The MIP-Solving-Framework SCIP
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1 The MIP-Solving-Framework SCIP Timo Berthold Zuse Institut Berlin DFG Research Center MATHEON Mathematics for key technologies Berlin,
2 What Is A MIP? Definition MIP The optimization problem is called a MIP (mixed integer program). 2 / 23
3 What Is A MIP? Definition MIP The optimization problem min c T x is called a MIP (mixed integer program). 2 / 23
4 What Is A MIP? Definition MIP The optimization problem min s.t. c T x Ax b is called a MIP (mixed integer program). 2 / 23
5 What Is A MIP? Definition MIP The optimization problem min s.t. c T x Ax b l x u is called a MIP (mixed integer program). 2 / 23
6 What Is A MIP? Definition MIP The optimization problem min s.t. c T x Ax b l x u is called a MIP (mixed integer program). 2 / 23
7 What Is A MIP? Definition MIP The optimization problem min s.t. c T x Ax b l x u x j Z j I is called a MIP (mixed integer program). 2 / 23
8 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
9 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
10 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
11 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
12 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
13 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
14 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
15 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
16 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
17 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
18 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
19 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
20 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
21 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
22 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
23 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
24 How Do We Solve MIPs? Exact methods Branch-And-Bound Cutting planes Combination: Branch-And-Cut 3 / 23
25 CP - A Further Concept Constraint Program (CP) Examples General constraints Integer variables (CIP) How do we solve CPs? 4 / 23
26 CP - A Further Concept Constraint Program (CP) General constraints Integer variables (CIP) Examples TSP as CP n-queens How do we solve CPs? 4 / 23
27 CP - A Further Concept Constraint Program (CP) General constraints Integer variables (CIP) Examples TSP as CP n-queens How do we solve CPs? Branching: Divide into subproblems, solve recursively Domain Propagation: Reductions in variables domains propagate E.g. x1 + 2x 2 5, x 1 2 x / 23
28 SCIP: Solver & Framework SCIP CP+MIP SCIP combines technologies Standalone-solver for MIP A bundle of MIP-solving-components as default plugins MIP-solver as fast as CPlex 9.0 Underlying LP-solver: treated as blackbox Branch-Cut-And-Price-Framework for MIP and CIP C++ wrapper classes for user plugins 5 / 23
29 SCIP: Solver & Framework SCIP CP+MIP SCIP combines technologies Standalone-solver for MIP A bundle of MIP-solving-components as default plugins MIP-solver as fast as CPlex 9.0 Underlying LP-solver: treated as blackbox Branch-Cut-And-Price-Framework for MIP and CIP C++ wrapper classes for user plugins 5 / 23
30 SCIP: Solver & Framework Key data Since October 2002 Achterberg, Wolter, B. et al. Approx lines of C code Free für academic use Available at July 2007: version / 23
31 SCIP: Solver & Framework Key data Since October 2002 Achterberg, Wolter, B. et al. Approx lines of C code Free für academic use Available at July 2007: version I hope so... 6 / 23
32 SCIP: Solver & Framework Types of plugins 7 / 23
33 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure 7 / 23
34 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? 7 / 23
35 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? Branching rule: how to divide the problem? 7 / 23
36 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? Branching rule: how to divide the problem? Separator: adds cuts, improves dual bound 7 / 23
37 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? Branching rule: how to divide the problem? Separator: adds cuts, improves dual bound Constraint handler: assures feasibility, strengthens formulation 7 / 23
38 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? Branching rule: how to divide the problem? Separator: adds cuts, improves dual bound Constraint handler: assures feasibility, strengthens formulation Conflict handler: learns from infeasibility, improves dual bound 7 / 23
39 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? Branching rule: how to divide the problem? Separator: adds cuts, improves dual bound Constraint handler: assures feasibility, strengthens formulation Conflict handler: learns from infeasibility, improves dual bound Heuristic: searches solutions, improves primal bound 7 / 23
40 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? Branching rule: how to divide the problem? Separator: adds cuts, improves dual bound Constraint handler: assures feasibility, strengthens formulation Conflict handler: learns from infeasibility, improves dual bound Heuristic: searches solutions, improves primal bound Propagator: simplifies problem, improves dual bound locally 7 / 23
41 SCIP: Solver & Framework Types of plugins Presolver: simplifies the problem in advance, strengthens structure Node selection: which subproblem should be regarded next? Branching rule: how to divide the problem? Separator: adds cuts, improves dual bound Constraint handler: assures feasibility, strengthens formulation Conflict handler: learns from infeasibility, improves dual bound Heuristic: searches solutions, improves primal bound Propagator: simplifies problem, improves dual bound locally Pricer: allows dynamic generation of variables 7 / 23
42 Flow Chart SCIP Start Init Presolving Stop Node selection Conflict analysis LP inf. Processing IP feas. Primal heuristics IP inf. LP feas. Branching 8 / 23
43 Flow Chart SCIP Start Init Presolving Stop Domain Propagation Conflict analysis Node selection IP feas. Solve LP Pricing LP inf. Processing Cuts Primal heuristics IP inf. LP feas. Enforce Constraints Branching 8 / 23
44 Living By Numbers Default plugins 5 presolvers 5 node selection rules 14 constraint handlers 8 separators 8 branching rules 4 conflict handlers 2 propagators 23 primal heuristics SCIP as a framework for a TSP-solver 9 / 23
45 Living By Numbers Default plugins SCIP as a framework for a TSP-solver main program: TSP file reader: graph structure: subtour constraint: Gomory-Hu algo.: FarInsert heuristic: 2-Opt heuristic: altogether: 196 lines 407 lines 80 lines 793 lines 658 lines 354 lines 304 lines 2792 lines 9 / 23
46 Comparison With Other Free MIP-Solvers 10 / 23
47 Primal Heuristics Characteristics Highest priority to feasibility Distinguish: Start heuristics Improvement heuristics Keep control of effort! Use available information 11 / 23
48 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
49 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
50 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
51 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
52 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
53 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
54 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
55 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
56 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
57 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
58 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
59 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
60 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
61 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
62 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
63 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
64 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
65 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
66 Primal Heuristics Used Information Variables locking numbers: Potentially violated rows Variables pseudocosts: Average objective change Special points: LP optimum at root node Current LP optimum Current best solution Other known solutions 12 / 23
67 Primal Heuristics Approaches Rounding Diving: simulate DFS in the Branch-And-Bound-tree using some special branching rule Objective diving: manipulate objective function LNS: solve some sub-mip Pivoting: manipulate simplex algorithm 13 / 23
68 Primal Heuristics Approaches Rounding Diving: simulate DFS in the Branch-And-Bound-tree using some special branching rule Objective diving: manipulate objective function LNS: solve some sub-mip Pivoting: manipulate simplex algorithm 13 / 23
69 Overview Implemented into SCIP 5 Rounding heuristics 8 Diving heuristics 3 Objective divers 4 LNS improvement heuristics 14 / 23
70 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
71 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
72 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
73 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
74 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
75 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
76 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
77 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
78 Rounding Heuristics Ideas Simple Rounding always stays feasible, Rounding may violate constraints, Shifting may unfix integers. 15 / 23
79 Diving Heuristics Idea: iteratively solve the LP and round a variable 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 16 / 23
80 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
81 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
82 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
83 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
84 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
85 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
86 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
87 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
88 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
89 The Feasibility Pump Algorithm 1. Solve LP; 2. Round LP optimum; 3. If feasible: 4. Stop! 5. Else: 6. Change Objective; 7. Go to 1; 17 / 23
90 The Objective Feasibility Pump Improvements Objective c T x regarded at each step Algorithm able to resolve from cycling Quality of solutions much better 18 / 23
91 The Objective Feasibility Pump Improvements Objective c T x regarded at each step 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% 18 / 23
92 Rens An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i Z; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting sub-mip 19 / 23
93 Rens An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i Z; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting sub-mip 19 / 23
94 Rens An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i Z; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting sub-mip 19 / 23
95 Rens An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i Z; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting sub-mip 19 / 23
96 Rens An LNS Rounding Heuristic Algorithm 1. x LP optimum; 2. Fix all integral variables: x i := x i i : x i Z; 3. Reduce domain of fractional variables: x i { x i ; x i }; 4. Solve the resulting sub-mip 19 / 23
97 Rens An LNS Rounding Heuristic Observations Solutions found by Rens are roundings of x Yields best possible rounding Yields certificate, if no rounding exists 20 / 23
98 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 Approx. 2 of the test instances are roundable 3 Rens finds optimum for 20%! Dominates all other rounding heuristics 20 / 23
99 An Example Instanz aflow30a: performance of SCIP with and without heuristics 21 / 23
100 Thesis Conclusions Results Coordination important Positive side effects Improvement of overall performance SCIP with heuristics twice as fast 22 / 23
101 The MIP-Solving-Framework SCIP Timo Berthold Zuse Institut Berlin DFG Research Center MATHEON Mathematics for key technologies Berlin,
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