Linear Programming, tropical convexity, and repeated games

Linear Programming, tropical convexity, and

repeated games

[email protected]

INRIA and CMAP, Ecole Polytechnique, CNRS

LAAS, June 18

Based on joint work with Akian, Guterman (IJAC 2012) andAllamigeon, Benchimol, Joswig arXiv:1308.0454, arXiv:1309.5925,arXiv:1405.4161.

Stephane Gaubert (INRIA and CMAP) Linear Programming and the Tropics. . . LAAS 1 / 68

http://www.arXiv.org/abs/1308.0454



Connections between:

linear programming (simplex algorithm, interiorpoints)

mean payoff games

through tropical geometry.


Some open questions in linear programming

A linear program is an optimization problem:

min c · x ; Ax 6 b, x ∈ Rn ,

where c ∈ Rn, A ∈ Rm×n, b ∈ Rm.3 methods

simplex algorithm (Dantzig), sometimes exponentialtime, efficient

ellipsoid (Khachyan), polynomial time, inefficient

interior points (Karmakar. . . ), polynomial time,efficient


Smale’s problem for LP

Question

Can linear programming be solved in strongly polynomialtime?

polynomial time: = execution time bounded by apolynomial P(m, n, L), L = number of bits to code theAij , bi , cj (sum of their log2’s).

6= strongly polynomial: number of arithmetic operationsbounded by a polynomial P(m, n), and the size ofoperands of arithmetic operations is bounded by apolynomial in L



Question






Question





The simplex method

Principle: iterate over adjacent vertices (basic points) of thepolyhedron while improving the objective function

c>v 1 > c>v 2 > . . . > c>vN

At a basic point, there may be several edges along which theobjective function is improved. =⇒the algorithm is parametrized by a pivoting rule, which selectsthe next edge to be followed.

Example: Dantzig’s rule, steepest edge rule, Bland’s rule, randomized rules, etc


The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2

v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2

v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4

At a basic point, there may be several edges along which theobjective function is improved.

=⇒the algorithm is parametrized by a pivoting rule, which selectsthe next edge to be followed.



The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




The simplex method


c>v 1 > c>v 2 > . . . > c>vN

v 1

v 2v 3

v 4




Complexity of the simplex method?

the number of iterations depends on choice of the pivot-ing rule and every iteration (pivoting from a basic pointto the next one) can be done with a strongly polynomialcomplexity

=⇒ if the number N of iterations is polynomial (in m and n), theoverall complexity is strongly polynomial.

for most (all?) pivoting rules, there are some counterexampleswith super polynomial number of iterations (Klee-Minty cube,etc)

is there a pivoting rule ensuring that the number of iterations inthe worst case is polynomially bounded?

or, equivalently, is there a strongly polynomial simplex algorithm?
















Ideal version of Dantzig’s problem:

Conjecture (Hirsch)

Any two vertices of the graph of a polytope with m facetsin dimension n can be joined by a path of length m− n atmost.

Disproved by F. Santos (2012, Ann. Math.), but theconjecture fails only by one unit. Weaker forms of theconjecture, like length = poly(m), still hold.



Interior points

For all µ > 0, consider the barrier problem

minµ−1c · x −n∑

j=1

log xj −m∑i=1

log wi , Ax + w = b, x > 0,w > 0

log strictly concave =⇒ optimal solution x(µ) is unique.µ 7→ x(µ) is the central path. x(0) is the solution of the LP.


Inductive step of interior points:

x ← Newton(x , µ);

reduce µ not too quickly so that x remains in an appropriateattraction bassin of Newton’s method. Size of attraction bassinis bounded by the inverse of a condition number (Shub-Smale).

Several path following methods. Complexity bounded by the lengthof the path in a degenerate Riemannian metric (locally inverse ofcondition number) or by a special curvature integral (Sonnevend).


Continuous analogue of Hirsch’s conjecture

An intrinsic complexity measure of the central path is:

total curvature =

∫ `

0

‖κ(τ)‖dτ

κ = Φ′′

Φ = central path parametrized by arclength

Conjecture (Deza, Terlaky and Zinchenko, 2008)

The total curvature of the central path of a polytope with m facets indimension n is bounded by O(m).

Dedieu, Malajovich, and Shub showed that the total curvature averagedover the 2m LP’s with sign conditions ±si 6 0 is O(n).

Dedieu and Shub first conjectured that the total curvature is O(n).

Contradicted by Deza, Terlaky, Zinchenko, redundant Klee-Minty cube.Stephane Gaubert (INRIA and CMAP) Linear Programming and the Tropics. . . LAAS 11 / 68

The mean payoff problem

Question (Gurvich, Karzanov, Khachyan 88)

Is there a polynomial time algorithm to solve a mean payoffdeterministic game?


Mean payoff (deterministic) games

G = (V ,E ) bipartite graph. rij price of the arc (i , j) ∈ E .

“Max” and “Min” move a token. The player making themove receives from the other player the paiement writtenon the arc.


v ki value of MAX, initial state (i ,MIN).

v k1 = min(−2 + 1 + v k−1

1 ,−8 + max(−3 + v k−11 ,−12 + v k−1

2 ))

v k2 = 0 + max(−9 + v k−1

1 , 5 + v k−12 )

2

1

8

−3

−12

0

53

2

1

1

2

−9MIN

MAX



v k1 = min(−2 + 1 + v k−1

1 ,−8 + max(−3 + v k−11 ,−12 + v k−1

2 ))

v k2 = 0 + max(−9 + v k−1

1 , 5 + v k−12 )

2

1

8

−3

−12

0

53

2

1

1

2

−9MIN

MAX

limk v k/k = (−1, 5)



v k1 = min(−2 + 1 + v k−1

1 ,−8 + max(−3 + v k−11 ,−12 + v k−1

2 ))

v k2 = 0 + max(−9 + v k−1

1 , 5 + v k−12 )

2

1

8

−3

−12

0

53

2

1

1

2

−9MIN

MAX

limk v k/k = (−1, 5)


Shapley operator

v k = T (v k−1), v 0 = 0, T : Rn → Rn

[T (x)]j = mini∈[m]

(− rji + max

l∈[n](r ′il + xl)

)Mean payoff vector

χ(T ) = limk→∞

T k(0)/k = limk→∞

v k/k .

T is order preserving, additively homogeneous ⇒sup-norm nonexpansive:

x 6 y =⇒ T (x) 6 T (y)

T (α + x) = α + T (x), ∀α ∈ R‖T (x)− T (y)‖ 6 ‖x − y‖


Shapley operator

v k = T (v k−1), v 0 = 0, T : Rn → Rn

[T (x)]j = mini∈[m]

(− rji + max

l∈[n](r ′il + xl)

)Mean payoff vector

χ(T ) = limk→∞

T k(0)/k = limk→∞

v k/k .

T is order preserving, additively homogeneous ⇒sup-norm nonexpansive:

x 6 y =⇒ T (x) 6 T (y)

T (α + x) = α + T (x), ∀α ∈ R‖T (x)− T (y)‖ 6 ‖x − y‖


The mean payoff χ(T ) := limk T k(0)/k does exist.

True for any nonexpansive map T : Rn → Rn with asemi-algebraic graph (Neyman 04, extending Bewley and

Kohlberg 76).

Proof: introduce a discount factor α < 1, letvα = T (αvα) be the discounted value, limα→1−(1− α)vαexists (a bounded semi-algebraic function of one variablehas a limit) and is equal to χ(T ).




Kohlberg 76).





Kohlberg 76).





Is there a polynomial time algorithm to solve thefollowing problem for deterministic games: is state iwinning for MAX? i.e., does χi(T ) > 0 ?

Problem in NP ∩ coNP (Zwick-Paterson 96)

value iteration is pseudo polynomial: compute T k(0)/k fork ∼ (n + m)3W , W = max |paiement of an arc| (assume integerpaiements), ibid.

policy iteration performs well, but super-polynomial counterexample by Friedmann 10























Theorem (Allamigeon, Benchimol, SG, Joswig arXiv:1309.5925)

Any strongly polynomial simplex algorithm equipped witha combinatorial pivoting rule yields a strongly polynomialalgorithm to solve mean payoff games.

Combinatorial pivoting rule: the choice of the next vertexdepends on (A, b, c) only through signs of minors of thematrix (

A bc> 0

)This includes signs of reduced costs. Bland’s rule iscombinatorial.→ positive answer to Dantzig implies positive answer toGurvich, Karzanov, Khachyan.


The central path can be tortuous


There is a family of linear programs with 3r + 4inequalities in dimension 2r + 2 where the central pathhas a total curvature greater than 2r/(3r).

This disproves the continuous analogue of Hirsch’s conjecture of Deza,

Terlaky, Zinchenko.


The proof uses max-plus or tropical algebra, actually,tropical linear programming.


Max-plus or tropical algebra

In an exotic country, children are taught that:

“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” =

“2× 3” =

“5/2” =

“23” =

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =

“5/2” =

“23” =

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =

“5/2” =

“23” =

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =5

“5/2” =

“23” =

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =5

“5/2” =

“23” =

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =5

“5/2” =3

“23” =

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =5

“5/2” =3

“23” =

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =5

“5/2” =3

“23” =“2× 2× 2” = 6

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3

“2× 3” =5

“5/2” =3

“23” =“2× 2× 2” = 6

“√−1” =




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3 “

(7 0−∞ 3

)(21

)”=

“2× 3” =5

“5/2” =3

“23” =“2× 2× 2” = 6

“√−1” =−0.5




“a + b” = max(a, b) “a × b” = a + b

So

“2 + 3” = 3 “

(7 0−∞ 3

)(21

)”=

(94

)“2× 3” =5

“5/2” =3

“23” =“2× 2× 2” = 6

“√−1” =−0.5


Max-plus / tropical semiring

Rmax = (R ∪ {−∞},max,+)


Cuninghame-Green 1960- OR (scheduling, optimization)

Vorobyev ∼65 . . . Zimmerman, Butkovic; Optimization

Maslov ∼ 80’- . . . Kolokoltsov, Litvinov, Samborskii, Shpiz. . . Quasi-classicanalysis, variations calculus

Simon ∼ 78- . . . Hashiguchi, Leung, Pin, Krob, . . . Automata theory

Gondran, Minoux ∼ 77 Operations research

Cohen, Quadrat, Viot ∼ 83- . . . Olsder, Baccelli, S.G., Akian discrete eventsystems, optimal control, idempotent probabilities, linear algebra

Nussbaum 86- Nonlinear analysis, dynamical systems, also related work inlinear algebra, Friedland 88, Bapat ˜94

Kim, Roush 84 Incline algebras

Fleming, McEneaney ∼00- max-plus approximation of HJB

Del Moral ∼95 Puhalskii ∼99, idempotent probabilities.Since 2000’ in pure maths, tropical geometry: Viro, Mikhalkin, Passare,Sturmfels . . . , recent work by Connes, Consani


The sister algebra: min-plus

“a + b” = min(a, b) “a × b” = a + b

“2 + 3” = 2

“2× 3” = 5


Some elementary tropical geometry

A tropical line in the plane is the set of (x , y) such thatthe max in

“ax + by + c”

is attained at least twice.

max(x , y , 0)


Some elementary tropical geometry

A tropical line in the plane is the set of (x , y) such thatthe max in

max(a + x , b + y , c)

is attained at least twice.

max(x , y , 0)


Two generic tropical lines meet at a unique point


By two generic points passes a unique tropical line


non generic case


non generic case resolved by perturbation


non generic case resolved by perturbation


Nonarchimedean valuation point of view

Let C{{t}} denote the field of Puiseux series, equippedwith the valuation val s = − smallest exponent of s,C{{t}} → Rmax;

E.g., val(t−1/2 − t + 7t3/2 + . . . ) = 1/2

val(z1 + z2) 6 max(val(z1), val(z2)), with equality whenleading coeffs dont cancel (e.g. if val(z1) 6= val(z2)).

val(z1z2) = val(z1) + val(z2)


The rational points of tropical hyperplanes are images ofhyperplanes of (C{{t}})n by the valuation, see:

Theorem (Kapranov)

Given p =∑

α pαzα ∈ C{{t}}[z1, . . . , zn], and Z ∈ Qn,

∃z ∈ (C{{t}})n, p(z) = 0, Z = val z

iffmaxα

val pα + 〈α,Z 〉 attained twice

Restriction to Q can be avoided by working with Puiseux series with real

exponents (Hahn / Hardy / van den Dries / Markwig), definable in Ran,∗

o-minimal model (Van Den Dries, Alessandrini).


Tropical segments:

f

g

[f , g ] := {“λf + µg” | λ, µ ∈ R∪ {−∞}, “λ+ µ = 1”}.

(The condition “λ, µ > 0” is automatic.)


Tropical segments:

f

g

[f , g ] := { sup(λ + f , µ + g) | λ, µ ∈R ∪ {−∞}, max(λ, µ) = 0}.

(The condition λ, µ > −∞ is automatic.)


Tropical convex set: f , g ∈ C =⇒ [f , g ] ∈ C


Tropical convex set: f , g ∈ C =⇒ [f , g ] ∈ C

Tropical convex cone: ommit “λ + µ = 1”, i.e., replace[f , g ] by {sup(λ + f , µ + g) | λ, µ ∈ R ∪ {−∞}}


Homogeneization

A convex set C in Rnmax is a cross section of a convex

cone C in Rn+1max ,

C := {(λ + u, λ) | u ∈ C , λ ∈ Rmax}


A tropical polytope with four vertices

Structure of the polyhedral complex: Develin, SturmfelsStephane Gaubert (INRIA and CMAP) Linear Programming and the Tropics. . . LAAS 32 / 68

Tropical half-spaces

Given a, b ∈ Rnmax, a, b 6≡ −∞,

H := {x ∈ Rnmax | “ax 6 bx”}




H := {x ∈ Rnmax | max

16i6nai + xi 6 max

16i6nbi + xi}

x2x1

x3

max(x1, x2,−2 + x3)





16i6nai + xi 6 max

16i6nbi + xi}

x2x1

x3

max(x1,−2 + x3) 6 x2





16i6nai + xi 6 max

16i6nbi + xi}

x2x1

x3

x1 6 max(x2 − 2 + x3)





16i6nai + xi 6 max

16i6nbi + xi}

x2x1

x3

max(x2 − 2 + x3) 6 x1


Tropical polyhedral cones

can be defined as intersections of finitely many half-spaces

x2x1

V

x3




x2x1

V

x3

x2x1

x3

2 + x1 6 max(x2, 3 + x3)




V



or as the tropical linear combinations of finitely manyvectors

V


Viro’s log-glasses, related to Maslov’s dequantization

a +h b := h log(ea/h + eb/h), h→ 0+

max(a, b) 6 a +h b 6 h log 2 + max(a, b)


Tropical convex sets are deformations of classical convexsets

∑i

taixi >∑i

tbixi → maxi

ai + Xi > maxi

bi + Xi ,

Xi = log xi/ log t, t →∞.


Tropical linear programming

Problem (Feasibility, conical form)

Given A,B ∈ Rm×nmax , is there a vector x 6≡ −∞ such that

“Ax 6 Bx”.

Problem (Tropical LP)

Given A,C ∈ Rm×nmax , b, d , f ∈ Rm

max,

min “f >x”, “Ax + b > Cx + d”, x ∈ Rnmax


Theorem (Akian, SG, Guterman, IJAC 2012)

Mean payoff games are equivalent to arrangements oftropical half-spaces (i.e., inequalities “Ax 6 Bx”).

State i is winning iff there is a vector x such that

“Ax 6 Bx”, xi 6= −∞ .

Corollary

Mean payoff games are equivalent to feasibility problemsin tropical linear programming.

By homogeneity, xi 6= −∞ can be replaced by xi > 0.


Consider the tropical polyhedral cone

C =⋂

16i6m

Hi

where (Hi)16i6m is a family of tropical half-spaces.

Hi : “Aix 6 Bix”

Let:[T (x)]j = min

16i6m−aij + max

16k6nbik + xk .

x 6 T (x) ⇐⇒ max16j6n

aij+xj 6 max16k6n

bik+xk , ∀1 6 i 6 m .



C =⋂

16i6m

Hi


Hi : max16j6n

aij +xj 6 max16k6n

bik +xk , aij , bik ∈ R∪{−∞}

Let:[T (x)]j = min

16i6m−aij + max

16k6nbik + xk .

x 6 T (x) ⇐⇒ max16j6n

aij+xj 6 max16k6n

bik+xk , ∀1 6 i 6 m .



C =⋂

16i6m

Hi


Hi : max16j6n

aij +xj 6 max16k6n


Let:[T (x)]j = min

16i6m−aij + max

16k6nbik + xk .

x 6 T (x) ⇐⇒ max16j6n

aij+xj 6 max16k6n

bik+xk , ∀1 6 i 6 m .



C =⋂

16i6m

Hi


Hi : max16j6n

aij +xj 6 max16k6n


Let:[T (x)]j = min

16i6m−aij + max

16k6nbik + xk .

x 6 T (x) ⇐⇒ max16j6n

aij+xj 6 max16k6n

bik+xk , ∀1 6 i 6 m .


Hi : max16j6n

aij + xj 6 max16k6n

bik + xk

[T (x)]j = min16i6m

−aij + max16k6n

bik + xk .

Interpretation of the game

State of MIN: variable xj , j ∈ {1, . . . , n}State of MAX: half-space Hi , i ∈ {1, . . . ,m}In state xj , Player MIN chooses a tropical half-spaceHi with xj in the LHS

In state Hi , player MAX chooses a variable xk at theRHS of Hi

Payment −aij + bik .


Assume that “Ax 6 Bx” for some x such that xi 6= −∞.

Then T (x) > xWLOG 0 > x (homogeneity).

T order preserving =⇒ T k(0) > T k(x) > x

and so

χi(T ) = limk→∞

[T k(0)]i/k > limk→∞

xi/k = 0

I.e., i is winning for MAX.Conversely, asssume i is winning for MAX. We use aninvariant half-line (Kohlberg theorem):

∃v , η, T (v + sη) = v + (s + 1)η, ∀s > 0

Then, η = χ(T ), and x is constructed from v .


Assume that “Ax 6 Bx” for some x such that xi 6= −∞.Then T (x) > x

WLOG 0 > x (homogeneity).


and so



xi/k = 0


∃v , η, T (v + sη) = v + (s + 1)η, ∀s > 0



Assume that “Ax 6 Bx” for some x such that xi 6= −∞.Then T (x) > xWLOG 0 > x (homogeneity).


and so



xi/k = 0


∃v , η, T (v + sη) = v + (s + 1)η, ∀s > 0





and so



xi/k = 0


∃v , η, T (v + sη) = v + (s + 1)η, ∀s > 0





and so



xi/k = 0

I.e., i is winning for MAX.

Conversely, asssume i is winning for MAX. We use aninvariant half-line (Kohlberg theorem):

∃v , η, T (v + sη) = v + (s + 1)η, ∀s > 0





and so



xi/k = 0


∃v , η, T (v + sη) = v + (s + 1)η, ∀s > 0

Then, η = χ(T ), and x is constructed from v .Stephane Gaubert (INRIA and CMAP) Linear Programming and the Tropics. . . LAAS 41 / 68

x1 x2

x3

x1 x2

x3

states 1,2,3 winning states 2,3 winning


Tropical simplex

A tropical LP

min “f · x”; “Ax + c 6 Bx + d”

A,B ∈ Rm×nmax , b, c ∈ Rm

max, f ∈ Rnmax, the inequalities

“x > 0” being included in “Ax + c 6 Bx + d”, can belifted to a classical LP over Hahn series

min f · x; Ax + c 6 Bx + d

A,B ∈ Km×n, b, c ∈ Km, f ∈ Kn,meaning that valA = A, valB = B , etc. Recall thatval 7t−1/2 − 1 + t1/2 + 7t + · · · = 1/2.


(0, 0, 0)

(0, 0, 4)

(4, 0, 0)

(4, 4, 0)

(4, 4, 4)


(t0, t0, t0)

(t0, t0, t−4)

(t−4, t0, t0)

(t−4, t−4, t0)

(t−4, t−4, t−4)


The simplex algorithm makes sense over any (totally)ordered field, in particular Hahn series.

Question

Can we compute the image of the path followed by thesimplex algorithm over Hahn series by the valuation,working only “tropically” (use only the information of thevaluation, dont use arithmetic operations on the series).

Question

What are tropical basic points?


A point of “Ax + c 6 Bx + d” is expected to be(tropically) basic if it saturates n “independent”inequalities.

What does independent mean?

This can be formalized using the tropical Cramer theorem.


Two generic tropical lines meet at a unique point


Every n affine tropical hyperplanes of Rnmax in general

position meet at a unique point. Richter-Gebert,

Sturmfels, Theobalt, 05 complex version, real version inMax Plus, 90, Akian, SG, Guterman 09, 13.

x being in such an intersection reads

“Mx + g = 0”, M ∈ Rn×nmax , g ∈ Rn

max .

One has xi = “D−1Di” where D = “ det M” andDi = “ det Mi”; replace column i of M by g to get Mi .


How are Cramer det defined?

det A = “∑σ

sgn∏i

Aiσ(i)” = maxσ

∑i

Aiσ(i)

This is an optimal assignment problem (O(n3) time).General position means that there is only one optimalpermutation.

The sign of the determinant is the sign of the optimalpermutation.

(Tropical Cramer determinant revisited, arXiv1309.6298)


Exercise

Solve

max(2 + x , y , 3) reached 2 times

max(x , y , 2) reached 2 times

D =

∣∣∣∣2 00 0

∣∣∣∣ = 2

Dx =

∣∣∣∣3 02 0

∣∣∣∣ = 3 Dy =

∣∣∣∣2 20 2

∣∣∣∣ = 4

x = “Dx/D” = 1, y = “Dy/D” = 2 .


Assume that the data are in general position. This can bedefined in terms of tropical Cramer subdeterminants of“(A + B , c + d)”.A tropical basic point is obtained by saturating ninequalities.

Theorem (Allamigeon, Benchimol, SG, JoswigarXiv:1308.0454)

The valuation of the path of the simplex algorithm overHahn series can be computed tropically (with acompatible pivoting rule). One iteration takesO(n(m + n)) time.

Tropical Cramer determinants = opt. assignment used tocompute reduce costs.



Example of compatible pivoting rule. A rule iscombinatorial if any entering/leaving inequalities arefunctions of the history (sequence of bases) and of thesigns of the minors of the matrix

M =( “A− B” “c − d”

f > “0”

).

(eg signs of reduced costs).

Corollary (Allamigeon, Benchimol, SG, JoswigarXiv:1309.5925)

If any combinatorial rule in classical linear programmingwould run in strongly polynomial time, then, mean payoffgames could be solved in strongly polynomial time.



Sketch of Proof

1 Mean payoff games are equivalent to tropical linearprograms (Akian, SG, Guterman)

2 Tropical linear programs can be lifted to a subclass ofclassical linear programs over Hahn series.

3 The set of runs (sequences of bases) of the classicalsimplex algorithm equipped with a combinatorialpivoting rule is independent of the real closed field.Being a run is a first order property, apply Tarski’s theorem.

4 Can simulate the classical simplex on Hahn seriestropically, every pivot being strongly polynomial(Allamigeon, Benchimol, SG, Joswig)


A key technical difficulty is to relax the general positioncondition.

→ work with higher valuation groups

Replace field of series K in the parameter t with realexponents by R[[tG ]], G totally ordered group

G = (Rk ,+,6lex)

R[[tG ]] is sent to G by the valuation.

R[[tG ]] is known to be real closed, Ribenboim.


Tropicalization of interior points ?


Given a positive µ ∈ R, the barrier problem is

minimizec>x

µ−

n∑j=1

log(xj)−m∑i=1

log(wi)

subject to Ax + w = b, x > 0,w > 0.

(1)

Ax + w = b

−A>y + s = c

wiyi = µ for all i ∈ [m]

xjsj = µ for all j ∈ [n]

x ,w , y , s > 0 .

(2)

For any µ > 0, ∃! (xµ,wµ, yµ, sµ) ∈ Rn × Rm × Rm × Rn. Thecentral path is the image of the map CA,b,c : R>0 → R2m+2n whichsends a positive real number µ to the vector (xµ,wµ, yµ, sµ).


The tropical central path

Assume now that A(t),b(t), c(t) have entries in K (absolutelyconverging series in t, with real exponents).

The tropical central path is the log-limit:

Ctrop : λ 7→ limt→+∞

logt C(t, λ) . (3)

The poinwise limit does exist since C(·, λ) is definable in apolynomially bounded o-minimal structure.


Theorem

The family of maps (logt C(t, ·))t converges uniformly on any closedinterval [a, b] ⊂ R to the tropical central path Ctrop.



Let (xµ,wµ) be the point on the primal central path of the linearprogram LP(A,b, c) at µ ∈ K with µ > 0, and let ν be that LP’soptimal value. Then val(xµ,wµ) is the greatest element of val(Pµ)where

Pµ :={(x,w) ∈ Kn+m | Ax+w = b, cx 6 ν+(n+m)µ, x > 0, w > 0} .


x1 + x2 6 2

tx1 6 1 + t2x2

tx2 6 1 + t3x1

x1 6 t2x2

x1, x2 > 0 .

(4)

Its value val(P) is the tropical set described by the inequalities:

max(x1, x2) 6 0

1 + x1 6 max(0, 2 + x2)

1 + x2 6 max(0, 3 + x1)

x1 6 2 + x2 .

(5)


−4 −3 −2 −1 0−4

−3

−2

−1

0

x1

x2

−4 −3 −2 −1 0−4

−3

−2

−1

0

x1

x2

Figure: Tropical central paths on the Hardy polyhedron (4) for theobjective function min x1 (left) and min tx1 + x2 (right).


Figure: Tropical central paths on the full-dimensional cells included in thepositive orthant induced by the arrangement of hyperplanes associatedwith (4); for the objective function min tx1 + x2 (in blue) andmax tx1 + x2 (in red). The parts of the paths that lie on the boundariesare slightly shifted inside their respective cell.


Bezem, Nieuwenhuis and Rodrıguez-Carbonell (2008) constructed aclass of tropical linear programs for which an algorithm of Butkovicand Zimmermann (2006) exhibits an exponential running time.

On this example, the tropical central path passes through anexponential number of basic points.

By dequantization, C(t, ·) for t large enough, yields a counterexample to the continuous analogue of the Hirsch conjecture.


The counter example

min v0

s.t. u0 6 t

v0 6 t2

vi 6 t1− 1

2i (ui−1 + vi−1) for 1 6 i 6 r

ui 6 tui−1 for 1 6 i 6 r

ui 6 tvi−1 for 1 6 i 6 r

ur > 0, vr > 0

r


The tropical central path is determined by the following dynamicalsystems

u0 = 1

v0 = min(2, λ)

vi = 1− 1

2i+ max(ui−1, vi−1)

ui = 1 + min(ui−1, vi−1)


0 1 20

1

2

3

4

5

λ

u1

v1

u2

v2

u3

v3

u4

v4

Figure: A tropical central path with many segments


Concluding remarks

Complexity of tropical LP = deterministic mean payoff games isopen

Transfer theorem: some classes of pivoting rules for the simplexalgorithm also solve mean payoff games. Worst case number ofiterations is smaller (or equal) for mean payoff games as thereare fewer instances.

Use higher value groups G = Rk , K = R[[tG ]].

Tropical interior points can be as bad as tropical simplex.

Are tortuous central paths really an obstruction to a“strong-polynomialization” of interior point / Shub-Smalehomotopy methods ?

Thank you !


Documents

Linear Programming, tropical convexity, and repeated games