using mixed volume theory to compute the convex hull ... · • the facet description was given by...

Emily Speakman Gennadiy Averkov

Using mixed volume theory to computethe convex hull volume for trilinearmonomials23rd Combinatorial Optimization Workshop, Aussois

January 9, 2019

Institute of Mathematical Optimization,

Otto-von-Guericke-University,

Magdeburg, Germany.

Talk Outline

• Using volume to compare relaxations for mixed-integernonlinear optimization

• The convex hull of the graph of a trilinear monomial over a box

• Use techniques from mixed volume theory to obtain analternative proof

1 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions ishard!

minx∈Rn,z∈Zm

f (x, z) : (x, z) ∈ F

• Global optimization of a mixed integer non-linear optimization(MINLO) problem

• Not necessarily convex sets/functions

Software: Baron, Couenne, Scip, Antigone...

• Each of these perform some variation on the algorithm knownas Spatial Branch-and-Bound (sBB)

• sBB has a daunting complexity

• How can we effectively tune/engineer this software?• experimentally• mathematically

minx∈Rn,z∈Zm

f (x, z) : (x, z) ∈ F

minx∈Rn,z∈Zm

f (x, z) : (x, z) ∈ F

minx∈Rn,z∈Zm

f (x, z) : (x, z) ∈ F

minx∈Rn,z∈Zm

f (x, z) : (x, z) ∈ F

Key algorithm: sBB

To find optimal solutionswe use Spatial Branch-and-Bound:

• Create sub-problemsby branching onindividual variables

• Generate convexrelaxations of thegraph of the functionat each node

The choice of convexification method is important

Computational tractability of sBB depends on the quality of theconvexifications we use.

We want both:

• Tight relaxations (good bounds)• Simple algebraic representations of feasible regions (solve

quickly)

We have a trade off:

We compare the tightness of convexifications vian-dimensional volume

• Allows us to quantify the differencebetween formulations analytically

• The optimal solution could occuranywhere in the feasible region, andtherefore the volume measurecorresponds to a uniform distribution onthe location of the optimal solution

• Volume was introduced as a means of comparing formulationsby Lee and Morris (1994)

• Recent survey on volumetric comparison of polyhedralrelaxations for optimization Lee, Skipper, Speakman (2018)

Trilinear monomials: some notationAssume we have a monomial of the form: f = x1x2x3.

• xi ∈ [ai, bi], where 0 ≤ ai < bi, for i = 1, 2, 3

• Label the variables x1, x2 and x3 such that:a1

b1≤ a2

b2≤ a3

b3(Ω)

The convex hull of the graph f = x1x2x3 (on the domain xi ∈ [ai, bi]) ispolyhedral, we refer to this polytope as PH.

• The facet description was given by Meyer and Floudas (2004)

• There are alternative convexifications for trilinear monomials(based on the well-know McCormick inequalities)

• S. and Lee (2017) consider these alternatives and computetheir volumes

• Here we focus on the convex hull i.e. PH

b1≤ a2

b2≤ a3

b3(Ω)

b1≤ a2

b2≤ a3

b3(Ω)

b1≤ a2

b2≤ a3

b3(Ω)

b1≤ a2

b2≤ a3

b3(Ω)

Convex hull of the graph f = x1x2x3 over a box• The extreme points of PH are the 8 points that correspond to the

23 = 8 choices of each x-variable at its upper or lower bound:

b1a2a3

a1a2a3

a1a2b3

a1b2a3

a1b2b3

b1b2b3

b1b2a3

b1a2b3

• Meyer and Floudas (2004) completely characterized the facetsof PH

• They did this for our special case (non-negative) and also ingeneral

Convex hull of the graph f = x1x2x3 over a box• The extreme points of PH are the 8 points that correspond to the

23 = 8 choices of each x-variable at its upper or lower bound:

b1a2a3

a1a2a3

a1a2b3

a1b2a3

a1b2b3

b1b2b3

b1b2a3

b1a2b3

• Meyer and Floudas (2004) completely characterized the facetsof PH

• They did this for our special case (non-negative) and also ingeneral

Facet DescriptionPH (Meyer and Floudas, 2004)

f − a2a3x1 − a1a3x2 − a1a2x3 + 2a1a2a3 ≥ 0

f − b2b3x1 − b1b3x2 − b1b2x3 + 2b1b2b3 ≥ 0

f − a2b3x1 − a1b3x2 − b1a2x3 + a1a2b3 + b1a2b3 ≥ 0

f − b2a3x1 − b1a3x2 − a1b2x3 + b1b2a3 + a1b2a3 ≥ 0

f −η1

b1 − a1x1 − b1a3x2 − b1a2x3 +

(η1a1

b1 − a1+ b1b2a3 + b1a2b3 − a1b2b3

)≥ 0

f −η2

a1 − b1x1 − a1b3x2 − a1b2x3 +

(η2b1

a1 − b1+ a1a2b3 + a1b2a3 − b1a2a3

)≥ 0

− f + a2a3x1 + b1a3x2 + b1b2x3 − b1b2a3 − b1a2a3 ≥ 0

− f + b2a3x1 + a1a3x2 + b1b2x3 − b1b2a3 − a1b2a3 ≥ 0

− f + a2a3x1 + b1b3x2 + b1a2x3 − b1a2b3 − b1a2a3 ≥ 0

− f + b2b3x1 + a1a3x2 + a1b2x3 − a1b2b3 − a1b2a3 ≥ 0

− f + a2b3x1 + b1b3x2 + a1a2x3 − b1a2b3 − a1a2b3 ≥ 0

− f + b2b3x1 + a1b3x2 + a1a2x3 − a1b2b3 − a1a2b3 ≥ 0

ai ≤ xi ≤ bi, i = 1..3

where η1 = b1b2a3 − a1b2b3 − b1a2a3 + b1a2b3 and η2 = a1a2b3 − b1a2a3 − a1b2b3 + a1b2a3.

Volume of PH

Theorem (S. and Lee, 2017)

Under Ω, the volume of PH is given by:1

24(b1 − a1)(b2 − a2)(b3 − a3) ×(

b1(5b2b3 − a2b3 − b2a3 − 3a2a3) + a1(5a2a3 − b2a3 − a2b3 − 3b2b3)).

Our contribution

• We provide an alternative way to obtain the formula for theconvex hull volume

• Observe a special structure in the convex hull polytope

• Allows us to use theory from so-called Mixed Volumes

Mixed Volumes

Kn is the set of all nonempty compact convex sets in Rn

TheoremThere is a unique, nonnegative function, V : (Kn)n → R, the mixedvolume, which is invariant under permutation of its arguments, suchthat, for every positive integer m > 0, one has

Vol(t1K1 + t2K2 + · · ·+ tmKm) =

m∑i1,...,in=1

ti1 . . . tinV(Ki1 , . . . ,Kin),

for arbitrary K1, . . .Km ∈ Kn and t1, t2, . . . , tn ∈ R+.

Mixed Volumes

Theorem

The mixed volume function satisfies the following properties:

(i) Vol(K1) = V(K1, . . . ,K1).

(ii) V(t′K′1 + t′′K′′

1 ,K2, . . . ,Kn) = t′V(K′1,K2, . . . ,Kn)

+ t′′V(K′′1 ,K2, . . . ,Kn),

for K1, . . .Kn ∈ Kn and t′, t′′ ∈ R+.

There is a great deal of rich theory, see Schneider (2014).

Mixed Volumes

Theorem

The mixed volume function satisfies the following properties:

(i) Vol(K1) = V(K1, . . . ,K1).

(ii) V(t′K′1 + t′′K′′

1 ,K2, . . . ,Kn) = t′V(K′1,K2, . . . ,Kn)

+ t′′V(K′′1 ,K2, . . . ,Kn),

for K1, . . .Kn ∈ Kn and t′, t′′ ∈ R+.

There is a great deal of rich theory, see Schneider (2014).

Visualizing four dimensions

A 2d projection of astandard 4d cube thatpreserves the usual 2dprojection of a 3d cube

Visualizing the convex hull• 2d representation

of the extremepoints of PH ∈ R4

• 23 = 8 extreme points

• R4 since points have form:(f = x1x2x3, x1, x2, x3)

• f variable represented by‘4th dimension’, inner cubeto outer cube

• v2 = [a1a2a3, a1, a2, a3]v6 = [b1b2b3, b1, b2, b3]

• The original proofconstructed a triangulationof the polytope

Another way of viewing the polytope• Consider the x3 component

(could be x1 or x2)

• Observe that fourof the points lie in thex3 = a3 hyperplane andform a 3d simplex, S

• Furthermore theremaining four points liein the x3 = b3 hyperplaneand form a simplex, T

• Consider calculating thevolume ofPH = conv(S ∪ T)via an integral as x3varies from a3 to b3

Alternative volume computation

Vol(PH) = Vol(conv(S ∪ T))

∫ b3

(b3 − t

b3 − a3S +

t − a3

b3 − a3T)

= (b3 − a3)−3

∫ b3

Vol((b3 − t)S + (t − a3)T)dt

= (b3 − a3)−3

∫ b3

(b3 − t)3 Vol(S) + 3(b3 − t)2(t − a3)V(S, S,T)

+ 3(b3 − t)(t − a3)2V(S,T,T) + (t − a3)

3 Vol(T) dt,

Where V(S, S,T) and V(S,T,T) are the mixed volumes.

Calculating the mixed volumes

• S and T are simplicies, therefore we can compute their volumevia a simple determinant calculation

• All that remains is to calculate V(S, S,T) and V(S,T,T)• To do this we need a couple of definitions

Definition (Support function)For K ∈ Kn, the support function, hK : Rn → R, is defined by

hK(u) = supx∈K

DefinitionFor a full dimensional polytope, P ⊆ Rn,

U(P) = the set of all outer facet normals, u, such that the length ofu is the (n − 1)-dimensional volume of the respective facet.

hK(u) = supx∈K

Now, to calculate V(S, S,T) and V(S,T,T), we make use of thefollowing fact:

For P ⊆ Rn, a full-dimensional polytope, and K ∈ Kn,

V(P,P, . . . ,P,K) =1n

∑u∈U(P)

hK(u).

• Thus, we need: U(S), U(T), hS(·) and hT(·)

• Given that S, T are tetrahedra we are able to compute these• Four facet normals to compute• Four extreme points to check

• We can therefore evaluate the integral and in doing so weobtain the same formula as the triangulation method

V(P,P, . . . ,P,K) =1n

∑u∈U(P)

hK(u).

V(P,P, . . . ,P,K) =1n

∑u∈U(P)

hK(u).

V(P,P, . . . ,P,K) =1n

∑u∈U(P)

hK(u).

Conclusions and future work

• Because of some nice properties of the convex hull polytope, weare able to use mixed volume theory as an alternative methodfor computing the volume, this gives a more compact proof

• However, the method does not naturally extend to the othervolume proofs, (unlike the original triangulation method) andhere is where the majority of the technical difficulties lay

• Interestingly, in both proof methods for the convex hull, thetechnical difficulties were similar – establishing the sign ofpolynomial expressions in the parameters

Conclusions and future work

• It seems likely that using this method we can extend to the caseof negative bounds (current work)

• This method gives a more natural way to extend to n = 4 thanwe had before, however, it still seems like there will bedifficulties doing this in practice

• Another tool in the volume toolbox!

Thank you for listening!References:

• J. Lee, & W. Morris. Geometric comparison of combinatorial polytopes.Discrete Applied Mathematics 55 163-182 (1994)

• J. Lee, D. Skipper & E. Speakman. Algorithmic and modeling insights viavolumetric comparison of polyhedral relaxations. Math Programming 170(1)121–140 (2018)

• Meyer, C.A & C.A. Floudas. Trilinear monomials with mixed sign domains:Facets of the convex and concave envelopes. Journal of Global Optimization29 125-155 (2004)

• Schneider, R. Convex bodies: the Brunn-Minkowski theory. 2nd edn. Volume151 of Encyclopedia of Math. and its Apps. Cambridge University Press(2014)

• E. Speakman & G. Averkov. Computing the volume of the convex hull of thegraph of a trilinear monomial using mixed volumes. ArXiv:1810.12625 (2018)

• E. Speakman & J. Lee. Quantifying double McCormick. Mathematics ofOperations Research 42(4) 1230–1253 (2017)

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