# Adaptive Constructive Interval Disjunction

TL;DR: On a representative sample of instances, ACID appears to be the best approach in solving and optimization, and has been added to the default strategies of the Ibex interval solver.

Abstract: An operator called CID and an efficient variant 3BCID wereproposed in 2007. For numerical CSPs handled by interval methods, these operators compute a partial consistency equivalent to Partition-1-AC for discrete CSPs. The two main parameters of CID are the number of times the main CID procedure is called and the maximum number ofsub-intervals treated by the procedure. The 3BCID operator is state-of-the-art in numerical CSP solving, but not in constrained global optimization. This paper proposes an adaptive variant of 3BCID. The number of variables handled is auto-adapted during the search, the other parameters are fixed and robust to modifications. On a representative sample of instances, ACID appears to be the best approach in solving and optimization, and has been added to the default strategies of the Ibex interval solver.

## Summary (3 min read)

### Introduction

- This paper proposes an adaptive variant of 3BCID.
- The number of variables handled is auto-adapted during the search, the other parameters are fixed and robust to modifications.
- On a representative sample of instances, ACID appears to be the best approach in solving and optimization, and has been added to the default strategies of the Ibex interval solver.

### I. CONSTRUCTIVE INTERVAL DISJUNCTION (CID)

- A filtering/contracting operator for numerical CSPs called Constructive Interval Disjunction (in short CID) has been proposed in [13].
- Applied first to continuous constraint satisfaction problems handled by interval methods, it has been more recently applied to constrained global optimization problems.
- This algorithm is state-of-the-art in constraint satisfaction, but is generally dominated by constraint propagation algorithms like HC4 in optimization.
- The main practical contribution is that an adaptive version of CID becomes efficient for both real-valued satisfaction and optimization problems, while needing no additional parameter value from the user.

### B. Numerical CSP

- The constraints defined in an NCSP are numerical.
- They are equations and inequalities using mathematical operators like +, , /, exp, log, sin.
- NCSPs are generally solved by a Branch & Contract interval strategy: Branch: a variable xi is chosen and its interval [xi] is split into two sub-intervals, thus making the whole process combinatorial.
- The 2BRevise procedure works with all the projection functions of a given constraint.
- C. 3B algorithm Stronger interval partial consistencies have also been proposed.

### D. CID

- Constructive Interval Disjunction (CID) is a partial consistency stronger than 3B-consistency [13].
- CID-consistency is similar to Partition-1-AC (P-1-AC) in finite domain CSPs [4].
- The main procedure varCID handles a single variable xi.
- The subboxes are contracted by ctc and hulled, giving [Xcid].
- The procedure var3BCID has been deeply studied and experimented in the past.

### II. ADAPTIVE CID: LEARNING THE NUMBER OF

- Like for SAC or 3B, a quasi fixed-point in terms of contraction can be reached by 3BCID (or CID) by calling var3BCID inside two nested loops.
- An outer loop calls the inner loop until no interval is contracted more than a predefined precision (thus reaching a quasi-fixed point).
- The authors will write in the remaining part of the paper that a variable is varcided when the procedure var3BCID is called on that variable to contract the current box.
- This gives good results in satisfaction but is dominated by pure constraint propagation in optimization.
- All the policies measure the decrease in search space size after each call to var3BCID.

### A. ACID0: auto-adapting numVarCID during search

- The first version ACID0 adapts the number of shaved variables dynamically at each node of the search tree.
- First, the variables are sorted by their impact, computed by the same formula as the SmearSumRel function (used for branching).
- Variables are then varcided until the cumulative contraction ratio during the last nv calls to var3BCID becomes less than ctratio.
- This algorithm has thus 2 parameters nv and ctratio, and it was difficult to tune them.
- The experimental results are not bad but this policy prevents numVarCID from reaching 0, i.e. from calling only constraint propagation.

### B. ACID1: interleaving learning and exploitation phases

- A more sophisticated approach avoids this drawback.
- After the kvarCIDth call to var3BCID, the gain in current box size from a var3BCID call to the next one, computed by the gainRatio formula, never exceeded a small given ratio, called ctratio.
- During the exploitation phase following the previous learning phase, the average of the different kvarCID values (obtained in the nodes of the learning phase) provides the new value of numVarCID.
- Numerous variants of this schema were tested.
- The authors fixed experimentally the 3 parameters of the ACID1 procedure learnLength, cycleLength and ctratio, respectively to 50, 1000 and 0.002.

### C. ACID2: taking into account the level in the search tree

- A criticism against ACID1 is that the authors average kvarCID values obtained at different levels of the search tree.
- This drawback is partially corrected by the successive learning phases of ACID1, where each learning phase corresponds to a part of the search tree.
- A value corresponds to one order of magnitude in the box width.
- This approach, called ACID2, gave in general results similar to those of ACID1 and appeared to be less robust.
- Indeed, only a few nodes sometimes fall at certain width levels, which renders the statistics not significant.

### III. EXPERIMENTS

- All the algorithms were implemented in the C++ interval library Ibex (Interval Based EXplorer) [6].
- All the experiments were run on the same computer (Intel X86 3GHz).
- The authors tested the algorithms on square NCSP solving and constrained global optimization.
- NCSP solving consists in finding all the solutions of a square system of n nonlinear equations with n real-values variables with bounded domains.
- Global optimization consists in finding the global minimum of a function over n variables subject to constraints (equations and inequalities), the objective function and/or the constraints being non-convex.

### A. Experiments in constraint satisfaction

- The authors selected from the COPRIN benchmark1 all the systems that were solved by one of the tested algorithms in a time comprised between 2 s and 3600 s.
- The authors compared their ACID method and its variants with the well known filtering techniques: a simple constraint propagation HC4, 3BCID-n (see Section II) and 3BCID-fp (fixed-point) in which a new iteration on all the variables is run when a variable domain width is reduced by more than 1%.
- In particular, setting s3b to 10 gives results better than with smaller values (s3b = 5) and with greater values.
- ACID1 obtains better gains w.r.t 3BCID-n in total time than on average because the best gains were obtained on difficult instances with more variables.
- In the right part of the table, the authors report the solving time ratios obtained when X-Newton is removed (¬ XN) from the contractor sequence (4 problems could not be solved in 10,000s).

### B. Experiments in constrained global optimization

- The authors used the IbexOpt strategy of Ibex that performs a Best First Branch & Bound.
- The precision required on the objective is 10−8.
- In fact, the more recent Mohc constraint propagation algorithm [1] is better than HC4.
- It is significant because the CP contraction is only a part of the IbexOpt algorithm [12] (linear relaxation and the search of feasible points are other important parts, not studied in this paper and set to their default algorithms in IbexOpt).
- ACID2 obtains results slightly worse than ACID1, rendering this refinement not promising in practice.

### IV. CONCLUSION

- The authors have presented in this paper an adaptive version of the 3BCID contraction operator used by interval methods and close to partition-1-AC.
- The best variant of this Adaptive CID operator (ACID1 in the paper) interleaves learning phases and exploitation phases to auto-adapt the number of variables handled.
- These variables are selected by an efficient branching heuristic and all the other parameters are fixed and robust to modifications.
- Overall, ACID1 adds no parameter to the solving or optimization strategies.
- It offers the best results on average and is the best or close to the best on every tested instance, even in presence of the best Ibex devices (Interval-Newton, X-Newton).

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##### Citations

18 citations

### Cites background from "Adaptive Constructive Interval Disj..."

...|) for approximating the search space, as done in (Neveu and Trombettoni 2013)....

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...In (Trombettoni and Chabert 2007; Neveu and Trombettoni 2013) a consistency called Constructive Interval Disjunction (CID), close to POAC in its principle, gave good results by simply calling the main procedure once on each variable or by adapting during search the number of times it is called....

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15 citations

### Cites background from "Adaptive Constructive Interval Disj..."

...Another adaptive interval-based contractor is proposed in [95]....

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9 citations

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##### References

1,690 citations

### "Adaptive Constructive Interval Disj..." refers methods in this paper

...The experimental protocol is the same as the NCSP experimental protocol, except that we do not use Interval-Newton that is only implemented for square systems....

[...]

...At each node of the search tree, we used the following sequence of contractors : HC4, shaving, Interval-Newton [8], and X-Newton [2]....

[...]

...It offers the best results on average and is the best or close to the best on every tested instance, even in presence of the best Ibex devices (Interval-Newton, X-Newton)....

[...]

...At each node of the search tree, we used the following sequence of contractors : HC4, shaving, Interval-Newton [8], and X-Newton [2]. shaving denotes a variant of ACID, 3BCID-n, 3BCID-fp or nothing when only HC4 is tested....

[...]

497 citations

### "Adaptive Constructive Interval Disj..." refers background or methods in this paper

...NCSPs are generally solved by a Branch & Contract interval strategy: • Branch: a variable xi is chosen and its interval [xi] is split into two sub-intervals, thus making the whole process combinatorial....

[...]

...A similar idea can be followed on numerical CSPs (NCSPs)....

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...3B-consistency [9] is a theoretical partial consistency similar to SAC for CSP although limited to the bounds of the domains....

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...Let us mention the constraint propagation algorithm called HC4 [3], [10], an efficient implementation of 2B [9], that can enforce the optimal local consistency (called hull-consistency) only if strong hypotheses are met (in particular, each variable 2013 IEEE 25th International Conference on Tools with Artificial Intelligence...

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...CID-consistency is similar to Partition-1-AC (P-1-AC) in finite domain CSPs [4]....

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404 citations

### "Adaptive Constructive Interval Disj..." refers methods in this paper

...of the SATZ algorithm [11] used to prove the satisfiability of Boolean formula....

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403 citations

### "Adaptive Constructive Interval Disj..." refers methods in this paper

...Let us mention the constraint propagation algorithm called HC4 [3], [10], an efficient implementation of 2B [9], that can enforce the optimal local consistency (called hull-consistency) only if strong hypotheses are met (in particular, each variable 1082-3409/13 $31.00 © 2013 IEEE DOI 10.1109/ICTAI.2013.138 900 must occur at most once in a same constraint)....

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...Therefore, we report in the penultimate column a comparison between ACID1 and HC4....

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...The main parameters of varCID are xi, a number scid of sub-intervals (accuracy) and a contraction algorithm ctc like HC4....

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...At each node of the search tree, we used the following sequence of contractors : HC4, shaving, Interval-Newton [8], and X-Newton [2]. shaving denotes a variant of ACID, 3BCID-n, 3BCID-fp or nothing when only HC4 is tested....

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...Table IV shows that we obtained an average gain of 10% over HC4....

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