Monomial Tropical Cones for Multicriteria Optimization Georg Loho joint work with Michael Joswig
Multicriteria optimization • multicriteria optimization problem: � � min f ( x ) = f 1 ( x ) , . . . , f d ( x ) subject to x ∈ X . • outcome space: Z = f ( X ) ⊆ R d • w ≦ z : if w i ≤ z i for all i ∈ [ d ] , partial ordering on R d . • for S ⊂ R d the minimal elements with respect to ≦ form the nondominated points. Observation The set dominated by N ⊆ R d is exactly N + R d � � \ N . ≥ 0 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 ( n d ) 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 + R d ≥ 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 T max = R ∪ {−∞} Tropical numbers Addition s ⊕ t := max( s, t ) = − min( − s, − t ) s ⊙ t := s + t Multiplication Operations are extended componentwise to T d +1 max
Tropical basics Definition T max = R ∪ {−∞} Tropical numbers Addition s ⊕ t := max( s, t ) = − min( − s, − t ) s ⊙ t := s + t Multiplication Operations are extended componentwise to T d +1 max Also: Dual tropical numbers T min = R ∪ {∞} with dual addition s ⊕ t = min( s, t ) .
Tropical basics Definition T max = R ∪ {−∞} Tropical numbers Addition s ⊕ t := max( s, t ) = − min( − s, − t ) s ⊙ t := s + t Multiplication Operations are extended componentwise to T d +1 max Also: Dual tropical numbers T min = 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 x ≤ 12 valid for every
Determining nondominated points with scalarizations M ( h ) h = ( − 3 , 2) M ( h )
Determining nondominated points with scalarizations M ( h ) h = ( − 3 , 2) M ( h )
Determining nondominated points with scalarizations M ( h ) h = ( − 3 , 2) M ( h )
Determining nondominated points with scalarizations M ( h ) h = ( − 3 , 2) g = (0 , 0) M ( h )
Determining nondominated points with scalarizations M ( g, h ) h = ( − 3 , 2) g = (0 , 0) M ( g, h )
Determining nondominated points with scalarizations M ( g, h ) h = ( − 3 , 2) g = (0 , 0) M ( g, h )
Determining nondominated points with scalarizations M ( g, h ) h = ( − 3 , 2) g = (0 , 0) M ( g, h )
Determining nondominated points with scalarizations M ( g, h ) h = ( − 3 , 2) g = (0 , 0) M ( g, h )
Adaptation of the generic method (Klamroth et al. 2015, Joswig, L 2017) Input: Image of the feasible set Z ⊂ R d Output: The set of nondominated points. 1: A ← E min ∪ e (0) 2: G ← ∅ 3: Ω ← E min 4: while A � = Ω do pick a in A \ Ω 5: g ← NextNonDominated ( Z, a ) 6: if g � = None then 7: A ← NewUpperBounds ( G, A, g ) 8: G ← G ∪ { g } 9: else 10: Ω ← Ω ∪ { a } 11: end if 12: 13: end while 14: return G
Tropical Inner and Outer Description • max -tropical cone: subset C ⊆ T d +1 max with � � max( λ + x 0 , µ + y 0 ) , . . . , max( λ + x d , µ + y d ) ∈ C for all λ, µ ∈ T max and x, y ∈ C . • G generates C if the latter is the minimal max -tropical cone containing G ; explicitly { λ 1 ⊙ g 1 ⊕ · · · ⊕ λ k ⊙ g k | g 1 , . . . , g k ∈ G, λ 1 , . . . , λ k ∈ T max } . • extremal generators: minimal generating set consists of extremal generators; an element g ∈ G is extremal iff, for g 1 , g 2 ∈ G g = µg 1 ⊕ νg 2 ⇒ g = µg 1 or g = νg 2 .
Tropical Inner and Outer Description • for vector a ∈ T d +1 max the set supp( a ) = { i | a i � = −∞} is its support. • closed max -tropical halfspace: a set of the form � � � x ∈ T d +1 � max( x i + a i | i ∈ I ) ≤ max( x j + a j | j ∈ J ) � max 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 x 2 0 − 1 − 1 0 2 2 g 3 G = a 3 0 − 1 3 − 2 0 1 a 4 g 2 0 − 1 1 ∞ 0 0 A = g 4 a 1 0 2 3 x 1 0 − 2 2 x 0 ≤ max( x 1 + 1 , x 2 − 1) g 1 x 1 ≤ x 2 a 2 max( x 1 − 2 , x 2 − 3) ≤ x 0 dehomogenized with x 0 = 0 max( x 0 , x 2 − 2) ≤ x 1 + 2
Tropical Inner and Outer Description x 2 0 − 1 − 1 0 2 2 g 3 G = a 3 0 − 1 3 − 2 0 1 a 4 g 2 0 − 1 1 ∞ 0 0 A = g 4 a 1 0 2 3 x 1 0 − 2 2 x 0 ≤ max( x 1 + 1 , x 2 − 1) g 1 x 1 ≤ x 2 a 2 max( x 1 − 2 , x 2 − 3) ≤ x 0 dehomogenized with x 0 = 0 max( x 0 , x 2 − 2) ≤ x 1 + 2
Tropical Inner and Outer Description x 2 0 − 1 − 1 0 2 2 g 3 G = a 3 0 − 1 3 − 2 0 1 a 4 g 2 0 − 1 1 ∞ 0 0 A = g 4 a 1 0 2 3 x 1 0 − 2 2 x 0 ≤ max( x 1 + 1 , x 2 − 1) g 1 x 1 ≤ x 2 a 2 max( x 1 − 2 , x 2 − 3) ≤ x 0 dehomogenized with x 0 = 0 max( x 0 , x 2 − 2) ≤ x 1 + 2
Tropical Inner and Outer Description x 2 0 − 1 − 1 0 2 2 g 3 G = a 3 0 − 1 3 − 2 0 1 a 4 g 2 0 − 1 1 ∞ 0 0 A = g 4 a 1 0 2 3 x 1 0 − 2 2 x 0 ≤ max( x 1 + 1 , x 2 − 1) g 1 x 1 ≤ x 2 a 2 max( x 1 − 2 , x 2 − 3) ≤ x 0 dehomogenized with x 0 = 0 max( x 0 , x 2 − 2) ≤ x 1 + 2
Tropical Inner and Outer Description x 2 0 − 1 − 1 0 2 2 g 3 G = a 3 0 − 1 3 − 2 0 1 a 4 g 2 0 − 1 1 ∞ 0 0 A = g 4 a 1 0 2 3 x 1 0 − 2 2 x 0 ≤ max( x 1 + 1 , x 2 − 1) g 1 x 1 ≤ x 2 a 2 max( x 1 − 2 , x 2 − 3) ≤ x 0 dehomogenized with x 0 = 0 max( x 0 , x 2 − 2) ≤ x 1 + 2
Monomial tropical cones Fix G ⊆ T d +1 max . • Monomial tropical cone M ( G ) (Allamigeon et al. 2010, JL 2017): � � � � x ∈ T d +1 � x 0 − g 0 ≤ min( x j − g j | j ∈ supp( g ) \ { 0 } ) � max g ∈ G • M ( G ) = M ( G ) ∩ R d +1 = � { 0 } × R d + R · 1 � � g + ≥ 0 g ∈ G • max -tropical cone in T d +1 max generated by the finite set G ∪ E max , where E max = { ( −∞ , 0 , −∞ , . . . , −∞ ) , . . . , ( −∞ , −∞ , . . . , −∞ , 0) }
Monomial tropical cones c = (0 , − 3 , ∞ ) ( −∞ , −∞ , 0) T 3 max R 3 M ( f, g, h ) h = (0 , − 3 , 2) b = (0 , 0 , 2) g = (0 , 0 , 0) a = (0 , 1 , 0) ( −∞ , 0 , −∞ ) f = (0 , 1 , −∞ )
Duality of monomial tropical cones M ( G ) : the complement of the interior of the max -tropical cone M ( G ) � R d +1 \ � ��� � { 0 } × R d � g + + R · 1 > 0 g ∈ G M M ( G ) in T d +1 ( G ) : closure of min with respect to min (add points with ∞ ). Theorem (Joswig, L 2017) M ( G ) is a min-tropical cone in R d +1 . More precisely, if H is a set The set of max-tropical halfspaces such that � H = M ( G ) , then M ( G ) = − M ( − A ) , where A ⊂ T d +1 min is the set of apices of the tropical halfspaces in H . In M particular, the set A ∪ E min generates ( G ) .
Duality of monomial tropical cones c = (0 , − 3 , ∞ ) ( −∞ , −∞ , 0) T 3 max R 3 M ( f, g, h ) h = (0 , − 3 , 2) b = (0 , 0 , 2) g = (0 , 0 , 0) ( ∞ , 0 , ∞ ) a = (0 , 1 , 0) ( −∞ , 0 , −∞ ) M ( f, g, h ) = − M ( − a, − b, − c ) T 3 ( ∞ , ∞ , 0) f = (0 , 1 , −∞ ) min
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) M Input: A set G ⊂ T d +1 max , the set A of extremal generators of ( G ) , and a point h ∈ T d +1 max with h 0 = 0 . M ( G ∪ { h } ) . Output: The set of extremal generators of • determine the set A ≥ of points in A which fulfill the “monomial inequality” x 0 ≥ min i ∈ [ d ] ( x i − h i ) given by h M • initialize the set B of generators of ( 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 } ) .
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