The Heuristic (Dark) Side of MIP Solvers. Asja Derviskadic, EPFL Vit Prochazka, NHH Christoph Schaefer, EPFL

Transcription

1 The Heuristic (Dark) Side of MIP Solvers Asja Derviskadic, EPFL Vit Prochazka, NHH Christoph Schaefer, EPFL 1

2 Table of content [Lodi], The Heuristic (Dark) Side of MIP Solvers, Hybrid Metaheuristics, , Integer Programming Heuristic nature of MIP solvers Key Features of MIP Conclusion 2

3 Integer programming Mathematical model -> Solver (Meta) Heuristics - do not guarantee optimality Random choice Go to the nearest city that you have not visited yet Metaheuristics tabu search, genetic algorithms (population of solutions) 3

4 Integer programming Mathematical model -> Solver (Meta) Heuristics - do not guarantee optimality Formulation of the model is (sometimes) difficult MIP solver will not solve it anyway 4

5 Heuristic nature of MIP solvers Trivial facts User imposes limits on computation time, number of nodes, number of feasible solutions, percentage gap Reason: Optimal solution is not required or not possible Time-critical Overall problem split into pieces Tolerance Floating point operations introduce errors MIP solvers work with tolerances to check against errors: Feasibility = tight tolerance Optimality = not so tight tolerance (Popular default value 10 2 ) 5

6 Heuristic nature of MIP solvers Less trivial facts Ineffective algorithmic decision N P-hard: there is always a polynomial path to the optimal solution Worst case: path followed will be exponentially long Ineffective (unlucky) choice of the first branching variable potentially leads to a search tree twice as big 6

7 Heuristic nature of MIP solvers Less trivial facts Benchmarking Test sets are made of hundred of instances classified from very easy to very difficult Instances can target specific difficulties New methods have to Improve on the sub-set specific to the difficulty that the new method targets Not significantly deteriorate the performance on the other test sets, mainly the easy ones 7

8 Heuristic Nature of MIP Solvers Less trivial facts Performance Variability Performance Variability Unexpected changes in performance when solving the same model with the same solver while applying apparently harmless actions, such as: Using a different computational environment Changing the constraints order Adding or removing redundant constraints or variable bounds Floating-point computations & Rounding errors Arbitrary path selection in the branch-and-cut tree & Imperfect tie-breaking Open questions: Is it a disturbing (or deteriorating) factor? What is the importance of each factor to variability? How can we change MIP solvers and MIP models so as to reduce variability? 8

9 Heuristic Nature of MIP Solvers Less trivial facts Performance Variability For each model and for each solver [Koch et Al.] define the Performance Variability score for time to optimality (VS), estimated on a sample of observations provided by solving multiple permutations of the model The coefficient of variation of X, being X the random variable representing the performance of the given solver for the given model VS = 1 n t i i=1 n i=1 t i i=1 n n t i 2 Where: n t i number of permutations corresponding running times 9

10 Heuristic Nature of MIP Solvers Less trivial facts Performance Variability Results for the BENCHMARK-Set with 87 instances (time limit 1 h solver) SCIP/SPX 100 permutations model, instance No instance for which all 100 permutations showed the same behavior Min 0.05 (mik ) Max 2.23 (neos ) 10

11 Heuristic Nature of MIP Solvers Less trivial facts Performance Variability Distribution of performance for specific instances Performance of one permutation when being solved with SCIP/SPX (sorted by non-decreasing solution time). Black dot corresponds to the performance of the original formulation. 11

12 Heuristic Nature of MIP Solvers Less trivial facts Performance Variability PV as a function of the number of threads. Instance roll3000 on a 32 core computer. Filled bar indicates minimum. Total number of nodes explored (CPLEX) Wall clock solution time (GUROBI) 12

13 The Heuristic Nature of MIP Solvers Less trivial facts Performance variability The question is How likely is it that the performance change is created by variability rather than by genuine algorithmic changes? PV affects all standard objectives of benchmarking Comparing different solvers, comparing different parameter settings for the same solver, or comparing a new algorithm to an existing algorithm. PV is not deterministic random variable that can be assessed with computational experiments Needs to be studied for each model and for each solver when studying performance & correctness of computation: Over a large data sample By running experiments on a large set of models By running experiments on permuted models for multiple permutations (artificially multiplying the number of observations)

14 Key Features of MIP Solvers Introduction Key components of any MIP solver Preprocessing Cutting plane generation Branching Primal heuristics 14

15 Constraints Constraints Key Features of MIP Solvers Preprocessing The MIP solver tries to detect changes in the input that will probably lead to a better performance of the solution process by model or algorithmic preprocessing Locally compare and analyze constraints and variables in a heuristic way Random permutations of rows/columns of the MIP generally lead to a performance deterioration of the solvers Variables Variables

16 Key Features of MIP Solvers Cutting Plane Phase of strengthening through adding new linear inequalities (cuts) Preparation for LP relaxation phase Inequalities are computed using 2 steps heuristically aggregating the entire MIP into a mixed integer set applying the cut separation to the mixed integer set 16

17 Key Features of MIP Solvers Cutting Plane - Heuristic decision Aggregation into a mixed integer set heuristic decision Cut selection Multiple possible cuts for each iteration Selection of which cut to use for the next LP relaxation is a heuristic decision 17

18 Key Features of MIP Solvers Branching Cutting planes Trade-off between cutting and branching Heuristic rule 18

19 Key Features of MIP Solvers Branching LB > Best Bound Best Bound Heuristic 19

20 Key Features of MIP Solvers Branching Splitting the region into smaller sub-mips: In the case of Z: x LP = => x > 4 or x < 3 y LP = => y > 7 or x < 6 You can try both (all fractional) and solve all sub-mips or you can choose only a subset of them to solve LP (Do you need to solve it to optimality? ) Good branching submips are easier (fast) to solve than original MIP Heuristic decision based on some a priori proxy measures 20

21 Key Features of MIP Solvers Primal heuristic Aim: getting a good feasible integer solution from the LP relaxation From simple (rounding) to more sophisticated ones (metaheuristics) Importance: Perception of the solver quality (you never know when a user will stop the process) Branch and Bound 21

22 Conclusions MIP solvers (i) Can be used as heuristics (ii) Are developed and tested using heuristic criteria (iii) Use heuristic decisions in each of their basic components (iv) Incorporate heuristic algorithms (used to find good solutions or as building blocks of sophisticated algorithmic strategies) (v) Benefit from ideas originated in different communities Constraint Programming propagation algorithms Metaheuristics neighborhood exploitation Are open frameworks for effective and sophisticated algorithmic development Are open to ideas originated in different areas 22

23 References [Koch et Al.] T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R.E. Bixby, E. Danna, G. Gamrath, A.M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D.E. Steffy, K. Wolter, MILPLIB 2010, Mathematical Programming Computation 3, ,

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25 Table of content Intro (Vit) Explanation of heuristic and MIP, fig Definition of MIP Objectif: bridge between heuristic and MIP [exact] (chapter1) Heuristic nature of MIP solvers (chapter 2)(Christoph) Trivial facts: float, computational constraint= time limitation, tolerance, nb of branches Non trivial facts: ineffective algorithmic decision, performance variability and non-trivial benchmarking Key Features of MIP Preprocessing: Non global constraint programming => heuristic component? (Asja) Cutting Plane: depending on the method heuristic determination of where to cut (Christoph) Branch and cut: branching decisions about is heuristic on how deep and wide to go(img) (Vit) Primal heuristic: (Vit) Conclusion (Asja) Using transport example for all chapters 25