Stephane.Gaubert@inria.fr 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
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
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
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
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
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
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 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 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 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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
: 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
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
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
(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
“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. 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
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
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
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
[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
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
+ 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 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
convenient choice of non-archimedean ﬁeld in tropical geometry . . . Stephane Gaubert (INRIA and CMAP) Long and winding central paths. . . GDR MOA, Dijon 45 / 76
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
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
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
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
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
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
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
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
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
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
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
(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
(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
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
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
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
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
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
) 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
(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
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
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
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 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
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
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
\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
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
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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