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
2 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
2 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
2 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
2 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
2 Speakman & Averkov // Mixed Volume
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
3 Speakman & Averkov // Mixed Volume
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:
4 Speakman & Averkov // Mixed Volume
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)
5 Speakman & Averkov // Mixed Volume
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
6 Speakman & Averkov // Mixed Volume
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
6 Speakman & Averkov // Mixed Volume
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
6 Speakman & Averkov // Mixed Volume
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
6 Speakman & Averkov // Mixed Volume
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
6 Speakman & Averkov // Mixed Volume
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:
v1 :=
b1a2a3
b1a2a3
v2 :=
a1a2a3
a1a2a3
v3 :=
a1a2b3
a1a2b3
v4 :=
a1b2a3
a1b2a3
v5 :=
a1b2b3
a1b2b3
v6 :=
b1b2b3
b1b2b3
v7 :=
b1b2a3
b1b2a3
v8 :=
b1a2b3
b1a2b3
• Meyer and Floudas (2004) completely characterized the facetsof PH
• They did this for our special case (non-negative) and also ingeneral
7 Speakman & Averkov // Mixed Volume
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:
v1 :=
b1a2a3
b1a2a3
v2 :=
a1a2a3
a1a2a3
v3 :=
a1a2b3
a1a2b3
v4 :=
a1b2a3
a1b2a3
v5 :=
a1b2b3
a1b2b3
v6 :=
b1b2b3
b1b2b3
v7 :=
b1b2a3
b1b2a3
v8 :=
b1a2b3
b1a2b3
• Meyer and Floudas (2004) completely characterized the facetsof PH
• They did this for our special case (non-negative) and also ingeneral
7 Speakman & Averkov // Mixed Volume
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.
8 Speakman & Averkov // Mixed Volume
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)).
9 Speakman & Averkov // Mixed Volume
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
10 Speakman & Averkov // Mixed Volume
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+.
11 Speakman & Averkov // Mixed Volume
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).
12 Speakman & Averkov // Mixed Volume
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).
12 Speakman & Averkov // Mixed Volume
Visualizing four dimensions
A 2d projection of astandard 4d cube thatpreserves the usual 2dprojection of a 3d cube
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1101
1100
1010
1110
1011
1111
13 Speakman & Averkov // Mixed Volume
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
v2
v6
v1
v3
v4
v5
v7
v8
14 Speakman & Averkov // Mixed Volume
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
v2
v6
v1
v3
v4
v5
v7
v8
15 Speakman & Averkov // Mixed Volume
Alternative volume computation
Vol(PH) = Vol(conv(S ∪ T))
=
∫ b3
a3
Vol
(b3 − t
b3 − a3S +
t − a3
b3 − a3T)
dt
= (b3 − a3)−3
∫ b3
a3
Vol((b3 − t)S + (t − a3)T)dt
= (b3 − a3)−3
∫ b3
a3
(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.
16 Speakman & Averkov // Mixed Volume
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
xTu.
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.
17 Speakman & Averkov // Mixed Volume
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
xTu.
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.
17 Speakman & Averkov // Mixed Volume
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
xTu.
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.
17 Speakman & Averkov // Mixed Volume
Calculating the mixed volumes
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
18 Speakman & Averkov // Mixed Volume
Calculating the mixed volumes
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
18 Speakman & Averkov // Mixed Volume
Calculating the mixed volumes
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
18 Speakman & Averkov // Mixed Volume
Calculating the mixed volumes
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
18 Speakman & Averkov // Mixed Volume
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
19 Speakman & Averkov // Mixed Volume
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!
20 Speakman & Averkov // Mixed Volume
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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