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