Chemins centraux sinueux, jeux et programmation linéaire non-archimédienne

Long and winding central paths, games and non-archimedean linear programming
[email protected] INRIA and CMAP, ´ Ecole polytechnique, CNRS GDR MOA, Dijon December 2-4, 2015 Based on work with Akian and Guterman (games) Allamigeon, Benchimol (PhD 2014) and Joswig (linear programming and games). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 1 / 76

This lecture: relations between Linear programming with large coe cients,
Smale Problem # 9 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 2 / 76

Smale Problem # 9 Linear programming over nonarchimedean ﬁelds Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 2 / 76

Smale Problem # 9 Linear programming over nonarchimedean ﬁelds Geometry of the central path in LP Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 2 / 76

Smale Problem # 9 Linear programming over nonarchimedean ﬁelds Geometry of the central path in LP Mean payo↵ games (one of the problems in NP \ coNP not known to be in P) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 2 / 76

Smale Problem # 9 Linear programming over nonarchimedean ﬁelds Geometry of the central path in LP Mean payo↵ games (one of the problems in NP \ coNP not known to be in P) via tropical geometry Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 2 / 76

Part I. Open problems concerning zero-sum games and linear programming
Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 3 / 76

The mean payo↵ problem Stephane Gaubert (INRIA and CMAP) Long
and winding central paths. . . GDR MOA, Dijon 4 / 76

Mean payo↵ games G = (V , E) bipartite graph.
rij 2 Z price of the arc (i, j) 2 E. MAX and MIN move a token, alternatively (square states: MAX plays; circle states: MIN plays). n MIN nodes, m MAX nodes. MIN always pays to MAX the price of the arc (having a negative fortune is allowed) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 5 / 76

vk i value of MAX, initial state (i, MIN). 1
−3 −12 0 5 3 2 1 1 2 −9 MIN MAX −2 −8 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 6 / 76

vk i value of MAX, initial state (i, MIN). 1
−3 −12 0 5 3 2 1 1 2 −9 MIN MAX −2 −8 limk!1 vk/k = ( 1, 5) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 6 / 76

Problem (Gurvich, Karzanov, Khachyan 88) Can we solve mean payo↵
games in polynomial time? Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 7 / 76

games in polynomial time? I.e., time 6 poly(L)? where L is the bitlength of the input L = X ij log2 (1 + |rij|) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 7 / 76

games in polynomial time? I.e., time 6 poly(L)? where L is the bitlength of the input L = X ij log2 (1 + |rij|) Mean payo↵ games in NP \ coNP, not known to be in P. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 7 / 76

games in polynomial time? I.e., time 6 poly(L)? where L is the bitlength of the input L = X ij log2 (1 + |rij|) Mean payo↵ games in NP \ coNP, not known to be in P. Zwick and Paterson [1996] showed that value iteration solves MPG in pseudo polynomial time O((n + m)5W ) where W = maxij |rij | = O(2L). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 7 / 76

Complexity issues in linear programming Stephane Gaubert (INRIA and CMAP)
Long and winding central paths. . . GDR MOA, Dijon 8 / 76

A linear program is an optimization problem: min c ·
x; Ax 6 b, x 2 Rn , where c 2 Qn, A 2 Qm⇥n, b 2 Qm. opt Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 9 / 76

Question (9th problem in Smale’s 21rst century list) Can linear
programming be solved in strongly polynomial time? [Smale, 2000], more on strongly polynomial algo. in [Gr¨ otschel, Lov´ asz, and Schrijver, 1993] Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 10 / 76

programming be solved in strongly polynomial time? polynomial time (Turing model): = execution time bounded by poly(L) or equivalently poly(n, m, L), L = number of bits to code the Aij, bi, cj [Smale, 2000], more on strongly polynomial algo. in [Gr¨ otschel, Lov´ asz, and Schrijver, 1993] Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 10 / 76

programming be solved in strongly polynomial time? polynomial time (Turing model): = execution time bounded by poly(L) or equivalently poly(n, m, L), L = number of bits to code the Aij, bi, cj 6= strongly polynomial (arithmetic model): number of arithmetic operations bounded by poly(m, n), and the size of operands of arithmetic operations is bounded by poly(L). [Smale, 2000], more on strongly polynomial algo. in [Gr¨ otschel, Lov´ asz, and Schrijver, 1993] Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 10 / 76

Three classical approaches to solve LP problem pivoting algorithms, simplex
Dantzig (1947) What do these approaches tell about Smale problem 9? Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 11 / 76

Dantzig (1947) ellipsoid, Khachyan (1979), polynomial time What do these approaches tell about Smale problem 9? Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 11 / 76

Dantzig (1947) ellipsoid, Khachyan (1979), polynomial time interior points, Karmarkar (1984) . . . , polynomial time. What do these approaches tell about Smale problem 9? Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 11 / 76

The simplex method (Dantzig, 1947) Iterate over adjacent vertices (basic
points) of the polyhedron while improving the objective function c>v1 > c>v2 > . . . > c>vN Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 12 / 76

points) of the polyhedron while improving the objective function c>v1 > c>v2 > . . . > c>vN v1 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 12 / 76

points) of the polyhedron while improving the objective function c>v1 > c>v2 > . . . > c>vN v1 v2 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 12 / 76

points) of the polyhedron while improving the objective function c>v1 > c>v2 > . . . > c>vN v1 v2 v3 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 12 / 76

points) of the polyhedron while improving the objective function c>v1 > c>v2 > . . . > c>vN v1 v2 v3 v4 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 12 / 76

points) of the polyhedron while improving the objective function c>v1 > c>v2 > . . . > c>vN v1 v2 v3 v4 the algorithm is parametrized by a pivoting rule, which selects the next edge to be followed. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 12 / 76

Complexity of pivoting algorithms? Every iteration (pivoting from a basic
point to the next one) can be done with a strongly polynomial complexity (linear system over Q). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 13 / 76

point to the next one) can be done with a strongly polynomial complexity (linear system over Q). if the number N of iterations is polynomial (in m and n), the overall complexity is strongly polynomial. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 13 / 76

point to the next one) can be done with a strongly polynomial complexity (linear system over Q). if the number N of iterations is polynomial (in m and n), the overall complexity is strongly polynomial. is there a pivoting rule ensuring that the number of iterations in the worst case is polynomially bounded? Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 13 / 76

point to the next one) can be done with a strongly polynomial complexity (linear system over Q). if the number N of iterations is polynomial (in m and n), the overall complexity is strongly polynomial. is there a pivoting rule ensuring that the number of iterations in the worst case is polynomially bounded? It is not even known that the graph of the polyhedron has polynomial diameter (polynomial Hirsch conjecture), ie that the perfectly lucid pivoting rule makes a polynomial number of steps. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 13 / 76

Interior points For all µ > 0, consider the barrier
problem min µ 1c · x m X i=1 log(bi Ai x), bi Ai x > 0 i 2 [m] log strictly concave + bounded feasible set =) optimal solution x(µ) is unique. µ 7! x(µ) is the central path. x(0) is the solution of the LP. x(0) x(1) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 14 / 76

min µ 1c · x m X i=1 log(bi Ai
x), bi Ai x > 0 i 2 [m] Taking @/@xj yields that x(µ) satisﬁes: µ 1cj + m X i=1 Aij bi Ai x = 0, j 2 [n] This deﬁnes an algebraic curve, the central curve. The points (µ, x) on this curve such that µ > 0 determine the central path. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 15 / 76

Inductive step of interior points: x NewtonStep(x, µ); reduce µ
so that x remains in an attraction bassin of Newton’s method. x(0) x(1) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 16 / 76

Inductive step of interior points: x NewtonStep(x, µ); reduce µ
so that x remains in an attraction bassin of Newton’s method. x(0) x(1) “the good convergence properties of Karmarkar’s algorithm arise from good geometric properties of the set of trajectories”, Bayer and Lagarias, 89. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 16 / 76

In order to get a complexity measure of the central
path (independent of the details of the path following method), Dedieu, Malajovich, and Shub considered the total curvature. . . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 17 / 76

Total curvature The total curvature of a path , parametrized
by arc length, so that k 0(s)k = 1, is given by  := Z L 0 k 00(s)kds or  = sup q>2 sup 06 0<···< q 6L \ ( k 1 ) ( k ) ( k+1 ) ↵1 ↵2 ↵3 ↵4 ↵5  > ↵1 + · · · + ↵5 > 5⇡ 2 Intuitively, a weakly curved central path should allow an interior point method to make big steps. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 18 / 76

Continuous analogue of Hirsch’s conjecture Dedieu and Shub (2005) initially
conjectured that the total curvature of the central path is O(n) (n number of variables). This was motivated by a theorem of Dedieu-Malajovich-Shub (2005): total curvature is O(n), averaged over all 2n+m LP’s (cells of the arrangement of hyperplanes), ✏i Ai x 6 bi , ⌘j xj > 0, ✏i , ⌘j = ±1. Illustration from Benchimol’s Phd Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 19 / 76

Deza, Terlaky and Zinchenko (2008) constructed a redundant Klee-Minty cube,
showing that a total curvature exponential in n is possible, and revised the conjecture of Dedieu and Shub: Conjecture (Continuous analogue of Hirsch conjecture, [Deza, Terlaky, and Zinchenko, 2008]) The total curvature of the central path is O(m), where m is the number of constraints. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 20 / 76

Theorem (Allamigeon, Benchimol, SG, Joswig) There is a LP with
2r + 2 variables and 3r + 4 inequalities such that the central path has a total curvature in ⌦(2r ). This disproves the conjecture of Deza, Terlaky and Zinchenko. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 21 / 76

2r + 2 variables and 3r + 4 inequalities such that the central path has a total curvature in ⌦(2r ). This disproves the conjecture of Deza, Terlaky and Zinchenko. Theorem (Allamigeon, Benchimol, SG, Joswig, MPG is “not more di cult” than LP) A strongly polynomial pivoting rule for LP would solve MPG in polynomial time (SIAM Opt, arXiv:1309.5925) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 21 / 76

2r + 2 variables and 3r + 4 inequalities such that the central path has a total curvature in ⌦(2r ). This disproves the conjecture of Deza, Terlaky and Zinchenko. Theorem (Allamigeon, Benchimol, SG, Joswig, MPG is “not more di cult” than LP) A strongly polynomial pivoting rule for LP would solve MPG in polynomial time (SIAM Opt, arXiv:1309.5925) The pivoting rule must satisfy mild technical conditions, in particular, combinatorial rules, depending on signs of minors of ( A b c 0 ) work. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 21 / 76

2r + 2 variables and 3r + 4 inequalities such that the central path has a total curvature in ⌦(2r ). This disproves the conjecture of Deza, Terlaky and Zinchenko. Theorem (Allamigeon, Benchimol, SG, Joswig, MPG is “not more di cult” than LP) A strongly polynomial pivoting rule for LP would solve MPG in polynomial time (SIAM Opt, arXiv:1309.5925) The pivoting rule must satisfy mild technical conditions, in particular, combinatorial rules, depending on signs of minors of ( A b c 0 ) work. Theorem (Allamigeon, Benchimol, SG, ICALP 2014, arXiv:1406.5433) Mean payo↵ games can be solved in polynomial time on average. This is derived as a consequence of the Adler-Karp-Shamir theorem in linear programming. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 21 / 76

Although the word “tropical” appears in none of these statements,
the proofs rely on tropical geometry in an essential way, through linear programming over non-archimedean ﬁelds. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 22 / 76

Part II. Operator approach to mean payo↵ games Stephane Gaubert
(INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 23 / 76

vk i value of MAX, initial state (i, MIN). vk
1 = min( 2 + 1 + vk 1 1 , 8 + max( 3 + vk 1 1 , 12 + vk 1 2 )) vk 2 = 0 + max( 9 + vk 1 1 , 5 + vk 1 2 ) 1 −3 −12 0 5 3 2 1 1 2 −9 MIN MAX −2 −8 v1 = (0, 0) v2 = ( 11, 5) v3 = ( 15, 10) v4 = ( 16, 15) limk vk/k = ( 1, 5) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 24 / 76

Theorem (Shapley) vk = T(vk 1), v0 = 0 .
The map T : Rn ! Rn is called Shapley operator. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 25 / 76

Theorem (Shapley) vk = T(vk 1), v0 = 0 .
The map T : Rn ! Rn is called Shapley operator. [T(v)]j = min i2[m], j!i ✓ rji + max k2[n], i!k (rik + xk ) ◆ Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 25 / 76

[Tk(0)]i is the value of the original game in horizon
k with initial state i. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 26 / 76

[Tk(0)]i is the value of the original game in horizon
k with initial state i. [Tk(u)]i is the value of a modiﬁed game in horizon k with initial state i, in which MAX receives an additional payment of uj in the terminal state j Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 26 / 76

Shapley operators of games are monotone (or order preserving) (M)
: x 6 y =) T(x) 6 T(y) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 27 / 76

: x 6 y =) T(x) 6 T(y) Undiscounted implies additively homogeneous (AH) : T(se + x) = se + T(x), 8s 2 R where e = (1, . . . , 1) is the unit vector. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 27 / 76

: x 6 y =) T(x) 6 T(y) Undiscounted implies additively homogeneous (AH) : T(se + x) = se + T(x), 8s 2 R where e = (1, . . . , 1) is the unit vector. AH and MH implies nonexpansive in the sup-norm (N) : kT(x) T(y)k1 6 kx yk1 , 8s 2 R Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 27 / 76

: x 6 y =) T(x) 6 T(y) Undiscounted implies additively homogeneous (AH) : T(se + x) = se + T(x), 8s 2 R where e = (1, . . . , 1) is the unit vector. AH and MH implies nonexpansive in the sup-norm (N) : kT(x) T(y)k1 6 kx yk1 , 8s 2 R Known axioms in non-linear Markov semigroup / PDE viscosity solutions theory, eg Crandall and Tartar, PAMS 80 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 27 / 76

Theorem (Bewley, Kohlerg 76, Neyman 03) The mean payo↵ vector
lim k!1 Tk(0)/k does exist if T : Rn ! Rn is semi-algebraic and nonexpansive in any norm. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 28 / 76

lim k!1 Tk(0)/k does exist if T : Rn ! Rn is semi-algebraic and nonexpansive in any norm. Same is true if T deﬁnable in a o-minimal structure, eg log-exp (risk sensitive) Bolte, SG, Vigeral, MOR 14. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 28 / 76

lim k!1 Tk(0)/k does exist if T : Rn ! Rn is semi-algebraic and nonexpansive in any norm. Same is true if T deﬁnable in a o-minimal structure, eg log-exp (risk sensitive) Bolte, SG, Vigeral, MOR 14. Finite action space and perfect information implies T piecewise linear. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 28 / 76

Theorem (Akian, SG, Guterman, “subharmonic vectors” IJAC 2012, arXiv:0912.2462) Let
T be the Shapley operator of a deterministic game. The following are equivalent. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 29 / 76

T be the Shapley operator of a deterministic game. The following are equivalent. initial state j is winning, meaning that 0 6 lim k!1 [Tk(0)]j /k Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 29 / 76

T be the Shapley operator of a deterministic game. The following are equivalent. initial state j is winning, meaning that 0 6 lim k!1 [Tk(0)]j /k there exists u 2 (R [ { 1})n, uj 6= 1, and u 6 T(u) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 29 / 76

T be the Shapley operator of a deterministic game. The following are equivalent. initial state j is winning, meaning that 0 6 lim k!1 [Tk(0)]j /k there exists u 2 (R [ { 1})n, uj 6= 1, and u 6 T(u) T : Rn ! Rn extends continuously (R [ { 1})n ! (R [ { 1})n. (Burbanks, Nussbaum, Sparrow). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 29 / 76

Proof idea Assume that u 2 Rn is such that
u 6 T(u). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 30 / 76

u 6 T(u). u 6 T(u) 6 T2(u) 6 · · · Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 30 / 76

u 6 T(u). u 6 T(u) 6 T2(u) 6 · · · u/k 6 Tk(u)/k Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 30 / 76

u 6 T(u). u 6 T(u) 6 T2(u) 6 · · · u/k 6 Tk(u)/k 0 6 limk!1 Tk(u)/k Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 30 / 76

u 6 T(u). u 6 T(u) 6 T2(u) 6 · · · u/k 6 Tk(u)/k 0 6 limk!1 Tk(u)/k kTk(u) Tk(0)k1 6 ku 0k1 = kuk1 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 30 / 76

u 6 T(u). u 6 T(u) 6 T2(u) 6 · · · u/k 6 Tk(u)/k 0 6 limk!1 Tk(u)/k kTk(u) Tk(0)k1 6 ku 0k1 = kuk1 0 6 limk!1 Tk(0)/k Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 30 / 76

u 6 T(u). u 6 T(u) 6 T2(u) 6 · · · u/k 6 Tk(u)/k 0 6 limk!1 Tk(u)/k kTk(u) Tk(0)k1 6 ku 0k1 = kuk1 0 6 limk!1 Tk(0)/k all states are winning. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 30 / 76

The converse follows from a ﬁxed point theorem of Kohlberg
(a nonexpansive piecewise linear map has an invariant half-line). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 31 / 76

Part III. Tropical geometry Stephane Gaubert (INRIA and CMAP) Long

In the tropical world “a + b” = max(a, b)
“a ⇥ b” = a + b The semiﬁeld of scalars is Rmax = R [ { 1}. “2 + 3” = Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 33 / 76

“a ⇥ b” = a + b The semiﬁeld of scalars is Rmax = R [ { 1}. “2 + 3” = 3 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 33 / 76

“a ⇥ b” = a + b The semiﬁeld of scalars is Rmax = R [ { 1}. “2 + 3” = 3 “2 ⇥ 3” = Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 33 / 76

“a ⇥ b” = a + b The semiﬁeld of scalars is Rmax = R [ { 1}. “2 + 3” = 3 “2 ⇥ 3” =5 “0” = 1, “1” = 0. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 33 / 76

“a ⇥ b” = a + b The semiﬁeld of scalars is Rmax = R [ { 1}. “2 + 3” = 3 “2 ⇥ 3” =5 “0” = 1, “1” = 0. For any totally ordered abelian group (G, +, 6), one can deﬁne Gmax . G = (RN, +, 6 lex ) specially useful. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 33 / 76

Tropical modules & tropical convex sets Scalars act on vectors
by “ x” = e + x. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 34 / 76

by “ x” = e + x. V ⇢ Rn max is a submodule, aka tropical convex cone, if for all x, y 2 V , , µ 2 Rmax , “ x + µy” = sup( e + x, µe + y) 2 V . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 34 / 76

by “ x” = e + x. V ⇢ Rn max is a submodule, aka tropical convex cone, if for all x, y 2 V , , µ 2 Rmax , “ x + µy” = sup( e + x, µe + y) 2 V . Since “ > 0” is automatic tropically, modules = cones. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 34 / 76

by “ x” = e + x. V ⇢ Rn max is a submodule, aka tropical convex cone, if for all x, y 2 V , , µ 2 Rmax , “ x + µy” = sup( e + x, µe + y) 2 V . Since “ > 0” is automatic tropically, modules = cones. V is a tropical convex set if the same is true conditionnally to “ + µ = 1”, i.e., max( , µ) = 0. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 34 / 76

By the subharmonic certiﬁcates theorem, the game is winning for
MAX, i.e., 9j, limk!1 [Tk(0)]j /k > 0, i↵ V = {v 2 Rn max | T(v) > v} 6⌘ “0” = ( 1, . . . , 1) . Proposition If T is a Shapley operator, then V is a tropical submodule of Rn max , closed in the Euclidean topology. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 35 / 76

MAX, i.e., 9j, limk!1 [Tk(0)]j /k > 0, i↵ V = {v 2 Rn max | T(v) > v} 6⌘ “0” = ( 1, . . . , 1) . Proposition If T is a Shapley operator, then V is a tropical submodule of Rn max , closed in the Euclidean topology. Proof. T M, u, v 2 V implies T(sup(u, v)) > sup(T(u), T(v)) > sup(u, v). T AH, v 2 V implies T( e + v) = e + T(v) > e + v. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 35 / 76

MAX, i.e., 9j, limk!1 [Tk(0)]j /k > 0, i↵ V = {v 2 Rn max | T(v) > v} 6⌘ “0” = ( 1, . . . , 1) . Proposition If T is a Shapley operator, then V is a tropical submodule of Rn max , closed in the Euclidean topology. Proof. T M, u, v 2 V implies T(sup(u, v)) > sup(T(u), T(v)) > sup(u, v). T AH, v 2 V implies T( e + v) = e + T(v) > e + v. All closed tropical submodules arise from a Shapley operator T (inﬁnite number of actions on one side allowed). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 35 / 76

Tropical adjoints Let A 2 Rm⇥n max , x 2
Rn max , y 2 Rn max (Ax)i = max j2[n] (Aij + xj ), i 2 [m] Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 36 / 76

Rn max , y 2 Rn max (Ax)i = max j2[n] (Aij + xj ), i 2 [m] Ax 6 y () x 6 A]y Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 36 / 76

Rn max , y 2 Rn max (Ax)i = max j2[n] (Aij + xj ), i 2 [m] Ax 6 y () x 6 A]y (A]y)j = min i2[m] ( Aij + yi ), j 2 [n] Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 36 / 76

Rn max , y 2 Rn max (Ax)i = max j2[n] (Aij + xj ), i 2 [m] Ax 6 y () x 6 A]y (A]y)j = min i2[m] ( Aij + yi ), j 2 [n] The adjoint A] is a priori deﬁned as a self-map of the completion (R [ {±1})n of Rn max , but it does preserve Rn as soon as the game has no states without actions. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 36 / 76

Rn max , y 2 Rn max (Ax)i = max j2[n] (Aij + xj ), i 2 [m] Ax 6 y () x 6 A]y (A]y)j = min i2[m] ( Aij + yi ), j 2 [n] The adjoint A] is a priori deﬁned as a self-map of the completion (R [ {±1})n of Rn max , but it does preserve Rn as soon as the game has no states without actions. More on adjoints: Cohen, SG, Quadrat, LAA 04 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 36 / 76

The Shapley operator of a MPG can be written as
[T(v)]j = min i2[m], j!i ✓ Aij + max k2[n], i!k (Bik + vk ) ◆ Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 37 / 76

[T(v)]j = min i2[m], j!i ✓ Aij + max k2[n], i!k (Bik + vk ) ◆ T(v) = A]Bv Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 37 / 76

[T(v)]j = min i2[m], j!i ✓ Aij + max k2[n], i!k (Bik + vk ) ◆ T(v) = A]Bv v 6 T(v) () Av 6 Bv Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 37 / 76

[T(v)]j = min i2[m], j!i ✓ Aij + max k2[n], i!k (Bik + vk ) ◆ T(v) = A]Bv v 6 T(v) () Av 6 Bv v 6 T(v) () max j2[n] (Aij + vj ) 6 max j2[n] (Bij + vj ), i 2 [m] Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 37 / 76

[T(v)]j = min i2[m], j!i ✓ Aij + max k2[n], i!k (Bik + vk ) ◆ T(v) = A]Bv v 6 T(v) () Av 6 Bv v 6 T(v) () max j2[n] (Aij + vj ) 6 max j2[n] (Bij + vj ), i 2 [m] The set of subharmonic certiﬁcates {v | Av 6 Bv} is a tropical convex polyhedral cone. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 37 / 76

Modules of subharmonic vectors x1 x2 x3 x1 x2 x3
states 1,2,3 winning states 2,3 winning Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 38 / 76

Part VI. Tropical convexity Stephane Gaubert (INRIA and CMAP) Long

Tropical half-spaces Given a, b 2 Rn max , a,
b 6⌘ 1, H := {x 2 Rn max | “ax 6 bx”} Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 40 / 76

b 6⌘ 1, H := {x 2 Rn max | max 16i6n ai + xi 6 max 16i6n bi + xi } x2 x1 x3 max(x1, x2, 2 + x3 ) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 40 / 76

b 6⌘ 1, H := {x 2 Rn max | max 16i6n ai + xi 6 max 16i6n bi + xi } x2 x1 x3 max(x1, 2 + x3 ) 6 x2 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 40 / 76

b 6⌘ 1, H := {x 2 Rn max | max 16i6n ai + xi 6 max 16i6n bi + xi } x2 x1 x3 x1 6 max(x2 2 + x3 ) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 40 / 76

b 6⌘ 1, H := {x 2 Rn max | max 16i6n ai + xi 6 max 16i6n bi + xi } x2 x1 x3 max(x2 2 + x3 ) 6 x1 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 40 / 76

A halfspace can always be written as: max i2I ai
+ xi 6 max j2I+ bj + xj , I \ I+ = ? . Apex: vi := max(ai , bi ). If v 2 Rn, H is the union of sectors of the tropical hyperplane with apex v: max 16i6n xi vi attained twice Halfspaces appeared in: Zimmermann 77; Cohen, Quadrat SG 00; Joswig 04; . . . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 41 / 76

Tropical polyhedral cones can be deﬁned as intersections of ﬁnitely
many half-spaces x2 x1 V x3 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 42 / 76

many half-spaces x2 x1 V x3 x2 x1 x3 2 + x1 6 max(x2, 3 + x3) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 42 / 76

many half-spaces V Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 42 / 76

A tropical polytope with four vertices Structure of the polyhedral
complex: Develin, Sturmfels Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 43 / 76

Tropical Minkowski-Weyl Theorem (SG, Katz, Relmics 06, JACO 2011) A
tropical polyhedral convex cone can be written as K = “ cone(Y )” with Y ﬁnite, and vice versa. Here, cone(·) denotes the set of tropical linear combinations. - Inequalities to generators: ﬁniteness can be proved by elimination Butkovic and Hegedus, 84; SG 92, recent improvements tropical double description Allamigeon, SG, Goubault. - generators to inequalities: use tropical separation theorem + the set of valid inequalities is itself a polyhedron (tropical polar; SG, Katz). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 44 / 76

Tropical objects arise by considering non-archimedean valuations. There is a
convenient choice of non-archimedean ﬁeld in tropical geometry . . . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 45 / 76

Puiseux series with real exponents C{{t R}}, f (t) =
X k2N ak tbk were ak 2 C, and the sequence bk 2 R tends to 1. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 46 / 76

X k2N ak tbk were ak 2 C, and the sequence bk 2 R tends to 1. The subﬁeld of series that converge absolutely for |t| large enough is the ﬁeld of generalized Dirichlet series of Hardy and Riesz. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 46 / 76

X k2N ak tbk were ak 2 C, and the sequence bk 2 R tends to 1. The subﬁeld of series that converge absolutely for |t| large enough is the ﬁeld of generalized Dirichlet series of Hardy and Riesz. Dirichlet series P k>1 ak k s correspond to bk = log k, t = exp(s). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 46 / 76

X k2N ak tbk were ak 2 C, and the sequence bk 2 R tends to 1. The subfield of series that converge absolutely for |t| large enough is the field of generalized Dirichlet series of Hardy and Riesz. Dirichlet series P k>1 ak k s correspond to bk = log k, t = exp(s). Real coe cients: totally ordered field wrt pointwise order near t = +1, i.e. lexicographic order on coe↵s. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 46 / 76

X k2N ak tbk were ak 2 C, and the sequence bk 2 R tends to 1. The subfield of series that converge absolutely for |t| large enough is the field of generalized Dirichlet series of Hardy and Riesz. Dirichlet series P k>1 ak k s correspond to bk = log k, t = exp(s). Real coe cients: totally ordered field wrt pointwise order near t = +1, i.e. lexicographic order on coe↵s. K := R{{t R}}cvg , absolutely convergent real generalized Puiseux series constitute a real closed field, van den Dries and Speissegger (TAMS) (definable functions in a variant of the o-minimal structure Ran,⇤ ). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 46 / 76

Alt. choice of ﬁeld: Hahn series (well ordered support). See
Hahn, Ribenboim, van den Dries and Speissegger, Markwig. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 47 / 76

Alt. choice of ﬁeld: Hahn series (well ordered support). See
Hahn, Ribenboim, van den Dries and Speissegger, Markwig. Non-archimedean valuation of f (t) = X k2N ak tbk val f = max k,ak 6=0 bk = lim t!1 | log f (t)| log t val(f + g) 6 max(val f , val g), and = holds if f , g > 0 val(fg) = val f + val g . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 47 / 76

Theorem (Combines Develin and Yu and Allamigeon, Benchimol, SG, Joswig
arXiv:1405.4161) 1 Every tropical polyhedron P can be written as P = val P where P is a polyhedron in Kn + , here K = R{{t R}}cvg . 2 Moreover, P is the uniform (Hausdor↵) limit of logt P := { log z log t | z 2 P} as t ! 1. Related deformation in Briec and Horvath. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 48 / 76

Tropical linear program min“c>x”; “A+x + b+ > A x
+ b ” min max j cj + xj max(max j (A+ ij + xj ), b+ i ) > max(max j (Aij + xj ), bi ) . x1 x2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 49 / 76

Correspondence classical $ tropical LP Theorem (Allamigeon, Benchimol, SG, Joswig,
arXiv:1308.0454, SIAM J. Disc. Math) Suppose that P = {x 2 Kn | Ax + b > 0} is included in the positive orthant of Kn and that the tropicalization of ( A , b ) is sign generic (to be deﬁned soon). Then, val(P) = {x 2 Rn max | “A+x + b+ > A x + b ”} , where (A+ b+) = val( A + b +) and (A b ) = val( A b ). Moreover the classical and tropical polyhedron have the same combinatorics: valuation sends basic points to basic points, edges to edges, etc. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 50 / 76

Correspondence classical $ tropical LP Theorem (Allamigeon, Benchimol, SG, Joswig,
arXiv:1308.0454, SIAM J. Disc. Math) Suppose that P = {x 2 Kn | Ax + b > 0} is included in the positive orthant of Kn and that the tropicalization of ( A , b ) is sign generic (to be deﬁned soon). Then, val(P) = {x 2 Rn max | “A+x + b+ > A x + b ”} , where (A+ b+) = val( A + b +) and (A b ) = val( A b ). Moreover the classical and tropical polyhedron have the same combinatorics: valuation sends basic points to basic points, edges to edges, etc. A point of a tropical polyhedron is basic if it saturates n inequalities. A tropically extreme point (member of a minimal generating family) is basic, but not vice versa. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 50 / 76

(0, 0, 0) (0, 0, 4) (4, 0, 0) (4,
4, 0) (4, 4, 4) Picture from [Allamigeon, Benchimol, Gaubert, and Joswig, 2015a]. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 51 / 76

(t0, t0, t0) (t0, t0, t4) (t4, t0, t0) (t4,
t4, t0) (t4, t4, t4) Picture from [Allamigeon, Benchimol, Gaubert, and Joswig, 2015a]. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 52 / 76

Tropical sign genericity and optimal assignments Let M 2 Rn⇥n
max and ✏ 2 {±1, 0}n⇥n. M 2 Kn⇥n is a lift of (✏, M) if val M = M and sgn M = ✏. We have val X 2Sn sgn( ) Y 16i6n Mi (i) 6 max 2Sn X 16i6n Mi (i) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 53 / 76

max and ✏ 2 {±1, 0}n⇥n. M 2 Kn⇥n is a lift of (✏, M) if val M = M and sgn M = ✏. We have val X 2Sn sgn( ) Y 16i6n Mi (i) 6 max 2Sn X 16i6n Mi (i) with = if (✏, M) is tropically sign generic, meaning that all optimal permutations yield the same sgn( ) Q 16i6n ✏i (i) . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 53 / 76

max and ✏ 2 {±1, 0}n⇥n. M 2 Kn⇥n is a lift of (✏, M) if val M = M and sgn M = ✏. We have val X 2Sn sgn( ) Y 16i6n Mi (i) 6 max 2Sn X 16i6n Mi (i) with = if (✏, M) is tropically sign generic, meaning that all optimal permutations yield the same sgn( ) Q 16i6n ✏i (i) . It is generic if there is only one optimal permutation. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 53 / 76

max and ✏ 2 {±1, 0}n⇥n. M 2 Kn⇥n is a lift of (✏, M) if val M = M and sgn M = ✏. We have val X 2Sn sgn( ) Y 16i6n Mi (i) 6 max 2Sn X 16i6n Mi (i) with = if (✏, M) is tropically sign generic, meaning that all optimal permutations yield the same sgn( ) Q 16i6n ✏i (i) . It is generic if there is only one optimal permutation. Sign-genericity is related to even cycle problem and Polya’s permanent problem. Checkable in polynomial time, as well as genericity (simpler). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 53 / 76

LP tropically sign generic means that every minor of “(A+
A , b+ b )” is sign generic. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 54 / 76

sign generic condition not satisﬁed, valuation does not commute with
the external representation. x1 + x2 6 1, t1x1 + x2 > 1, x1 + t1x2 > 1 Xi = log xi/ log t, t ! 0. max(X1 , X2 ) 6 0, max(1 + X1 , X2 ) > 0, max(X1 , 1 + X2 ) > 0 . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 55 / 76

sign generic condition satisﬁed, valuation does commute with the external
representation. x1 + x2 6 t✏, t1x1 + x2 > 1, x1 + t1x2 > 1, t2x1 > 1, t2x2 > 1 Xi = log xi / log t, t ! 0. max(X1 , X2 )6✏, max(1+X1 , X2 )>0, max(X1 , 1+X2 )>0, X1 >2, X2 >2 . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 55 / 76

sign generic condition satisﬁed, valuation does commute with the external
representation. Xi = log xi/ log t, t ! 0. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 55 / 76

Assume that the data are tropically in general position. Theorem
(Allamigeon, Benchimol, SG, Joswig arXiv:1308.0454, SIAM J. Disc. Math) The valuation of the path of the simplex algorithm over K can be computed tropically (with a compatible pivoting rule). One iteration takes O(n(m + n)) time. Tropical Cramer determinants = opt. assignment used to compute reduce costs. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 56 / 76

Assume that the data are tropically in general position. Theorem
(Allamigeon, Benchimol, SG, Joswig arXiv:1308.0454, SIAM J. Disc. Math) The valuation of the path of the simplex algorithm over K can be computed tropically (with a compatible pivoting rule). One iteration takes O(n(m + n)) time. Tropical Cramer determinants = opt. assignment used to compute reduce costs. Pivoting is more subttle tropically. Tropical general position assumption (only one optimal permutation) stronger than sign general position (all optimal permutations yield monomials with the same sign). The O(n(n + m)) bound arises by tracking the deformations of a hypergraph along a tropical edge. We can still pivot tropically if the data are only in sign general position, but then we (currently) loose a factor n in time. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 56 / 76

Example of compatible pivoting rule. A rule is combinatorial if
any entering/leaving inequalities are functions of the history (sequence of bases) and of the signs of the minors of the matrix M = A b c > 0 . (eg signs of reduced costs). Most known pivoting rules are combinatorial. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 57 / 76

Mean payo↵ games reduces to tropical LP The result of
Akian, SG, Guterman on subharmonic certiﬁcates yields Corollary State i of a mean payo↵ game is winning if the following tropical linear program is feasible “Ax 6 Bx”, “xi > 1” Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 58 / 76

Theorem (Allamigeon, Benchimol, SG, Joswig arXiv:1309.5925, SIAM J. Opt, +
reﬁnement in Benchimol’s PhD) If any combinatorial (or even “semialgebraic”) rule in classical linear programming would run in strongly polynomial time, then, mean payo↵ games could be solved in strongly polynomial time. The “semialgebraic” rule must satisfy a mild technical assumption (polynomial time solvability of LP’s over Newton polytopes). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 59 / 76

Sketch of Proof 1 Mean payo↵ games are equivalent to
feasibility problems in tropical linear programming (Akian, SG, Guterman) 2 Tropical linear programs can be lifted to a subclass of classical linear programs over K. 3 The set of runs (sequences of bases) of the classical simplex algorithm equipped with a combinatorial (or even semialgebraic) pivoting rule is independent of the real closed ﬁeld. Being a run is a ﬁrst order property, apply Tarski’s theorem. So, number of iterations of classical simplex over K is the same as over R. 4 Can simulate the classical simplex on K tropically, every pivot being strongly polynomial. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 60 / 76

Technicalities were hidden here. Instead of K, we eventually use
a ﬁeld of formal Hahn series R[[tRN ]], the value group RN is equipped with lex order to encode a symbolic perturbation scheme (needs to encode a MPG by a LP in general position). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 61 / 76

Part V Tropicalization of the central path Stephane Gaubert (INRIA
and CMAP) Long and winding central paths. . . GDR MOA, Dijon 62 / 76

Primal-dual central path minimize c>x µ n X j=1 log(xj
) m X i=1 log(wi ) subject to Ax + w = b, x > 0, w > 0. (1) Ax + w = b A>y + s = c wi yi = µ for all i 2 [m] xj sj = µ for all j 2 [n] x, w, y, s > 0 . (2) For any µ > 0, 9! (xµ, wµ, yµ, sµ) 2 Rn ⇥ Rm ⇥ Rm ⇥ Rn. The central path is the image of the map C : R >0 ! R2m+2n which sends µ > 0 to the vector (xµ, wµ, yµ, sµ). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 63 / 76

The tropical central path Assume now that A (t), b
(t), c (t) have entries in K (absolutely converging Puiseux series with real exponents, t ! 1). The tropical central path is the log-limit: Ctrop : 7! lim t!1 log C(t ) log t . (3) The pointwise limit does exist since C(·) is deﬁnable in a polynomially bounded o-minimal structure. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 64 / 76

Theorem The family of maps (logt C(t, ·))t converges uniformly
on any closed interval [a, b] ⇢ R to the tropical central path Ctrop . Proof of uniformity uses max(a, b) 6 logt (ta + tb) 6 logt 2 + max(a, b) . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 65 / 76

Computing the tropical central path P := {( x ,
w ) 2 Kn+m | Ax + w = b , x > 0, w > 0} Theorem (Allamigeon, Benchimol, SG, Joswig arXiv:1405.4161) Assume that b , c > 0. Then, for µ = t , val( x µ, w µ) = max val P \ {(x, w) 2 Rm+n max | c>x 6 } . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 66 / 76

Computing the tropical central path P := {( x ,
w ) 2 Kn+m | Ax + w = b , x > 0, w > 0} Theorem (Allamigeon, Benchimol, SG, Joswig arXiv:1405.4161) Assume that b , c > 0. Then, for µ = t , val( x µ, w µ) = max val P \ {(x, w) 2 Rm+n max | c>x 6 } . In particular, the valuation of the analytic center (take = 1) is the maximal element of val P. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 66 / 76

x1 + x2 6 2 t1 x1 6 1 +
t2 x2 t1 x2 6 1 + t3 x1 x1 6 t2 x2 x1, x2 > 0 . (4) Its value val(Pt) 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) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 67 / 76

4 3 2 1 0 4 3 2 1 0
x1 x2 4 3 2 1 0 4 3 2 1 0 x2 Figure : Tropical central paths on the Puiseux polyhedron (4) for the objective function min x1 (left) and min t1 x1 + x2 (right). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 68 / 76

The counter example . . . Stephane Gaubert (INRIA and
CMAP) Long and winding central paths. . . GDR MOA, Dijon 69 / 76

min v0 s.t. u0 6 t1 v0 6 t2 vi
6 t(1 1 2i )( ui 1 + vi 1 ) for 1 6 i 6 r ui 6 t1 ui 1 for 1 6 i 6 r ui 6 t1 vi 1 for 1 6 i 6 r ur > 0, vr > 0 LPr Theorem (Allamigeon, Benchimol, SG, Joswig arXiv:1405.4161) For t large enough, the total curvature of the central path is > (2r 1 1)⇡/2. Large enough: log2 t = ⌦(2r ). Need an exponential number of bits. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 70 / 76

u0 6 t1 u0 6 1 v0 6 t2 v0
6 2 vi 6 t(1 1 2i )(ui 1 + vi 1 ) vi 6 1 1 2i + max(ui 1 , vi 1 ) ui 6 t1 ui 1 ui 6 1 + ui 1 ui 6 t1 vi 1 ui 6 1 + vi 1 ur > 0, vr > 0 c>x = v0 6 The tropical central path is given by u0 = 1 v0 = min(2, ) vi = 1 1 2i + max(ui 1 , vi 1 ) ui = 1 + min(ui 1 , vi 1 ) Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 71 / 76

1 2 3 4 5 u1 v1 u2 v2 u3
v3 u4 v4 Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 72 / 76

How the counter example was found Bezem, Nieuwenhuis and Rodr´
ıguez-Carbonell (2008) constructed a class of tropical linear programs for which an algorithm of Butkoviˇ c and Zimmermann (2006) exhibits an exponential running time. Their algorithm is, loosely speaking, in the family of tropical simplex algorithms. There is a class of tropical LP for which the tropical central path degenerates to a simplex path (moves only on the edges). This is the case on this example. The tropical central path passes through an exponential number of basic points. The central path of Pt converges to the tropical central path as t ! 0 (dequantization). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 73 / 76

How we bound the classical curvature Can deﬁne tropical angle
\t 2 {0, ⇡/2, ⇡}, \tPQR := inf \ PQR , P , Q , R 2 Kn, val P = P, val Q = Q, val R = R Can deﬁne tropical total curvature t := sup k 1 X i=0 \tPi 1 Pi Pi+1 , P0 , . . . , Pk points on the path . ⇡/2 ⇡/2 Total curvature of the classical path is > than the tropical total curvature of its valuation, which is ⌦(2r ) here. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 74 / 76

Concluding remarks Mean payo↵ games are not harder than “pivoting
for LP” Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 75 / 76

for LP” The central path can degenerate to a tropical simplex path and have an exponentially large curvature. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 75 / 76

for LP” The central path can degenerate to a tropical simplex path and have an exponentially large curvature. Got polynomial time solvability result for MPG in average as a consequence of the approach (Allamigeon, Benchimol, SG). Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 75 / 76

for LP” The central path can degenerate to a tropical simplex path and have an exponentially large curvature. Got polynomial time solvability result for MPG in average as a consequence of the approach (Allamigeon, Benchimol, SG). Is the inequality MPG 6 LP strict? Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 75 / 76

for LP” The central path can degenerate to a tropical simplex path and have an exponentially large curvature. Got polynomial time solvability result for MPG in average as a consequence of the approach (Allamigeon, Benchimol, SG). Is the inequality MPG 6 LP strict? More precise complexity results in PhD of Benchimol. Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 75 / 76

for LP” The central path can degenerate to a tropical simplex path and have an exponentially large curvature. Got polynomial time solvability result for MPG in average as a consequence of the approach (Allamigeon, Benchimol, SG). Is the inequality MPG 6 LP strict? More precise complexity results in PhD of Benchimol. Tropical spectraedra, tropical SDP ? current investigation by M. Skomra Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 75 / 76

The tentative conclusion of the story is that “detropicalization” yields
unusual instances, combinatorially tractable: tropicalization is a machine to ﬁnd counter-examples. Thank you ! Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 76 / 76

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