Using mixed volume theory to compute the convex hull volume for - - PowerPoint PPT Presentation

Emily Speakman Gennadiy Averkov

Using mixed volume theory to compute the convex hull volume for trilinear monomials

23rd 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-integer

nonlinear optimization

• The convex hull of the graph of a trilinear monomial over a box
• Use techniques from mixed volume theory to obtain an

alternative proof

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Global optimization of non-convex functions is hard! min

x∈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

2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! min

x∈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...

2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! min

x∈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 known

as Spatial Branch-and-Bound (sBB)

2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! min

x∈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 known

as Spatial Branch-and-Bound (sBB)

• sBB has a daunting complexity

2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! min

x∈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 known

as 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 solutions we use Spatial Branch- and-Bound:

• Create sub-problems

by branching on individual variables

• Generate convex

relaxations of the graph of the function at each node

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The choice of convexification method is important

Computational tractability of sBB depends on the quality of the convexifications we use. We want both:

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

quickly) We have a trade off:

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We compare the tightness of convexifications via n-dimensional volume

• Allows us to quantify the difference

between formulations analytically

• The optimal solution could occur

anywhere in the feasible region, and therefore the volume measure corresponds to a uniform distribution on the location of the optimal solution

• Volume was introduced as a means of comparing formulations

by Lee and Morris (1994)

• Recent survey on volumetric comparison of polyhedral

relaxations for optimization Lee, Skipper, Speakman (2018)

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Trilinear monomials: some notation

Assume we have a monomial of the form: f = x1x2x3.

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Trilinear monomials: some notation

Assume we have a monomial of the form: f = x1x2x3.

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

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Trilinear monomials: some notation

Assume 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)

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Trilinear monomials: some notation

Assume 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]) is polyhedral, we refer to this polytope as PH.

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Trilinear monomials: some notation

Assume 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]) is polyhedral, 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 compute

their 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 b1 a2 a3     v2 :=     a1a2a3 a1 a2 a3     v3 :=     a1a2b3 a1 a2 b3     v4 :=     a1b2a3 a1 b2 a3     v5 :=     a1b2b3 a1 b2 b3     v6 :=     b1b2b3 b1 b2 b3     v7 :=     b1b2a3 b1 b2 a3     v8 :=     b1a2b3 b1 a2 b3    

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 b1 a2 a3     v2 :=     a1a2a3 a1 a2 a3     v3 :=     a1a2b3 a1 a2 b3     v4 :=     a1b2a3 a1 b2 a3     v5 :=     a1b2b3 a1 b2 b3     v6 :=     b1b2b3 b1 b2 b3     v7 :=     b1b2a3 b1 b2 a3     v8 :=     b1a2b3 b1 a2 b3    

• Meyer and Floudas (2004) completely characterized the facets
• f PH
• They did this for our special case (non-negative) and also in

general

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Facet Description

PH (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 − a1 x1 − b1a3x2 − b1a2x3 + ( η1a1 b1 − a1 + b1b2a3 + b1a2b3 − a1b2b3 ) ≥ 0 f − η2 a1 − b1 x1 − 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.

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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) ) .

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Our contribution

• We provide an alternative way to obtain the formula for the

convex hull volume

• Observe a special structure in the convex hull polytope
• Allows us to use theory from so-called Mixed Volumes

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Mixed Volumes

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

Theorem

There is a unique, nonnegative function, V : (Kn)n → R, the mixed volume, which is invariant under permutation of its arguments, such that, 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+.

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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+.

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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).

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Visualizing four dimensions

A 2d projection of a standard 4d cube that preserves the usual 2d projection of a 3d cube

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1101 1100 1010 1110 1011 1111

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Visualizing the convex hull

• 2d representation
• f the extreme

points of PH ∈ R4

• 23 = 8 extreme points
• R4 since points have form:

(f = x1x2x3, x1, x2, x3)

• f variable represented by

‘4th dimension’, inner cube to outer cube

• v2 = [a1a2a3, a1, a2, a3]

v6 = [b1b2b3, b1, b2, b3]

• The original proof

constructed a triangulation

• f 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 four
• f the points lie in the

x3 = a3 hyperplane and form a 3d simplex, S

• Furthermore the

remaining four points lie in the x3 = b3 hyperplane and form a simplex, T

• Consider calculating the

volume of PH = conv(S ∪ T) via an integral as x3 varies 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 − a3 S + t − a3 b3 − a3 T ) 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.

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Calculating the mixed volumes

• S and T are simplicies, therefore we can compute their volume

via 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

17 Speakman & Averkov // Mixed Volume

Calculating the mixed volumes

• S and T are simplicies, therefore we can compute their volume

via 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) = sup

x∈K

xTu.

17 Speakman & Averkov // Mixed Volume

Calculating the mixed volumes

• S and T are simplicies, therefore we can compute their volume

via 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) = sup

x∈K

xTu.

Definition

For a full dimensional polytope, P ⊆ Rn, U(P) = the set of all outer facet normals, u, such that the length of u is the (n − 1)-dimensional volume of the respective facet.

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Calculating the mixed volumes

Now, to calculate V(S, S, T) and V(S, T, T), we make use of the following fact: For P ⊆ Rn, a full-dimensional polytope, and K ∈ Kn, V(P, P, . . . , P, K) = 1 n ∑

u∈U(P)

hK(u).

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Calculating the mixed volumes

Now, to calculate V(S, S, T) and V(S, T, T), we make use of the following fact: For P ⊆ Rn, a full-dimensional polytope, and K ∈ Kn, V(P, P, . . . , P, K) = 1 n ∑

u∈U(P)

hK(u).

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

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 the following fact: For P ⊆ Rn, a full-dimensional polytope, and K ∈ Kn, V(P, P, . . . , P, K) = 1 n ∑

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

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 the following fact: For P ⊆ Rn, a full-dimensional polytope, and K ∈ Kn, V(P, P, . . . , P, K) = 1 n ∑

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 we
• btain 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, we

are able to use mixed volume theory as an alternative method for computing the volume, this gives a more compact proof

• However, the method does not naturally extend to the other

volume proofs, (unlike the original triangulation method) and here is where the majority of the technical difficulties lay

• Interestingly, in both proof methods for the convex hull, the

technical difficulties were similar – establishing the sign of polynomial expressions in the parameters

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Conclusions and future work

• It seems likely that using this method we can extend to the case
• f negative bounds (current work)
• This method gives a more natural way to extend to n = 4 than

we had before, however, it still seems like there will be difficulties 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 via

volumetric 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 Optimization 29 125-155 (2004)

• Schneider, R. Convex bodies: the Brunn-Minkowski theory. 2nd edn. Volume

151 of Encyclopedia of Math. and its Apps. Cambridge University Press (2014)

• E. Speakman & G. Averkov. Computing the volume of the convex hull of the

graph of a trilinear monomial using mixed volumes. ArXiv:1810.12625 (2018)

• E. Speakman & J. Lee. Quantifying double McCormick. Mathematics of

Operations Research 42(4) 1230–1253 (2017)

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