Monomial Tropical Cones for Multicriteria Optimization Georg Loho - - PowerPoint PPT Presentation

Monomial Tropical Cones for Multicriteria Optimization

Georg Loho

joint work with Michael Joswig

Multicriteria optimization

• multicriteria optimization problem:

min f(x) =

• f1(x), . . . , fd(x)
• subject to

x ∈ X .

• outcome space: Z = f(X) ⊆ Rd
• w ≦ z: if wi ≤ zi for all i ∈ [d], partial ordering on Rd.
• for S ⊂ Rd the minimal elements with respect to ≦ form the

nondominated points.

Observation

The set dominated by N ⊆ Rd is exactly

• N + Rd

≥0

• \ N.

We think of Z = f(X) as a discrete set.

Tropical Geometry and Optimization

Interplay of Tropical Geometry and Optimization

• Geometric methods for mean-payoff games and complexity of classical

linear programming through tropical linear programming (Akian et al. ’12, Allamigeon et al. ’14+)

• Computing tropical determinants by Hungarian method (Butkoviˇ

c ’10) We want to enumerate all nondominated points.

Previous Work

• testing different regions of the search space (Sylva, Crema 2008)
• generating all n non-dominated points with O(nd) scalarizations

(Kirlik, Sayın 2014)

• assymptotically tight number of scalarizations Θ(n⌊d/2⌋) (D¨

achert, Klamroth, Lacour, Vanderpooten 2016)

Key ideas

• Sets of the form G + Rd

≥0 are tropically convex

• Tropical double description for ’monomial tropical cones’ is an efficient

algorithm to generate nondominated points

• Works for arbitrary number of objective functions
• Duality between ’local upper bounds’ and nondominated points (local

minima)

Tropical basics

Definition

Tropical numbers Tmax = R ∪ {−∞} Addition s ⊕ t := max(s, t) = − min(−s, −t) Multiplication s ⊙ t := s + t Operations are extended componentwise to Td+1

max

Tropical basics

Definition

Tropical numbers Tmax = R ∪ {−∞} Addition s ⊕ t := max(s, t) = − min(−s, −t) Multiplication s ⊙ t := s + t Operations are extended componentwise to Td+1

max

Also: Dual tropical numbers Tmin = R ∪ {∞} with dual addition s ⊕ t = min(s, t).

Tropical basics

Definition

Tropical numbers Tmax = R ∪ {−∞} Addition s ⊕ t := max(s, t) = − min(−s, −t) Multiplication s ⊙ t := s + t Operations are extended componentwise to Td+1

max

Also: Dual tropical numbers Tmin = R ∪ {∞} with dual addition s ⊕ t = min(s, t).

Example

(5 ⊕ −7) ⊙ 10 ⊕ −100 = 15 (−3) ⊙ x ⊕ 1 = 9 valid for x = 12 But: (−3) ⊙ x ⊕ 9 = 9 valid for every x ≤ 12

Determining nondominated points with scalarizations

h = (−3, 2) M(h) M (h)

Determining nondominated points with scalarizations

h = (−3, 2) M(h) M (h)

Determining nondominated points with scalarizations

h = (−3, 2) M(h) M (h)

Determining nondominated points with scalarizations

g = (0, 0) h = (−3, 2) M(h) M (h)

Determining nondominated points with scalarizations

g = (0, 0) h = (−3, 2) M(g, h) M (g, h)

Determining nondominated points with scalarizations

g = (0, 0) h = (−3, 2) M(g, h) M (g, h)

Determining nondominated points with scalarizations

g = (0, 0) h = (−3, 2) M(g, h) M (g, h)

Determining nondominated points with scalarizations

g = (0, 0) h = (−3, 2) M(g, h) M (g, h)

Adaptation of the generic method

(Klamroth et al. 2015, Joswig, L 2017) Input: Image of the feasible set Z ⊂ Rd Output: The set of nondominated points.

1: A ← Emin ∪ e(0) 2: G ← ∅ 3: Ω ← Emin 4: while A = Ω do 5:

pick a in A \ Ω

6:

g ← NextNonDominated(Z, a)

7:

if g = None then

8:

A ← NewUpperBounds(G, A, g)

9:

G ← G ∪ {g}

10:

else

11:

Ω ← Ω ∪ {a}

12:

end if

13: end while 14: return G

Tropical Inner and Outer Description

• max-tropical cone: subset C ⊆ Td+1

max with

• max(λ + x0, µ + y0), . . . , max(λ + xd, µ + yd)
• ∈ C

for all λ, µ ∈ Tmax and x, y ∈ C.

• G generates C if the latter is the minimal max-tropical cone

containing G; explicitly {λ1 ⊙ g1 ⊕ · · · ⊕ λk ⊙ gk | g1, . . . , gk ∈ G, λ1, . . . , λk ∈ Tmax} .

• extremal generators: minimal generating set consists of extremal

generators; an element g ∈ G is extremal iff, for g1, g2 ∈ G g = µg1 ⊕ νg2 ⇒ g = µg1 or g = νg2 .

Tropical Inner and Outer Description

• for vector a ∈ Td+1

max the set supp(a) = {i | ai = −∞} is its support.

• closed max-tropical halfspace: a set of the form
• x ∈ Td+1

max

• max(xi + ai | i ∈ I) ≤ max(xj + aj | j ∈ J)
• for disjoint nonempty subsets I, J ⊂ supp(a)

Theorem (Tropical Weyl-Minkowski Theorem (Gaubert, Katz 2007))

C finitely generated max-tropical cone ⇔ C intersection of finitely many closed max-tropical halfspaces.

Tropical Inner and Outer Description

x1 x2 a1 a2 a3 a4 g1 g2 g3 g4 dehomogenized with x0 = 0 G =     −1 −1 2 2 −1 3 −2 1     A =     −1 1 ∞ 2 3 −2 2     x0 ≤ max(x1 + 1, x2 − 1) x1 ≤ x2 max(x1 − 2, x2 − 3) ≤ x0 max(x0, x2 − 2) ≤ x1 + 2

Tropical Inner and Outer Description

x1 x2 a1 a2 a3 a4 g1 g2 g3 g4 dehomogenized with x0 = 0 G =     −1 −1 2 2 −1 3 −2 1     A =     −1 1 ∞ 2 3 −2 2     x0 ≤ max(x1 + 1, x2 − 1) x1 ≤ x2 max(x1 − 2, x2 − 3) ≤ x0 max(x0, x2 − 2) ≤ x1 + 2

Tropical Inner and Outer Description

x1 x2 a1 a2 a3 a4 g1 g2 g3 g4 dehomogenized with x0 = 0 G =     −1 −1 2 2 −1 3 −2 1     A =     −1 1 ∞ 2 3 −2 2     x0 ≤ max(x1 + 1, x2 − 1) x1 ≤ x2 max(x1 − 2, x2 − 3) ≤ x0 max(x0, x2 − 2) ≤ x1 + 2

Tropical Inner and Outer Description

x1 x2 a1 a2 a3 a4 g1 g2 g3 g4 dehomogenized with x0 = 0 G =     −1 −1 2 2 −1 3 −2 1     A =     −1 1 ∞ 2 3 −2 2     x0 ≤ max(x1 + 1, x2 − 1) x1 ≤ x2 max(x1 − 2, x2 − 3) ≤ x0 max(x0, x2 − 2) ≤ x1 + 2

Tropical Inner and Outer Description

x1 x2 a1 a2 a3 a4 g1 g2 g3 g4 dehomogenized with x0 = 0 G =     −1 −1 2 2 −1 3 −2 1     A =     −1 1 ∞ 2 3 −2 2     x0 ≤ max(x1 + 1, x2 − 1) x1 ≤ x2 max(x1 − 2, x2 − 3) ≤ x0 max(x0, x2 − 2) ≤ x1 + 2

Monomial tropical cones

Fix G ⊆ Td+1

max.

• Monomial tropical cone M(G) (Allamigeon et al. 2010, JL 2017):
• g∈G
• x ∈ Td+1

max

• x0 − g0 ≤ min(xj − gj | j ∈ supp(g) \ {0})
• M(G) = M(G) ∩ Rd+1 =

 

g∈G

g +

• {0} × Rd

≥0

• 

 + R · 1

• max-tropical cone in Td+1

max generated by the finite set G ∪ Emax, where

Emax = {(−∞, 0, −∞, . . . , −∞), . . . , (−∞, −∞, . . . , −∞, 0)}

Monomial tropical cones

R3 g = (0, 0, 0) h = (0, −3, 2) f = (0, 1, −∞) M(f, g, h) b = (0, 0, 2) a = (0, 1, 0) c = (0, −3, ∞) (−∞, −∞, 0) (−∞, 0, −∞) T3

max

Duality of monomial tropical cones

M (G): the complement of the interior of the max-tropical cone M(G)

• g∈G
• Rd+1 \
• g +
• {0} × Rd

>0

• + R · 1

M (G): closure of M (G) in Td+1

min with respect to min (add points with ∞).

Theorem (Joswig, L 2017)

The set M (G) is a min-tropical cone in Rd+1. More precisely, if H is a set

• f max-tropical halfspaces such that H = M(G), then

M (G) = −M(−A) , where A ⊂ Td+1

min is the set of apices of the tropical halfspaces in H. In

particular, the set A ∪ Emin generates M (G).

Duality of monomial tropical cones

R3 g = (0, 0, 0) h = (0, −3, 2) f = (0, 1, −∞) M(f, g, h) b = (0, 0, 2) a = (0, 1, 0) c = (0, −3, ∞) M (f, g, h) = −M(−a, −b, −c) (−∞, −∞, 0) (−∞, 0, −∞) (∞, 0, ∞) (∞, ∞, 0) T3

min

T3

max

Analogy of dual classical and tropical convex hull

Iterative tropical double description for monomial cones

Adaptation of ’tropical double description’ by Allamigeon et al. (2010) Input: A set G ⊂ Td+1

max, the set A of extremal generators of

M (G), and a point h ∈ Td+1

max with h0 = 0.

Output: The set of extremal generators of M (G ∪ {h}).

• determine the set A≥ of points in A which fulfill the “monomial

inequality” x0 ≥ mini∈[d](xi − hi) given by h

• initialize the set B of generators of

M (G ∪ {h}) with A≥

• for each pair (b, c) ∈ A≥ ×
• A \ A≥

, determine the intersection of the tropical line through b and c with the boundary of the halfspace h

• discard the intersection points, which are not extremal in

M (G ∪ {h}).

Upper bound theorems – many objective functions

Allamigeon et al. 2010, JL 2017

The number of extremal generators of a tropical cone in Td+1

max defined by n

halfspaces is in O(n⌊d/2⌋).

Proof

For each tropical cone there is a classical cone with similar combinatorial

• structure. Now claim follows from McMullen’s Upper Bound Theorem.

Upper bound theorems – many objective functions

Allamigeon et al. 2010, JL 2017

The number of extremal generators of a tropical cone in Td+1

max defined by n

halfspaces is in O(n⌊d/2⌋).

Proof

For each tropical cone there is a classical cone with similar combinatorial

• structure. Now claim follows from McMullen’s Upper Bound Theorem.

Kaplan et al. 2008, JL 2017

The maximal number of extremal generators of monomial tropical cones in Td+1

max defined by n halfspaces is in Θ(n⌊d/2⌋).

Upper bound theorems – many objective functions

Allamigeon et al. 2010, JL 2017

The number of extremal generators of a tropical cone in Td+1

max defined by n

halfspaces is in O(n⌊d/2⌋).

Proof

For each tropical cone there is a classical cone with similar combinatorial

• structure. Now claim follows from McMullen’s Upper Bound Theorem.

Kaplan et al. 2008, JL 2017

The maximal number of extremal generators of monomial tropical cones in Td+1

max defined by n halfspaces is in Θ(n⌊d/2⌋).

Theorem (JL 2017)

All n nondominated points of a discrete d-criteria optimization problem can be determined in Θ(n⌊d/2⌋) with iterative tropical convex hull computation.

Conclusion

Summary

• generating all nondominated points with tropical convex hull
• study of monomial tropical cones and their duality
• promising connection between tropical algebra, multicriteria
• ptimization and monomial ideals
• arxiv: 1707.09305