International audienceWe prove that hull consistency for a system of equations or inequalities can be achieved in polynomial time providing that the underlying functions are monotone with respect to each variable. This result holds including when variables have multiple occurrences in the expressions of the functions, which is usually a pitfall for interval-based contractors. For a given constraint, an optimal contractor can thus be enforced quickly under monotonicity and the practical significance of this theoretical result is illustrated on a simple example

E. Hyvönen

E.R. Hansen

F. Benhamou

F. Delobel

G. Alefeld

H. Collavizza

J.G. Cleary

L. Granvilliers

L. Jaulin

P. Hentenryck Van

R. Moore

V. Kreinovich

Abstract. We prove that hull consistency for a system of equations or inequalities can be achieved in polynomial time providing that the underlying functions are monotone with respect to each variable. This result holds including when variables have multiple occurrences in the expressions of the functions, which is usually a pitfall for interval-based contractors. For a given constraint, an optimal contractor can thus be enforced quickly under monotonicity and the practical significance of this theoretical result is illustrated on a simple example. 

Gilles Chabert

Luc Jaulin

CiteSeerX

Hull consistency under monotonicity

Chabert, Gilles

Jaulin, Luc

HAL-Univ-Nantes

Hull Consistency Under MonotonicityGilles Chabert, Luc JaulinTo cite this version:Gilles Chabert, Luc Jaulin. Hull Consistency Under Monotonicity. Ian P. Gent. CP’09 (15thInternational Conference on Principles and Practice of Constraint Programming), Sep 2009,Lisbon, Portugal. Springer Verlag, 5732, p. 188-195, 2009, LNCS (Lecture Notes in ComputerScience). <hal-00428970>HAL Id: hal-00428970https://hal.archives-ouvertes.fr/hal-00428970Submitted on 30 Oct 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Hull Consistency Under MonotonicityGilles Chabert1 and Luc Jaulin21 Ecole des Mines de Nantes LINA CNRS UMR 6241,4, rue Alfred Kastler 44300 Nantes, Francegilles.chabert@emn.fr2 ENSIETA, 2, rue Franc¸ois Verny 29806 Brest Cedex 9, Franceluc.jaulin@ensieta.frAbstract. We prove that hull consistency for a system of equationsor inequalities can be achieved in polynomial time providing that theunderlying functions are monotone with respect to each variable. Thisresult holds including when variables have multiple occurrences in theexpressions of the functions, which is usually a pitfall for interval-basedcontractors. For a given constraint, an optimal contractor can thus beenforced quickly under monotonicity and the practical significance of thistheoretical result is illustrated on a simple example.1 IntroductionSolving constraint problems with real variables has been the subject of significantdevelopments since the early 90’s (see [3] for a comprehensive survey).One of the key contribution is the concept of hull consistency, which is the coun-terpart of bound consistency in discrete constraint programming, as Definition 1shows below.Let us briefly trace the history. The underlying concepts of interval propaga-tion appeared first in several pioneering papers [6, 11, 17, 12] while consistencytechniques for numerical CSP were formalized a few years later in [15, 5]. A the-oretical comparative study of consistencies was then conducted in [7, 8]. Finally,hull consistency was made operational in [2, 9] where the famous HC4 algorithmis described.Since hull consistency is based on the bound consistency of every isolated con-straint, enforcing hull consistency in the general case (i.e., for arbitrary nonlinearequations) is a NP-hard problem [14]. On the practical side, this results into theinability to give a sharp enclosure when variables occur more than once in theexpression of a constraint. This happens, in particular, with HC4.We show that hull consistency can be enforced in polynomial time if the functionsinvolved are all monotone.Monotonicity follows the very intuitive idea that a function varies either in thesame direction as a variable or in the opposite one (see Definition 2). It turnsout that usual functions, i.e., built with arithmetic operators (+,−,×,/) andelementary functions (sin, exp, etc.) are analytic and therefore most of the timestrictly monotone. In rigorous terms, most of the time means that, unless thefunction is flat (or defined piecewise), the set of points that do not satisfy localstrict monotonicity is of measure zero (in the sense of measure theory).As a consequence, if a better contraction (or filtering) can be achieved undermonotonicity, branch & prune algorithms should take advantage of it.Monotonicity has been considered from the beginning of interval analysis [16],but with a motivation slightly different from ours. One of the most fundamentalissues of interval analysis is the design of inclusion functions [13], i.e., methodsfor computing an enclosure of the range of a function on any given box. Ofcourse, the sharper the better. Now, an optimal inclusion function (i.e., a methodfor computing the exact range on any box) can be built straightforwardly formonotone functions (see §2).Hence, the main matter since that time has been to devise efficient way to detectmonotonicity of a function f over a box [x]. One simple way to proceed is bychecking that the gradient does not get null in [x] which, in turn, requires aninclusion function for the gradient. The latter can then be based either on a directinterval evaluation, Taylor forms or the monotonicity test itself in a reentrantfashion (leading to the calculation of second derivatives, and so on).Surprisingly enough, monotonicity has never been used so far in the design ofcontractors. Remember that, although related, computing a sharp enclosure for{f(x), x ∈ [x]} and for {x ∈ [x] | f(x) = 0} are quite different goals. Aswe already said, getting an optimal enclosure with a monotone function f isstraightforward in the first case. But it is not so in the second case, especiallywhen x is a vector of variables. Algorithm 1 below will provide an answer.In the following, we first define properly the different concepts. The main re-sult, a monotonicity-based polytime optimal contractor for a constraint, is thenpresented. Finally, we highlight the practical benefits of this contractor with asimple example.1.1 Notations and definitionsWe consider throughout this paper a constraint satisfaction problem (CSP) witha vector of n real variables x1, . . . , xn.Domains of variables are represented by real intervals and a Cartesian productof intervals is called a box. Intervals and boxes will be surrounded by brackets,e.g., [x]. If [x] is a box, x− and x+ will stand for the two opposite corners formedby the lower and upper bound respectively of each components (see Figure 2).Hence, x−i and x+i will stand for the lower and upper bound respectively of theinterval [x]i. The width of an interval [x] (width[x]) is x+ − x−.Furthermore, given a mapping f on Rn, we shall denote by {f = 0} the constraintf(x) = 0 viewed as the set of all solution tuples, i.e.,{f = 0} := {x ∈ Rn | f(x) = 0}.We can now give a definition of hull consistency.2Definition 1 (Hull consistency). Let P be a constraint satisfaction probleminvolving a vector x of n variables and let [x] be the domain of x.P is said to be hull consistent if for every constraint c and for all i (1 ≤ i ≤ n),there exists two points in [x] which satisfy c and whose ith coordinates are x−iand x+i respectively.The key property of hull consistency lies in the combination of local reasoningand interval representation of domains. This concept brought a decisive improve-ment to the traditional Newton-based numerical solvers that were basically onlyable to contract domains globally.Definition 2 (Monotonicity). A mapping f : R → R is increasing over aninterval [x] if ∀a ∈ [x],∀b ∈ [x] a ≤ b=⇒ f(a) ≤ f(b).A mapping f : R → R is decreasing if −f is increasing, and monotone if itis either decreasing or increasing.A mapping f : Rn → R is increasing (resp. decreasing, monotone) over abox [x] if ∀x˜ ∈ [x] and ∀i, 1 ≤ i ≤ n, xi 7→ f(x˜1, . . . , x˜i−1, xi, x˜i+1, . . . , x˜n) isincreasing (resp. decreasing, monotone) over [x]i.Strict monotonicity is satisfied when formulas hold with strict inequalities.Let f : [y] ⊆ R → R be an increasing function. For any interval [x] ⊆ [y],the infimum and the supremum of f on [x] are f(x−) and f(x+). Hence, thefollowing interval function:[x]→ [f(x−), f(x+)]is an optimal inclusion function for f . This result easily generalizes to monotonemultivariate functions, by a componentwise repetition of the same argument.2 Main resultEnforcing hull consistency on a CSP boils down to enforcing bound consistencyon every isolated constraint (cf. Definition 1). Giving an optimal contractor fora single constraint is thus the main issue, which we shall address below. We shalleven focus on an equation f(x) = 0 (inequalities will be discussed further).Consider first the univariate case (f has a single variable) and assume that f isdifferentiable. The set {f = 0} can easily be bracketed by an interval Newtoniteration (see, e.g., [16] for details on the operations involved):[y]←(y˜ − f(y˜)/[f ′]([y]))∩ [y]where [f ′] is a (convergent) inclusion function for f ′ and y˜ any point in [y] suchthat f(y˜) 6= 0. Henceforth, we assume that a procedure univ newton(f, [x], ε)is available. This procedure returns an interval [y] such that both [y−, y−+ε]3and [y+−ε, y+] intersect {f = 0}. It can refer to any implementation of theunivariate interval Newton iteration, such as the one given in [10].Let us state the complexity. As noticed in the introduction, an analytic functionis locally either strictly monotone or flat. Thus, it makes sense to assume strictmonotonicity when dealing with complexity. The interval Newton iteration hasa quadratic rate of convergence [1], i.e., the width of [y] at every step is up to aconstant factor less than the square of the width at the previous step. However,the quadratic rate is only achieved when the iteration is contracting, i.e., wheny˜ − f(y˜)/[f ′]([y]) ⊆ [y]. While this condition is not fulfilled, the progression canbe slow, as the following figure illustrates:min slopemax slope[x]l(0)l(1)l(2)l(3)· · ·Fig. 1: Slow progression of the Newton iteration (with the left bound as point of ex-pansion). The maximum slope on the interval [x] (on the right side) is repeatedlyencompassed in the interval computation of the derivative, which explains the slowprogression. The successive lower bounds of the interval [x] are l(0), l(1), . . ..When the point of expansion y˜ is the midpoint of [y], the width of the interval isat least divided by two (this is somehow a way to interleave a dichotomy withinthe Newton iteration). Thus, the worst-case complexity of univ newton with themidpoint heuristic is O(log(w/ε)), where w is the width of the initial domain.Finally, note that if f is not differentiable (or if no convergent inclusion functionis available for f ′), one can still resort to a simple dichotomy and achieve thesame complexity.The general algorithm (called OCTUM: optimal contractor under monotonicity)that works with a multivariate mapping f : Rn → R is given below. Note thatuniv newton is called on the restriction of f to (axis-aligned) edges of the inputbox [x]. Since (n− 1) coordinates are fixed on an edge, the restriction is indeeda function from R to R.To ease the description of the algorithm, we will assume that the multivariatefunction f is increasing (according to Definition 2). Once the algorithm is un-derstood, considering the other possible configurations makes no difficulty andjust require a case-by-case adaptation. Lines 1 to 5 initializes the two vectorsx⊖ and x⊕ that correspond to the vertices where f is minimized and maximizedrespectively. When f is increasing, x⊖ and x⊕ are just aliases for x− and x+.4Line 6 checks that the box [x] contains a solution (and otherwise, the algorithmreturns the empty set). The main loop relies on the following fact (see Figure2) that will be proven below. Remember that f is assumed to be increasing andthat [x] contains at least one solution. The minimum of xi when x describes thesolutions inside [x] is then either reached(1) on the edge where all the other variables are instantiated to their upperbound x+j or(2) on the face where xi = x−i (which means that no contraction can be made).faceedgeedgex−x+x1x3 x2{f = 0}xe1(x−1 , ·, ·)(·, x+2 , x+3 )Fig. 2: The first component of the solutions inside the box either reaches its minimumon the face (x−1 , ·, ·) or on the edge (·, x+2 , x+3 ).Furthermore, as soon as the minimum for a component xi is met on an edge,the solution on this edge makes all x+j (j 6= i) consistent in the domain of thejth variable. To skip useless filtering operations for the upper bounds of theremaining variables, we use a flag named sup in the algorithm (see lines 17 and18). Finally, when univ newton is called, either the lower or upper bound of theresulting interval is considered, depending on which bound of [x]i is contracted(see lines 15 and 16 or 21 and 22). This ensures that no solution is lost.Filtering the upper bound of xi is entirely symmetrical. The complexity of OCTUMis O(n× log(width[x]ε)), where width[x] stands for max1≤i≤n width[x]i. It can bequalified as a pseudo-linear complexity.The OCTUM algorithm can be very easily extended to an inequality f(x) ≤ 0 orf(x) ≥ 0 by simply skipping narrowing operations on y−i or y+i respectively.The completeness and optimality of OCTUM relies on the following proposition.Proposition 1. Let f : Rn → R be a continuous increasing mapping and [x] abox such that {f = 0} ∩ [x] 6= ∅. Given i, 1 ≤ i ≤ n put:x˜i := inf{xi, x ∈ {f = 0} ∩ [x]}.Then, one of the two options holds:5Algorithm 1: OCTUM(f, [x], ε)Input: a monotone function f , a n-dimensional box [x], ε > 0Output: the smallest box [y] enclosing [x] ∩ {f = 0}, up to the precision εfor i = 1 to n do1if (f ր xi) then x⊕i ← x+i ; // x⊕ is the vertex where f is maximized2else x⊕i ← x−i // (f ր xi) means “f is increasing w.r.t. xi”3if (f ր xi) then x⊖i ← x−i ; // x⊖ is the vertex where f is minimized4else x⊖i ← x+i5if f(x⊖) > 0 or f(x⊕) < 0 then return ∅ // check if a solution exists6[y]← [x]7sup ← false // true when x⊕i is consistent for all remaining i8inf ← false // true when x⊖i is consistent for all remaining i9for i = 1 to n do10curr sup ← sup ; // save the current value for the second block11if inf is false then12[s] ← univ newton(t 7→ f(x⊕1 , . . . , x⊕i−1, t, x⊕i+1, x⊕n ), [y]i, ε)13if [s] 6= ∅ then14if (f ր xi) then y−i ← s− // update lower bound of xi15else y+i ← s+ ; // update upper bound of xi16sup ← true // the edge x⊕j , j 6= i, contains a solution17if curr sup is false then18[s] ← univ newton(t 7→ f(x⊖1 , . . . , x⊖i−1, t, x⊖i+1, x⊖n ), [y]i, ε)19if [s] 6= ∅ then20if (f ր xi) then y+i ← s+21else y−i ← s−22inf ← true23return [y]241. x˜i = x−i ,2. there exists x ∈ [x] such that xi = x˜i and for every j 6= i, xj = x+j .Proof. Let x∗ be a solution point in [x] minimizing xi, i.e., f(x∗) = 0 andx∗i = x˜i. If x∗i = x−i , the first option holds and we are done. Assume x∗i > x−i . Ifn = 1, the second option holds trivially. If n > 1, consider the following vector:x := (x+1 , . . . , x+i−1, x∗i , x+i+1, . . . , x+n ).We will prove that f(x) = 0. By contradiction, assume f(x) > 0 (f beingincreasing). Since xi = x∗i > x−i , there exists by continuity ε > 0 such thatx∗i −ε > x−i and f(x−εei) > 0 (ei being the ith unit vector). Now, x∗ is the pointin [x] where f gets null with the smallest ith coordinate. Hence, f(x∗− εei) < 0.Since f is continuous, f gets null somewhere on the segment joining x∗ − εeiand x − εei because the sign of f is opposite at the two extremities. Since [x]is convex, the corresponding point is inside [x] and its ith component is x∗i − εi,which contradicts the fact that x∗i is the infimum among the solutions. N63 A first experimentConsider the problem of characterizing the set of points (x, y) in [−3, 0]× [0, 3]satisfying f(x, y) = 0 with f(x, y) = x2y2 − 9x2y + 6xy2 − 20xy − 1.Let us compare OCTUM with three other standard generic contractors (namelyHC4 [2, 9], BOX [4, 18] and 3B [15]) as pruning steps of a classical branch & prunesystem. We have implemented a very naive method for detecting monotonicity,using an interval evaluation of the gradient that is systematically computed forevery box (a better method would be to manage flags w.r.t. each variable in abacktrackable structure, each flag being set incrementally as soon as f is proven tobe monotone). Even with this naive implementation, OCTUM yields better results,both in terms of quality (see Figure 3) and quantity (see Table below).(a) Using HC4. (b) Using BOX.(c) Using 3B. (d) Using OCTUM.Fig. 3: Comparing the monotonicity-based contractor OCTUM with other standard oper-ators. Black surfaces encompass the solutions while grey boxes represent the contractedparts. The thinnest black surface is obtained with OCTUM. Note however the two littlemarks in (d) that correspond to points where one component of the gradient gets null.Running time Number of backtracks Size of solution setHC4 0.66s 28240 6928BOX 1.37s 9632 35953B 1.89s 9171 2564OCTUM 0.40s 6047 114374 ConclusionWe have proven that hull consistency can be achieved in polynomial time inthe case of constraints involving monotone functions. Hull consistency amountsto bound consistency for each isolated constraint. We have given an algorithmcalled OCTUM that enforces bound consistency for an equation under monotonic-ity (and explained how to adapt it to inequalities). Hull consistency based onOCTUM can then be programed by simply embedding OCTUM in a classical AC3propagation loop. A first experiment has illustrated the two nice properties ofOCTUM: optimality and (pseudo-)linear complexity.References1. G. 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In IEEE Canadian Conf. on Elec. and Comp. Engineering, 1990.18. P. Van Hentenryck, D. McAllester, and D. Kapur. Solving Polynomial SystemsUsing a Branch and Prune Approach. SIAM J. Numer. Anal., 34(2):797–827,1997.8

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