value of integer variable Solves LP afterwards 28 / 34
89 Other Heuristics Combinatorial heuristics 1-Opt Shifts value of integer variable Solves LP afterwards Octane Duality of cube and octahedron Ray shooting algorithm 28 / 34
90 Other Heuristics Combinatorial heuristics 1-Opt Shifts value of integer variable Solves LP afterwards Octane Duality of cube and octahedron Ray shooting algorithm 28 / 34
91 Other Heuristics Combinatorial heuristics 1-Opt Shifts value of integer variable Solves LP afterwards Octane Duality of cube and octahedron Ray shooting algorithm 28 / 34
92 Other Heuristics Combinatorial heuristics 1-Opt Shifts value of integer variable Solves LP afterwards Octane Duality of cube and octahedron Ray shooting algorithm 28 / 34
93 Other Heuristics Combinatorial heuristics 1-Opt Shifts value of integer variable Solves LP afterwards Octane Duality of cube and octahedron Ray shooting algorithm 28 / 34
94 Outline 1 Introduction Basics Integration Into SCIP 2 Available Heuristics Rounding Heuristics (Objective) Diving LNS & Others 3 Remarks & Results 29 / 34
95 Implementation Details Some tips and tricks Use limits which respect the problem size LNS: stalling node limit Diving: try simple rounding on the fly Favor binaries over general integers Avoid cycling without randomness 30 / 34
96 Implementation Details Some tips and tricks Use limits which respect the problem size LNS: stalling node limit Diving: try simple rounding on the fly Favor binaries over general integers Avoid cycling without randomness int nlpiterations = SCIPgetNNodeLPIterations(scip); int maxlpiterations = heurdata maxlpiterquot * nlpiterations; 30 / 34
97 Implementation Details Some tips and tricks Use limits which respect the problem size LNS: stalling node limit Diving: try simple rounding on the fly Favor binaries over general integers Avoid cycling without randomness SCIP_CALL( ); SCIPsetLongintParam(subscip,"limits/stallnodes",nstallnodes) 30 / 34
98 Optimal Objective Primal Bound With Heuristics Dual Bound With Heuristics Primal Bound Without Heuristics Dual Bound Without Heuristics Solution Found By: Relaxation Feaspump Crossover Rens An Example 1300 bound time (seconds) 31 / 34
99 Remarks & Conclusions Single heuristics Deactivating a single heuristic yields 1%-6% degradation No heuristic dominates the others Coordination important Overall effect (SCIP 0.82b) 32 / 34
100 Remarks & Conclusions Single heuristics Deactivating a single heuristic yields 1%-6% degradation No heuristic dominates the others Coordination important Overall effect (SCIP 0.82b) Better pruning, earlier fixing 7% less instances without any solution 5% more instances solved within one hour only half of the branch-and-bound-nodes only half of the solving time 32 / 34
101 Remarks & Conclusions Single heuristics Deactivating a single heuristic yields 1%-6% degradation No heuristic dominates the others Coordination important Overall effect (SCIP 0.82b) Better pruning, earlier fixing 7% less instances without any solution 5% more instances solved within one hour only half of the branch-and-bound-nodes Not this much only half of the solving time in SCIP / 34
102 Perspective & Further Research To be implemented DINS heuristic k-opt for k > 1 probing heuristics Known problems Coordination could be strengthened Often poor for pure combinatorial problems 33 / 34
103 Primal Heuristics in SCIP Timo Berthold Zuse Institute Berlin DFG Research Center MATHEON Mathematics for key technologies Berlin, 10/11/2007
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