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Condition numbers in nonarchimedean semidefinite programming . . . - - PowerPoint PPT Presentation

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Condition numbers in nonarchimedean semidefinite programming . . . and what they say about stochastic mean payoff games Xavier Allamigeon, St ephane Gaubert, Ricardo Katz, Mateusz Skomra INRIA and CMAP, Ecole polytechnique, CNRS January

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Condition numbers in nonarchimedean semidefinite programming . . . and what they say about stochastic mean payoff games

Xavier Allamigeon, St´ ephane Gaubert, Ricardo Katz, Mateusz Skomra

INRIA and CMAP, ´ Ecole polytechnique, CNRS

January 24, 2019, Birmingham

Stephane.Gaubert@inria.fr

Based on : arXiv:1603.06916 and arXiv:1801.02089 (both in J. Symb. Comp.) and arXiv:1610.06746, with Allamigeon and Skomra, and on arXiv:1802.07712 (proc. MTNS) with Allamigeon, Katz and Skomra, and Skomra’s thesis.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 1 / 58

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Feasibility semidefinite programmming problem

Definition (spectrahedron) Given symmetric matrices Q(0), . . . , Q(n) ∈ Rm×m, the associated spectrahedron is defined as S = {x ∈ Rn : Q(0) + x1Q(1) + · · · + xnQ(n) is positive semidefinite} .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 2 / 58

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Feasibility semidefinite programmming problem

Definition (spectrahedron) Given symmetric matrices Q(0), . . . , Q(n) ∈ Rm×m, the associated spectrahedron is defined as S = {x ∈ Rn : Q(0) + x1Q(1) + · · · + xnQ(n) is positive semidefinite} . The semidefinite feasibility problem (SFDP) consists in deciding whether S = ∅.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 2 / 58

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Feasibility semidefinite programmming problem

Definition (spectrahedron) Given symmetric matrices Q(0), . . . , Q(n) ∈ Rm×m, the associated spectrahedron is defined as S = {x ∈ Rn : Q(0) + x1Q(1) + · · · + xnQ(n) is positive semidefinite} . The semidefinite feasibility problem (SFDP) consists in deciding whether S = ∅. The semidefinite programming problem (SDP) consists in minimizing a linear form over S

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 2 / 58

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SDP can be solved in polynomial time by the ellipsoid or interior point methods in a restricted sense.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 3 / 58

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SDP can be solved in polynomial time by the ellipsoid or interior point methods in a restricted sense. We obtain ε-approximate solutions. Complexity bounds: Poly(n, m, log ε, log R, log r, . . . ) , where (R, r, . . . ) are metric estimates of the spectrahedron (log R can be exponential in n).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 3 / 58

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SDP can be solved in polynomial time by the ellipsoid or interior point methods in a restricted sense. We obtain ε-approximate solutions. Complexity bounds: Poly(n, m, log ε, log R, log r, . . . ) , where (R, r, . . . ) are metric estimates of the spectrahedron (log R can be exponential in n). S may not contain any rational points.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 3 / 58

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SDP can be solved in polynomial time by the ellipsoid or interior point methods in a restricted sense. We obtain ε-approximate solutions. Complexity bounds: Poly(n, m, log ε, log R, log r, . . . ) , where (R, r, . . . ) are metric estimates of the spectrahedron (log R can be exponential in n). S may not contain any rational points. The SDP feasibility problem is not known to be in NP (let alone P) in the Turing machine model.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 3 / 58

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SDP can be solved in polynomial time by the ellipsoid or interior point methods in a restricted sense. We obtain ε-approximate solutions. Complexity bounds: Poly(n, m, log ε, log R, log r, . . . ) , where (R, r, . . . ) are metric estimates of the spectrahedron (log R can be exponential in n). S may not contain any rational points. The SDP feasibility problem is not known to be in NP (let alone P) in the Turing machine model. Exact answers to SDFP can be obtained by quantifier elimination or critical points methods.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 3 / 58

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SDP can be solved in polynomial time by the ellipsoid or interior point methods in a restricted sense. We obtain ε-approximate solutions. Complexity bounds: Poly(n, m, log ε, log R, log r, . . . ) , where (R, r, . . . ) are metric estimates of the spectrahedron (log R can be exponential in n). S may not contain any rational points. The SDP feasibility problem is not known to be in NP (let alone P) in the Turing machine model. Exact answers to SDFP can be obtained by quantifier elimination or critical points methods.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 3 / 58

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SDP can be solved in polynomial time by the ellipsoid or interior point methods in a restricted sense. We obtain ε-approximate solutions. Complexity bounds: Poly(n, m, log ε, log R, log r, . . . ) , where (R, r, . . . ) are metric estimates of the spectrahedron (log R can be exponential in n). S may not contain any rational points. The SDP feasibility problem is not known to be in NP (let alone P) in the Turing machine model. Exact answers to SDFP can be obtained by quantifier elimination or critical points methods.

  • E. de Klerk and F. Vallentin. “On the Turing model complexity of interior point

methods for semidefinite programming”. In: SIAM J. Opt. 26.3 (2016),

  • pp. 1944–1961
  • D. Henrion, S. Naldi, and M. Safey El Din. “Exact algorithms for linear matrix

inequalities”. In: SIAM J. Opt. 26.4 (2016), pp. 2512–2539

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 3 / 58

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To better understand SDP over the reals. . .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 4 / 58

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To better understand SDP over the reals. . .

SDP over nonarchimedean fields

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 4 / 58

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To better understand SDP over the reals. . .

SDP over nonarchimedean fields equivalence between nonarchimedean SDP whose input has generic valuation and stochastic mean payoff games with perfect information (a problem in NP ∩ coNP not known to be in P)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 4 / 58

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To better understand SDP over the reals. . .

SDP over nonarchimedean fields equivalence between nonarchimedean SDP whose input has generic valuation and stochastic mean payoff games with perfect information (a problem in NP ∩ coNP not known to be in P) nonarchimedean condition number

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 4 / 58

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To better understand SDP over the reals. . .

SDP over nonarchimedean fields equivalence between nonarchimedean SDP whose input has generic valuation and stochastic mean payoff games with perfect information (a problem in NP ∩ coNP not known to be in P) nonarchimedean condition number use some metric geometry ideas

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 4 / 58

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Generalized Puiseux series

A (formal generalized) Puiseux series is a series of form x = x(t) =

∞

  • i=1

citαi , where the sequence (αi)i ⊂ R is strictly decreasing and either finite or unbounded and ci are real. Includes (generalized) Dirichlet series αi = − log i, t = exp(s). Hardy, Riesz 1915

  • L. van den Dries and P. Speissegger. “The real field with convergent

generalized power series”. In: Transactions of the AMS 350.11 (1998),

  • pp. 4377–4421.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 5 / 58

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Generalized Puiseux series

A (formal generalized) Puiseux series is a series of form x = x(t) =

∞

  • i=1

citαi , where the sequence (αi)i ⊂ R is strictly decreasing and either finite or unbounded and ci are real. Includes (generalized) Dirichlet series αi = − log i, t = exp(s). Hardy, Riesz 1915 The subset of absolutely converging (for t large enough) Puiseux series forms a real closed field, denoted here by K.

  • L. van den Dries and P. Speissegger. “The real field with convergent

generalized power series”. In: Transactions of the AMS 350.11 (1998),

  • pp. 4377–4421.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 5 / 58

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Generalized Puiseux series

A (formal generalized) Puiseux series is a series of form x = x(t) =

∞

  • i=1

citαi , where the sequence (αi)i ⊂ R is strictly decreasing and either finite or unbounded and ci are real. Includes (generalized) Dirichlet series αi = − log i, t = exp(s). Hardy, Riesz 1915 The subset of absolutely converging (for t large enough) Puiseux series forms a real closed field, denoted here by K. We say that x y if x(t) y(t) for all t large enough. This is a linear order on K.

  • L. van den Dries and P. Speissegger. “The real field with convergent

generalized power series”. In: Transactions of the AMS 350.11 (1998),

  • pp. 4377–4421.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 5 / 58

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Definition (SDFP over Puiseux series) Given symmetric matrices Q(0), Q(1), . . . , Q(n), denote Q(x) = Q(0) + x1Q(1) + · · · + xnQ(n) . Decide if the following spectrahedron is empty S = {x ∈ Kn

0 : Q(x) is positive semidefinite}

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 6 / 58

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Definition (SDFP over Puiseux series) Given symmetric matrices Q(0), Q(1), . . . , Q(n), denote Q(x) = Q(0) + x1Q(1) + · · · + xnQ(n) . Decide if the following spectrahedron is empty S = {x ∈ Kn

0 : Q(x) is positive semidefinite}

Proposition S = ∅ iff for all t large enough, the following real spectrahedron is non-empty S(t) = {x ∈ Rn

0 : Q(0)(t)+x1Q(1)(t)+· · ·+xnQ(n)(t) is pos. semidef.}

  • Proof. K is the field of germs of univariate functions definable in a
  • -minimal structure.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 6 / 58

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Theorem (Allamigeon, SG, Skomra) There is a correspondence between nonarchimedean semidefinite programming problems and zero-sum stochastic games with perfect

  • information. If the valuations of the matrices Q(i) are generic,

feasibility holds iff Player Max wins the game.

  • X. Allamigeon, S. Gaubert, and M. Skomra. “Solving Generic

Nonarchimedean Semidefinite Programs Using Stochastic Game Algorithms”. In: Journal of Symbolic Computation 85 (2018), pp. 25–54. doi: 10.1016/j.jsc.2017.07.002. eprint: 1603.06916.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 7 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 .

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information.

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Take the spectrahedral cone Q(x) :=   tx3 −x1 −t3/4x3 −x1 t−1x1 + t−5/4x3 − x2 −x3 −t3/4x3 −x3 t9/4x2   0 . We associate with Q(x) a stochastic game with perfect information. Circles: Min plays, Square: Max plays, Bullet: Nature flips coin, Payments made by Min to Max Max is winning implies that the cone is nontrivial, and yields a feasible point (t1.06, t0.02, t1.13).

x3 x3 x1 x2 1 2 3 9/4 9/4 −5/4 −1 −1 1 −3/4 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 8 / 58

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Benchmark

We tested our method on randomly chosen matrices Q(1), . . . , Q(n) ∈ Km×m with positive entries on diagonals and no zero entries. We used the value iteration algorithm.

(n, m) (50, 10) (50, 40) (50, 50) (50, 100) (50, 1000) time 0.000065 0.000049 0.000077 0.000279 0.026802 (n, m) (100, 10) (100, 15) (100, 80) (100, 100) (100, 1000) time 0.000025 0.000270 0.000366 0.000656 0.053944 (n, m) (1000, 10) (1000, 50) (1000, 100) (1000, 200) (1000, 500) time 0.000233 0.073544 0.015305 0.027762 0.148714 (n, m) (2000, 10) (2000, 70) (2000, 100) (10000, 150) (10000, 400) time 0.000487 1.852221 0.087536 19.919844 2.309174 Table: Execution time (in sec.) of Procedure CheckFeasibility on random instances.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 9 / 58

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Experimental phase transition for random nonarchimedean SDP

n = # variables, m = size matrices

200 400 600 800 1,000 20 40 60

n m

0 % 20 % 40 % 60 % 80 % 100 %

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 10 / 58

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The present work on tropical condition numbers grew to explain this picture.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 11 / 58

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Valuation of Puiseux series

x = x(t) =

∞

  • k=1

cktαk val(x) = lim

t→∞

log |x(t)| log t = α1 (and val(0) = −∞) .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 12 / 58

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Valuation of Puiseux series

x = x(t) =

∞

  • k=1

cktαk val(x) = lim

t→∞

log |x(t)| log t = α1 (and val(0) = −∞) . Lemma Suppose that x, y ∈ Kn

  • 0. Then

x y = ⇒ val(x) val(y) val(x + y) = max(val(x), val(y)) val(xy) = val(x) + val(y).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 12 / 58

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Valuation of Puiseux series

x = x(t) =

∞

  • k=1

cktαk val(x) = lim

t→∞

log |x(t)| log t = α1 (and val(0) = −∞) . Lemma Suppose that x, y ∈ Kn

  • 0. Then

x y = ⇒ val(x) val(y) val(x + y) = max(val(x), val(y)) val(xy) = val(x) + val(y). Thus, val is a morphism from K0 to a semifield of characteristic

  • ne, the tropical semifield T := (R ∪ {−∞}, max, +).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 12 / 58

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Tropical spectrahedra

Definition Suppose that S is a spectrahedron in Kn

  • 0. Then we say that val(S)

is a tropical spectrahedron. How can we study these creatures?

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 13 / 58

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A S ⊂ Kn is basic semialgebraic if S = {(x1, . . . , xn) ∈ Kn : Pi(x1, . . . , xn) ⋄ 0, ⋄ ∈ {>, =}, ∀i ∈ [q]} where P1, . . . , Pq ∈ K[x1, . . . , xn]. A semialgebraic set is a finite union of basic semialgebraic sets.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 14 / 58

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A S ⊂ Kn is basic semialgebraic if S = {(x1, . . . , xn) ∈ Kn : Pi(x1, . . . , xn) ⋄ 0, ⋄ ∈ {>, =}, ∀i ∈ [q]} where P1, . . . , Pq ∈ K[x1, . . . , xn]. A semialgebraic set is a finite union of basic semialgebraic sets. A set S ⊂ Rn is basic semilinear if it is of the form S = {(x1, . . . , xn) ∈ Rn : ℓi(x1, . . . , xn) ⋄ h(i), ⋄ ∈ {>, =}, ∀i ∈ [q]} where ℓ1, . . . , ℓq are linear forms with integer coefficients, h(1), . . . , h(q) ∈ R. A semilinear set is a finite union of basic semilinear sets.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 14 / 58

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A S ⊂ Kn is basic semialgebraic if S = {(x1, . . . , xn) ∈ Kn : Pi(x1, . . . , xn) ⋄ 0, ⋄ ∈ {>, =}, ∀i ∈ [q]} where P1, . . . , Pq ∈ K[x1, . . . , xn]. A semialgebraic set is a finite union of basic semialgebraic sets. A set S ⊂ Rn is basic semilinear if it is of the form S = {(x1, . . . , xn) ∈ Rn : ℓi(x1, . . . , xn) ⋄ h(i), ⋄ ∈ {>, =}, ∀i ∈ [q]} where ℓ1, . . . , ℓq are linear forms with integer coefficients, h(1), . . . , h(q) ∈ R. A semilinear set is a finite union of basic semilinear sets. Theorem (Alessandrini, Adv. in Geom. 2013) If S ⊂ Kn

>0 is semi-algebraic, then val(S) ⊂ Rn is semilinear and it is

closed.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 14 / 58

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A S ⊂ Kn is basic semialgebraic if S = {(x1, . . . , xn) ∈ Kn : Pi(x1, . . . , xn) ⋄ 0, ⋄ ∈ {>, =}, ∀i ∈ [q]} where P1, . . . , Pq ∈ K[x1, . . . , xn]. A semialgebraic set is a finite union of basic semialgebraic sets. A set S ⊂ Rn is basic semilinear if it is of the form S = {(x1, . . . , xn) ∈ Rn : ℓi(x1, . . . , xn) ⋄ h(i), ⋄ ∈ {>, =}, ∀i ∈ [q]} where ℓ1, . . . , ℓq are linear forms with integer coefficients, h(1), . . . , h(q) ∈ R. A semilinear set is a finite union of basic semilinear sets. Theorem (Alessandrini, Adv. in Geom. 2013) If S ⊂ Kn

>0 is semi-algebraic, then val(S) ⊂ Rn is semilinear and it is

closed. Constructive version in Allamigeon, SG, Skomra arXiv:1610.06746 using Denef-Pas quantifier elimination in valued fields.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 14 / 58

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SLIDE 45

S := val(S) is tropically convex max(α, β) = 0, u, v ∈ S = ⇒ sup(αe + u, βe + v) ∈ S , where e = (1, . . . , 1)⊤.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 15 / 58

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SLIDE 46

S := val(S) is tropically convex max(α, β) = 0, u, v ∈ S = ⇒ sup(αe + u, βe + v) ∈ S , where e = (1, . . . , 1)⊤.

Figure: Tropical spectrahedron.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 15 / 58

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SLIDE 47

Theorem (Semi-algebraic version of Kapranov theorem, Allamigeon, SG, Skomra arXiv:1610.06746) Consider a collection of m regions delimited by hypersurfaces: Si := {x ∈ Kn

0 | P− i (x) P+ i (x)},

i ∈ [m] where P±

i = α p± i,αxα ∈ K0[x], and let

Si := {x ∈ Rn | max

α (val p− i,α + α, x) max α (val p+ i,α + α, x)}

Then val(

  • i∈[m]

Si) ⊂

  • i∈[m]

val(Si) ⊂

  • i∈[m]

Si and the equality holds if

i∈[m] Si is the closure of its interior; in

particular if the valuations val p±

i,α are generic.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 16 / 58

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SLIDE 48

Example 1. S = {x ∈ K3

>0 | x2 1 tx2 + t4x2x3}

val S = {x ∈ R3 | 2x1 max(1 + x2, 4 + x2 + x3)} Example 2.

Figure: This set is the closure of its interior.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 17 / 58

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SLIDE 49

The correspondence between stochastic mean payoff games and nonarchimedean spectrahedra explained

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 18 / 58

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SLIDE 50

Stochastic mean payoff games

Two player, Min and Max, and a half-player, Nature, move a token

  • n a digraph, alternating moves in a cyclic way:

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 19 / 58

slide-51
SLIDE 51

Stochastic mean payoff games

Two player, Min and Max, and a half-player, Nature, move a token

  • n a digraph, alternating moves in a cyclic way:

If the current state i belongs to Player Min, this player chooses and arc i → j, and receives Aji from Player Max.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 19 / 58

slide-52
SLIDE 52

Stochastic mean payoff games

Two player, Min and Max, and a half-player, Nature, move a token

  • n a digraph, alternating moves in a cyclic way:

If the current state i belongs to Player Min, this player chooses and arc i → j, and receives Aji from Player Max. The current state j now belongs to the half-player Nature, Nature throws a dice and next state becomes r with probability Pjr.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 19 / 58

slide-53
SLIDE 53

Stochastic mean payoff games

Two player, Min and Max, and a half-player, Nature, move a token

  • n a digraph, alternating moves in a cyclic way:

If the current state i belongs to Player Min, this player chooses and arc i → j, and receives Aji from Player Max. The current state j now belongs to the half-player Nature, Nature throws a dice and next state becomes r with probability Pjr. The current state r now belongs Player Max, this player chosses an arc r → s, and receives Brs from Player Max.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 19 / 58

slide-54
SLIDE 54

Stochastic mean payoff games

Two player, Min and Max, and a half-player, Nature, move a token

  • n a digraph, alternating moves in a cyclic way:

If the current state i belongs to Player Min, this player chooses and arc i → j, and receives Aji from Player Max. The current state j now belongs to the half-player Nature, Nature throws a dice and next state becomes r with probability Pjr. The current state r now belongs Player Max, this player chosses an arc r → s, and receives Brs from Player Max. the current state s now belongs to Player Min, and so on.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 19 / 58

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SLIDE 55

If Min/Max play k turns according to strategies σ, τ, the payment of the game starting from state i ∈ [n] := {Min states} is denoted by Rk

i (σ, τ).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 20 / 58

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SLIDE 56

If Min/Max play k turns according to strategies σ, τ, the payment of the game starting from state i ∈ [n] := {Min states} is denoted by Rk

i (σ, τ).

v k

i is the value of the game in horizon k, starting from state i, and

σ∗, τ ∗ are optimal strategies if ERk

i (σ∗, τ) v k i = ERk i (σ∗, τ ∗) ERk i (σ, τ ∗),

∀σ, τ

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 20 / 58

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SLIDE 57

If Min/Max play k turns according to strategies σ, τ, the payment of the game starting from state i ∈ [n] := {Min states} is denoted by Rk

i (σ, τ).

v k

i is the value of the game in horizon k, starting from state i, and

σ∗, τ ∗ are optimal strategies if ERk

i (σ∗, τ) v k i = ERk i (σ∗, τ ∗) ERk i (σ, τ ∗),

∀σ, τ Theorem (Shapley) v k

i =

min

j∈Nature states

  • −Aji+
  • r∈Max states

Pjr max

s∈Min states(Brs+v k−1 s

)

  • , v 0 ≡ 0

v k = F(v k−1), F : Rn → Rn Shapley operator

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 20 / 58

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SLIDE 58

If Min/Max play k turns according to strategies σ, τ, the payment of the game starting from state i ∈ [n] := {Min states} is denoted by Rk

i (σ, τ).

v k

i is the value of the game in horizon k, starting from state i, and

σ∗, τ ∗ are optimal strategies if ERk

i (σ∗, τ) v k i = ERk i (σ∗, τ ∗) ERk i (σ, τ ∗),

∀σ, τ Theorem (Shapley) v k

i =

min

j∈Nature states

  • −Aji+
  • r∈Max states

Pjr max

s∈Min states(Brs+v k−1 s

)

  • , v 0 ≡ 0

v k = F(v k−1), F : Rn → Rn Shapley operator F(x) = (−A⊤) ⊙min,+ (P × (B ⊙max,+ x)) = A♯ ◦ P ◦ B(x)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 20 / 58

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SLIDE 59

The mean payoff vector ¯ v := lim

k→∞ v k/k = lim k→∞ F k(0)/k ∈ Rn

does exist and it is achieved by positional stationnary strategies (coro

  • f Kohlberg 1980).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 21 / 58

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SLIDE 60

The mean payoff vector ¯ v := lim

k→∞ v k/k = lim k→∞ F k(0)/k ∈ Rn

does exist and it is achieved by positional stationnary strategies (coro

  • f Kohlberg 1980).

Mean payoff games: compute the mean payoff vector

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 21 / 58

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SLIDE 61

The mean payoff vector ¯ v := lim

k→∞ v k/k = lim k→∞ F k(0)/k ∈ Rn

does exist and it is achieved by positional stationnary strategies (coro

  • f Kohlberg 1980).

Mean payoff games: compute the mean payoff vector We say that the mean payoff game with initial state i is (weakly) winning for Max if limk v k

i /k 0.

Gurvich, Karzanov and Khachyan asked in 1988 whether the determinisitic version is in P. Still open. Their argument implies membership in NP ∩ coNP, see also Zwick, Paterson. Same is true in the stochastic case (Condon).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 21 / 58

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SLIDE 62

Collatz-Wielandt property / winning certificates

T := R ∪ {−∞}, Theorem (Akian, SG, Guterman IJAC 2912, coro of Nussbaum) max

i∈n ¯

vi = cw(R) cw(F) := max

  • λ ∈ R | ∃x ∈ Tn, x ≡ −∞: λe + x F(x)
  • Corollary

Player Max has at least one winning state (i.e., 0 maxi ¯ vi) iff ∃x ∈ Tn, x ≡ −∞, x F(x)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 22 / 58

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SLIDE 63

Definition A square matrix is called a Metzler matrix if its off-diagonal entries are nonpositive.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 23 / 58

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SLIDE 64

Definition A square matrix is called a Metzler matrix if its off-diagonal entries are nonpositive. We suppose Q(1), . . . , Q(n) ∈ Km×m are Metzler — the general case will reduce to this one.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 23 / 58

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SLIDE 65

Definition A square matrix is called a Metzler matrix if its off-diagonal entries are nonpositive. We suppose Q(1), . . . , Q(n) ∈ Km×m are Metzler — the general case will reduce to this one. Want to decide whether Q(x) = x1Q(1) + · · · + xnQ(n) 0 for some x ∈ Kn

0, x = 0.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 23 / 58

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SLIDE 66

If Q 0 is a m × m symmetric matrix, then, the 1 × 1 and 2 × 2 principal minors of Q are nonnegative: Qii 0, QiiQjj Q2

ij.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 24 / 58

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SLIDE 67

If Q 0 is a m × m symmetric matrix, then, the 1 × 1 and 2 × 2 principal minors of Q are nonnegative: Qii 0, QiiQjj Q2

ij.

Is there a “converse”?

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 24 / 58

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SLIDE 68

If Q 0 is a m × m symmetric matrix, then, the 1 × 1 and 2 × 2 principal minors of Q are nonnegative: Qii 0, QiiQjj Q2

ij.

Is there a “converse”? Lemma Assume that Qii 0, QiiQjj (m − 1)2Q2

  • ij. Then Q 0.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 24 / 58

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SLIDE 69

If Q 0 is a m × m symmetric matrix, then, the 1 × 1 and 2 × 2 principal minors of Q are nonnegative: Qii 0, QiiQjj Q2

ij.

Is there a “converse”? Lemma Assume that Qii 0, QiiQjj (m − 1)2Q2

  • ij. Then Q 0.

Proof. Can assume that Qii ≡ 1 (consider diag(Q)−1/2Q diag(Q)−1/2). Then, |Qij| 1/(m − 1), and so Qii

j=i |Qij| implies Q 0.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 24 / 58

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SLIDE 70

If Q 0 is a m × m symmetric matrix, then, the 1 × 1 and 2 × 2 principal minors of Q are nonnegative: Qii 0, QiiQjj Q2

ij.

Is there a “converse”? Lemma Assume that Qii 0, QiiQjj (m − 1)2Q2

  • ij. Then Q 0.

Proof. Can assume that Qii ≡ 1 (consider diag(Q)−1/2Q diag(Q)−1/2). Then, |Qij| 1/(m − 1), and so Qii

j=i |Qij| implies Q 0.

Archimedean modification of Yu’s theorem, that the image by the nonarchimedean valuation of the SDP cone is given by 1 × 1 and 2 × 2 minor conditions.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 24 / 58

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SLIDE 71

Let S := {x ∈ Kn

0 : Q(x) 0}

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 25 / 58

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SLIDE 72

Let S := {x ∈ Kn

0 : Q(x) 0}

Let Sout be defined by the 1 × 1 and 2 × 2 principal minor conditions Qii(x) 0, Qii(x)Qjj(x) (Qij(x))2

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 25 / 58

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SLIDE 73

Let S := {x ∈ Kn

0 : Q(x) 0}

Let Sout be defined by the 1 × 1 and 2 × 2 principal minor conditions Qii(x) 0, Qii(x)Qjj(x) (Qij(x))2 Let Sin be defined by the reinforced minor conditions Qii(x) 0, Qii(x)Qjj(x) (m − 1)2(Qij(x))2

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 25 / 58

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SLIDE 74

Let S := {x ∈ Kn

0 : Q(x) 0}

Let Sout be defined by the 1 × 1 and 2 × 2 principal minor conditions Qii(x) 0, Qii(x)Qjj(x) (Qij(x))2 Let Sin be defined by the reinforced minor conditions Qii(x) 0, Qii(x)Qjj(x) (m − 1)2(Qij(x))2 Theorem (Allamigeon, SG, Skomra) Sin ⊆ S ⊆ Sout and if Q is tropically generic (valuations of coeffs are generic), val(Sin) = val(S) = val(Sout) .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 25 / 58

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SLIDE 75

Let S := {x ∈ Kn

0 : Q(x) 0}

Let Sout be defined by the 1 × 1 and 2 × 2 principal minor conditions Qii(x) 0, Qii(x)Qjj(x) (Qij(x))2 Let Sin be defined by the reinforced minor conditions Qii(x) 0, Qii(x)Qjj(x) (m − 1)2(Qij(x))2 Theorem (Allamigeon, SG, Skomra) Sin ⊆ S ⊆ Sout and if Q is tropically generic (valuations of coeffs are generic), val(Sin) = val(S) = val(Sout) . We show that if X = ∩k{x | Pk(x) 0}, then val X = ∩k val{x | Pk(x) 0} if the polynomials Pk are tropically generic

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 25 / 58

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SLIDE 76

Let S := {x ∈ Kn

0 : Q(x) 0}

Let Sout be defined by the 1 × 1 and 2 × 2 principal minor conditions Qii(x) 0, Qii(x)Qjj(x) (Qij(x))2 Let Sin be defined by the reinforced minor conditions Qii(x) 0, Qii(x)Qjj(x) (m − 1)2(Qij(x))2 Theorem (Allamigeon, SG, Skomra) Sin ⊆ S ⊆ Sout and if Q is tropically generic (valuations of coeffs are generic), val(Sin) = val(S) = val(Sout) . We show that if X = ∩k{x | Pk(x) 0}, then val X = ∩k val{x | Pk(x) 0} if the polynomials Pk are tropically generic (apply semi-algebraic version of Kapranov theorem)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 25 / 58

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SLIDE 77

Can we describe combinatorially val S?

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 26 / 58

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SLIDE 78

Suppose Qii(x) 0, write Qii = Q+

ii − Q− ii .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 27 / 58

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SLIDE 79

Suppose Qii(x) 0, write Qii = Q+

ii − Q− ii .

Then val Q+

ii (x) val Q− ii (x)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 27 / 58

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SLIDE 80

Suppose Qii(x) 0, write Qii = Q+

ii − Q− ii .

Then val Q+

ii (x) val Q− ii (x)

Moreover, if Qii(x)Qjj(x) (Qij(x))2

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 27 / 58

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SLIDE 81

Suppose Qii(x) 0, write Qii = Q+

ii − Q− ii .

Then val Q+

ii (x) val Q− ii (x)

Moreover, if Qii(x)Qjj(x) (Qij(x))2 then Q+

ii (x)Q+ jj (x) + Q− ii (x)Q− jj (x) Q+ ii (x)Q− jj (x) + Q− ii (x)Q+ jj (x)

+ (Qij(x))2

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 27 / 58

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SLIDE 82

Suppose Qii(x) 0, write Qii = Q+

ii − Q− ii .

Then val Q+

ii (x) val Q− ii (x)

Moreover, if Qii(x)Qjj(x) (Qij(x))2 then Q+

ii (x)Q+ jj (x) + Q− ii (x)Q− jj (x) Q+ ii (x)Q− jj (x) + Q− ii (x)Q+ jj (x)

+ (Qij(x))2 and so val Q+

ii (x) + val Q+ jj (x) 2 val Qij(x)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 27 / 58

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SLIDE 83

Tropical Metzler spectrahedra

Theorem (tropical Metzler spectrahedra) For tropically generic Metzler matrices (Q(k))k the set val(S) is described by the tropical minor inequalities of order 1 and 2, ∀i, max

Q(k)

ii >0

(xk + val(Q(k)

ii )) max Q(l)

jj <0

(xl + val(Q(l)

jj ))

and ∀i = j, max

Q(k)

ii >0

(xk + val(Q(k)

ii )) + max Q(k)

jj >0

(xk + val(Q(k)

jj ))

2 max

Q(l)

ij <0

(xl + val(Q(l)

ij )) .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 28 / 58

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SLIDE 84

Tropical Metzler spectrahedra

Theorem (tropical Metzler spectrahedra) For tropically generic Metzler matrices (Q(k))k the set val(S) is described by the tropical minor inequalities of order 1 and 2, ∀i, max

Q(k)

ii >0

(xk + val(Q(k)

ii )) max Q(l)

jj <0

(xl + val(Q(l)

jj ))

and ∀i = j, max

Q(k)

ii >0

(xk + val(Q(k)

ii )) + max Q(k)

jj >0

(xk + val(Q(k)

jj ))

2 max

Q(l)

ij <0

(xl + val(Q(l)

ij )) .

Extends the characterization of val(SDPCONE) by Yu. .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 28 / 58

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SLIDE 85

From spectrahedra to Shapley operators

Lemma The set val(S) can be equivalently defined as the set of all x such that for all k we have xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

In other words, we have val(S) = {x ∈ (R ∪ {−∞})n : x F(x)} , where F is a Shapley operator of a stochastic mean payoff game. We denote this game by Γ.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 29 / 58

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SLIDE 86

Reading the Game on the Shapley Operator

xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 30 / 58

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SLIDE 87

Reading the Game on the Shapley Operator

xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

MIN wants to show infeasibility, MAX feasibility

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 30 / 58

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SLIDE 88

Reading the Game on the Shapley Operator

xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

MIN wants to show infeasibility, MAX feasibility state of MIN, xk, 1 k n

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 30 / 58

slide-89
SLIDE 89

Reading the Game on the Shapley Operator

xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

MIN wants to show infeasibility, MAX feasibility state of MIN, xk, 1 k n MIN chooses {i, j}, 1 i = j m or {i} with Qk

ii < 0, MAX

pays to MIN val Q(k)

ij

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 30 / 58

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SLIDE 90

Reading the Game on the Shapley Operator

xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

MIN wants to show infeasibility, MAX feasibility state of MIN, xk, 1 k n MIN chooses {i, j}, 1 i = j m or {i} with Qk

ii < 0, MAX

pays to MIN val Q(k)

ij

NATURE throws a dice to decide whether i or j is the next state

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 30 / 58

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SLIDE 91

Reading the Game on the Shapley Operator

xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

MIN wants to show infeasibility, MAX feasibility state of MIN, xk, 1 k n MIN chooses {i, j}, 1 i = j m or {i} with Qk

ii < 0, MAX

pays to MIN val Q(k)

ij

NATURE throws a dice to decide whether i or j is the next state suppose next state of MAX, i, 1 i m,

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 30 / 58

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SLIDE 92

Reading the Game on the Shapley Operator

xk min

Q(k)

ij <0

  • − val(Q(k)

ij ) + 1

2

  • max

Q(l)

ii >0

(val(Q(l)

ii ) + xl)

+ max

Q(l)

jj >0

(val(Q(l)

jj ) + xl)

  • .

MIN wants to show infeasibility, MAX feasibility state of MIN, xk, 1 k n MIN chooses {i, j}, 1 i = j m or {i} with Qk

ii < 0, MAX

pays to MIN val Q(k)

ij

NATURE throws a dice to decide whether i or j is the next state suppose next state of MAX, i, 1 i m, MAX moves to xl such that Q(l)

ii > 0, MIN pays to MAX

val Q(l)

ii .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 30 / 58

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SLIDE 93

Main example revisited

Q(1) :=   −1 −1 t−1   , Q(2) :=   −1 t9/4   , Q(3) :=   t −t3/4 t−5/4 −1 −t3/4 −1   . x3 x1 x2 1 2 3

9/4 9/4 −5/4 −5/4 −1 −1 1 1 −3/4

Construction of Γ We construct Γ as follows:

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 31 / 58

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SLIDE 94

Main example revisited

Q(1) :=   −1 −1 t−1   , Q(2) :=   −1 t9/4   , Q(3) :=   t −t3/4 t−5/4 −1 −t3/4 −1   . x3 x1 x2 1 2 3

9/4 9/4 −5/4 −5/4 −1 −1 1 1 −3/4

Construction of Γ The number of matrices (here: 3) defines the number of states controlled by Player Min.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 31 / 58

slide-95
SLIDE 95

Main example revisited

Q(1) :=   −1 −1 t−1   , Q(2) :=   −1 t9/4   , Q(3) :=   t −t3/4 t−5/4 −1 −t3/4 −1   . x3 x1 x2 1 2 3

9/4 9/4 −5/4 −5/4 −1 −1 1 1 −3/4

Construction of Γ The size of matrices (here: 3 × 3) defines the number of states controlled by Player Max (here: 3).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 31 / 58

slide-96
SLIDE 96

Main example revisited

Q(1) :=   −1 −1 t−1   , Q(2) :=   −1 t9/4   , Q(3) :=   t −t3/4 t−5/4 −1 −t3/4 −1   . x3 x1 x2 1 2 3

9/4 9/4 −5/4 −5/4 −1 −1 1 1 −3/4

Construction of Γ If Q(k)

ii

is negative, then Player Min can move from state k to state i. After this move Player Max receives − val(Q(k)

ii ).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 31 / 58

slide-97
SLIDE 97

Main example revisited

Q(1) :=   −1 −1 t−1   , Q(2) :=   −1 t9/4   , Q(3) :=   t −t3/4 t−5/4 −1 −t3/4 −1   . x3 x1 x2 1 2 3

9/4 9/4 −5/4 −5/4 −1 −1 1 1 −3/4

Construction of Γ If Q(k)

ii

is positive, then Player Max can move from state i to state k. After this move Player Max receives val(Q(k)

ii ).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 31 / 58

slide-98
SLIDE 98

Main example revisited

Q(1) :=   −1 −1 t−1   , Q(2) :=   −1 t9/4   , Q(3) :=   t −t3/4 t−5/4 −1 −t3/4 −1   . x3 x1 x2 1 2 3

9/4 9/4 −5/4 −5/4 −1 −1 1 1 −3/4

Construction of Γ If Q(k)

ij

is nonzero, i = j, then Player Min have a coin-toss move from state k to states (i, j) and Player Max receives − val(Q(k)

ij ).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 31 / 58

slide-99
SLIDE 99

Main example revisited

Q(1) :=   −1 −1 t−1   , Q(2) :=   −1 t9/4   , Q(3) :=   t −t3/4 t−5/4 −1 −t3/4 −1   . x3 x1 x2 1 2 3

9/4 9/4 −5/4 −5/4 −1 −1 1 1 −3/4

Construction of Γ If Q(k)

ij

is nonzero, i = j, then Player Min have a coin-toss move from state k to states (i, j) and Player Max receives − val(Q(k)

ij ).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 31 / 58

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SLIDE 100

Example

There is only one pair of optimal policies 3 →

  • 1 , 3
  • ,

2 → 1 . 3 1 2 1 2 3

9/4 −5/4 −1 1 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 32 / 58

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SLIDE 101

Example

There is only one pair of optimal policies 3 →

  • 1 , 3
  • ,

2 → 1 . The value equals 3/40 > 0. 3 1 2 1 2 3

9/4 −5/4 −1 1 −3/4

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 32 / 58

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SLIDE 102

Example

There is only one pair of optimal policies 3 →

  • 1 , 3
  • ,

2 → 1 . The value equals 3/40 > 0. 3 1 2 1 2 3

9/4 −5/4 −1 1 −3/4

Corollary The spectrahedral cone S has a nontrivial point in the positive

  • rthant K3

0.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 32 / 58

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SLIDE 103

Example

The Shapley operator is given by F(x) = (x1 + x3 2 , x1−1, x2 + x3 2 +7 8) and u = (1.06, 0.02, 1.13) is a bias vector, F(u) = λe + u, λ= value 3 1 2 1 2 3

9/4 −5/4 −1 1 −3/4

Corollary The spectrahedral cone S has a nontrivial point in the positive

  • rthant K3
  • 0. For example, it contains the point (t1.06, t0.02, t1.13).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 32 / 58

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SLIDE 104

Tropical analogue of Helton-Nie conjecture

Helton-Nie conjectured that every convex semialgebraic set is the projection of a spectrahedron.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 33 / 58

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SLIDE 105

Tropical analogue of Helton-Nie conjecture

Helton-Nie conjectured that every convex semialgebraic set is the projection of a spectrahedron. Scheiderer (SIAGA, 2018) showed that the cone of nonnegative forms

  • f degree 2d in n variables is not representable in this way unless

2d = 2 or n 2 or (n, 2d) = (3, 4), disproving the conjecture. His result implies the conjecture is also false over K. However. . .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 33 / 58

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SLIDE 106

Tropical analogue of Helton-Nie conjecture, cont.

Theorem (Allamigeon, Gaubert, and Skomra, MEGA2017+JSC.) Fix a set S ⊂ Rn. TFAE S is the image by val of a convex semialgebraic set of Kn

>0

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 34 / 58

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SLIDE 107

Tropical analogue of Helton-Nie conjecture, cont.

Theorem (Allamigeon, Gaubert, and Skomra, MEGA2017+JSC.) Fix a set S ⊂ Rn. TFAE S is the image by val of a convex semialgebraic set of Kn

>0

S is the image by val of the image by proj : Rp → Rn (p n) of a spectrahedron of Kp

>0

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 34 / 58

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SLIDE 108

Tropical analogue of Helton-Nie conjecture, cont.

Theorem (Allamigeon, Gaubert, and Skomra, MEGA2017+JSC.) Fix a set S ⊂ Rn. TFAE S is the image by val of a convex semialgebraic set of Kn

>0

S is the image by val of the image by proj : Rp → Rn (p n) of a spectrahedron of Kp

>0

S is tropically convex, closed and semilinear

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 34 / 58

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SLIDE 109

Tropical analogue of Helton-Nie conjecture, cont.

Theorem (Allamigeon, Gaubert, and Skomra, MEGA2017+JSC.) Fix a set S ⊂ Rn. TFAE S is the image by val of a convex semialgebraic set of Kn

>0

S is the image by val of the image by proj : Rp → Rn (p n) of a spectrahedron of Kp

>0

S is tropically convex, closed and semilinear There exists a stochastic game with Shapley operator F : Rn → Rn such that S = {x ∈ Rn | x F(x)} ,

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 34 / 58

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SLIDE 110

Tropical analogue of Helton-Nie conjecture, cont.

Theorem (Allamigeon, Gaubert, and Skomra, MEGA2017+JSC.) Fix a set S ⊂ Rn. TFAE S is the image by val of a convex semialgebraic set of Kn

>0

S is the image by val of the image by proj : Rp → Rn (p n) of a spectrahedron of Kp

>0

S is tropically convex, closed and semilinear There exists a stochastic game with Shapley operator F : Rn → Rn such that S = {x ∈ Rn | x F(x)} , There exists a stochastic game with transition probabilities 0, 1

2, 1 and Shapley operator F : Rp → Rp, with p n, such

that S = proj{x ∈ Rp | x F(x)}

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 34 / 58

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SLIDE 111

How to solve the game in practice

Gurvich, Karzanov and Khachyan pumping algorithm (1988) iterative algorithm with hard (discontinuous) thresholds, generalized to the stochastic case by Boros, Elbassioni, Gurvich and Makino (2015, hard complexity estimates)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 35 / 58

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SLIDE 112

How to solve the game in practice

Gurvich, Karzanov and Khachyan pumping algorithm (1988) iterative algorithm with hard (discontinuous) thresholds, generalized to the stochastic case by Boros, Elbassioni, Gurvich and Makino (2015, hard complexity estimates) policy iteration Hoffman-Karp 66 irreducible case, Denardo 67 discounted case (strongly polynomial by Ye, Hansen, Miltersen, Zwick 2011), stochastic mean payoff case Akian, Cochet, Detournay, SG (2006, 2012).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 35 / 58

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SLIDE 113

How to solve the game in practice

Gurvich, Karzanov and Khachyan pumping algorithm (1988) iterative algorithm with hard (discontinuous) thresholds, generalized to the stochastic case by Boros, Elbassioni, Gurvich and Makino (2015, hard complexity estimates) policy iteration Hoffman-Karp 66 irreducible case, Denardo 67 discounted case (strongly polynomial by Ye, Hansen, Miltersen, Zwick 2011), stochastic mean payoff case Akian, Cochet, Detournay, SG (2006, 2012). value iteration, Zwick Paterson (1996) in the deterministic case.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 35 / 58

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SLIDE 114

How to solve the game in practice

Gurvich, Karzanov and Khachyan pumping algorithm (1988) iterative algorithm with hard (discontinuous) thresholds, generalized to the stochastic case by Boros, Elbassioni, Gurvich and Makino (2015, hard complexity estimates) policy iteration Hoffman-Karp 66 irreducible case, Denardo 67 discounted case (strongly polynomial by Ye, Hansen, Miltersen, Zwick 2011), stochastic mean payoff case Akian, Cochet, Detournay, SG (2006, 2012). value iteration, Zwick Paterson (1996) in the deterministic case. more refined value type iteration, special case of simple stochastic games Ibsen-Jensen, Miltersen (2012)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 35 / 58

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SLIDE 115

Basic value iteration

tx := maxi xi (read “top”), bx := mini xi (read “bot”)

1: procedure ValueIteration(F) 2:

⊲ F a Shapley operator from Rn to Rn

3:

⊲ The algorithm will report whether Player Max or Player Min wins the mean payoff game represented by F

4:

u := 0 ∈ Rn

5:

while t(u) > 0 and b(u) < 0 do u := F(u) ⊲ At iteration ℓ, u = F ℓ(0) is the value vector of the game in finite horizon ℓ

6:

done

7:

if t(u) 0 then return “Player Min wins”

8:

else return “Player Max wins”

9:

end

10: end

This is what we implemented to solve the benchmarks of large scale nonarchimedean SDP.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 36 / 58

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SLIDE 116

Complexity analysis?

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 37 / 58

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SLIDE 117

Complexity analysis? Answer: Metric geometry tool

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 37 / 58

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SLIDE 118

Funk, Hilbert and Thompson metric

C closed convex pointed cone, x y if y − x ∈ C, Funk reverse metric (Papadopoulos, Troyanov): RFunk(x, y) := log inf{λ > 0|λx y}

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 38 / 58

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SLIDE 119

Funk, Hilbert and Thompson metric

C closed convex pointed cone, x y if y − x ∈ C, Funk reverse metric (Papadopoulos, Troyanov): RFunk(x, y) := log inf{λ > 0|λx y} C = Rn

+, RFunk(x, y) = log maxi yi/xi (tropical sesquilinear form)

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 38 / 58

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SLIDE 120

Funk, Hilbert and Thompson metric

C closed convex pointed cone, x y if y − x ∈ C, Funk reverse metric (Papadopoulos, Troyanov): RFunk(x, y) := log inf{λ > 0|λx y} C = Rn

+, RFunk(x, y) = log maxi yi/xi (tropical sesquilinear form)

C = S+

n = positive semidefinite matrices,

RFunk(x, y) = log max spec(x−1y).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 38 / 58

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SLIDE 121

Funk, Hilbert and Thompson metric

C closed convex pointed cone, x y if y − x ∈ C, Funk reverse metric (Papadopoulos, Troyanov): RFunk(x, y) := log inf{λ > 0|λx y} C = Rn

+, RFunk(x, y) = log maxi yi/xi (tropical sesquilinear form)

C = S+

n = positive semidefinite matrices,

RFunk(x, y) = log max spec(x−1y). Lemma F : int C → int C is order preserving and homogeneous of degree 1 iff RFunk(F(x), F(y)) RFunk(x, y), ∀x, y ∈ int C .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 38 / 58

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SLIDE 122

We can symmetrize Funk’s metric in two ways dT(x, y) = max(RFunk(x, y), RFunk(y, x)) Thompsons’ part metric dH(x, y) := RFunk(x, y)+RFunk(y, x) Hilbert’s projective metric (plays the role of Euclidean metric in tropical convexity Cohen, SG, Quadrat 2004) dH(x, y) = log x − log yH where zH := max

i∈[n] zi − min i∈[n] zi .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 39 / 58

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SLIDE 123

e1 e2 e3

A ball in Hilbert’s projective metric is classically and tropically convex.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 40 / 58

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SLIDE 124

S(F) := {x ∈ Tn : x F(x)}, T := R ∪ {−∞} cw(F) = max

i

¯ vi, cw(F) = min

i

¯ vi (best and worst mean payoffs).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 41 / 58

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SLIDE 125

S(F) := {x ∈ Tn : x F(x)}, T := R ∪ {−∞} cw(F) = max

i

¯ vi, cw(F) = min

i

¯ vi (best and worst mean payoffs). We say that u ∈ Rn is a bias (tropical eigenvector) if F(u) = λe + u Then, λ = cw(F) = cw(F), denoted by ρ(F) for “spectral radius”, it is unique.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 41 / 58

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SLIDE 126

S(F) := {x ∈ Tn : x F(x)}, T := R ∪ {−∞} cw(F) = max

i

¯ vi, cw(F) = min

i

¯ vi (best and worst mean payoffs). We say that u ∈ Rn is a bias (tropical eigenvector) if F(u) = λe + u Then, λ = cw(F) = cw(F), denoted by ρ(F) for “spectral radius”, it is unique. Existence of u guaranteed by ergodicity conditions, Akian, SG, Hochart, DCSD A.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 41 / 58

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SLIDE 127

Definition An order-preserving and additively homogeneous self-map F of Tn is said to be diagonal free when Fi(x) is independent of xi for all i ∈ [n].

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 42 / 58

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SLIDE 128

Definition An order-preserving and additively homogeneous self-map F of Tn is said to be diagonal free when Fi(x) is independent of xi for all i ∈ [n]. Theorem Let F be a diagonal free self-map of Tn. Then, S(F) contains a Hilbert ball of positive radius if and only if cw(F) > 0. Moreover, when S(F) contains a Hilbert ball of positive radius, the supremum of the radii of the Hilbert balls contained in S(F) coincides with cw(F).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 42 / 58

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SLIDE 129

Biggest Hilbert ball in a tropical polyhedra

Extends a theorem of Sergeev, showing that the tropical eigenvalue

  • f A gives the inner radius of the polytropes {x | x Ax}.
slide-130
SLIDE 130

Biggest Hilbert ball in a tropical polyhedra

Extends a theorem of Sergeev, showing that the tropical eigenvalue

  • f A gives the inner radius of the polytropes {x | x Ax}.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 43 / 58

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SLIDE 131

C :=

  • x ∈ Kn : Q(0) + x1Q(1) + · · · + xnQ(n) is PSD
  • F : Tn → Tn Shapley operator of C.

P(F): does there exist x ∈ Tn such that x ≡ −∞ and x F(x)?

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 44 / 58

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SLIDE 132

C :=

  • x ∈ Kn : Q(0) + x1Q(1) + · · · + xnQ(n) is PSD
  • F : Tn → Tn Shapley operator of C.

P(F): does there exist x ∈ Tn such that x ≡ −∞ and x F(x)? PR(F): does there exist x ∈ Rn such that x ≪ F(x)? Theorem (Allamigeon, SG, Skomra)

(i)

if P(F) is infeasible, or equivalently, S(F) is trivial, then C is trivial.

(ii)

if PR(F) is feasible, or equivalently, S(F) is strictly nontrivial, then C is strictly nontrivial, meaning that there exists x ∈ Kn

>0

such that the matrix x1Q(1) + · · · + xnQ(n) is positive definite.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 44 / 58

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SLIDE 133

We define the condition number cond(F) of the above problem P(F) by: (inf{u∞ : u ∈ Rn , P(u + F) is infeasible})−1 (1) if P(F) is feasible, and (inf{u∞ : u ∈ Rn , P(u + F) is feasible})−1 (2) if P(F) is infeasible.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 45 / 58

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SLIDE 134

We define the condition number cond(F) of the above problem P(F) by: (inf{u∞ : u ∈ Rn , P(u + F) is infeasible})−1 (1) if P(F) is feasible, and (inf{u∞ : u ∈ Rn , P(u + F) is feasible})−1 (2) if P(F) is infeasible. u + F: Shapley operator of a game in which in state i, Max receives an additional payment of ui.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 45 / 58

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SLIDE 135

condR(F) is defined as cond(F), considering PR(F). Proposition Let F be a continuous, order-preserving, and additively homogeneous self-map of Tn. Then, condR(F) = |cw(F)|−1 and cond(F) = |cw(F)|−1.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 46 / 58

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SLIDE 136

R(F) := inf {uH : u ∈ Rn, F(u) = ρ(F) + u} . If F is assumed to have a bias vector v ∈ Rn, i.e. F(v) = ρ(F) + v, |ρ(F)|−1 = |cw(F)|−1 = |cw(F)|−1 = condR(F) = cond(F) . Theorem (Allamigeon, SG, Katz, Skomra) Suppose that the Shapley operator F has a bias vector and that ρ(F) = 0. Then ValueIteration terminates after Nvi R(F) cond(F) iterations and returns the correct answer. Compare with log(R/r) in the ellipsoid / interior point methods.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 47 / 58

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SLIDE 137

F = A♯ ◦ B ◦ P (3) where A ∈ Tm×n, B ∈ Tm×q, integer entries, P ∈ Rq×n row-stochastic W := max {|Aij − Bih|: Aij = −∞, Bih = −∞, i ∈ [m], j ∈ [n], h ∈ [q]} Probabilities Pil rational with a common denominator M ∈ N>0, Pil = Qil/M, where Qil ∈ [M] for all i ∈ [q] and l ∈ [n]. A state i ∈ [q] is nondeterministic if there are at least two indices l, l′ ∈ [n] such that Pil > 0 and Pil′ > 0.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 48 / 58

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SLIDE 138

Theorem Let F be a Shapley operator as above, still supposing that F has a bias vector and that ρ(F) is nonzero. If k is the number of nondeterministic states of the game, then cond(F) nMmin{k,n−1}.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 49 / 58

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SLIDE 139

Theorem Let F be a Shapley operator as above, still supposing that F has a bias vector and that ρ(F) is nonzero. If k is the number of nondeterministic states of the game, then cond(F) nMmin{k,n−1}. Relies on an estimate of Skomra of denominators of invariant measures, obtained from Tutte matrix tree theorem, improves Boros, Elbassioni, Gurvich and Makino

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 49 / 58

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SLIDE 140

Theorem (Allamigeon, SG, Katz, Skomra) R(F) 10n2WMmin{k,n−1} . We construct a bias by vanishing discount, which yields of the bound

  • n R(F).

Corollary Let F be the above Shapley operator, still supposing that it has a bias vector and that ρ(F) is nonzero. Then, procedure ValueIteration stops after Nvi 10n3WM2 min{k,n−1} (4) iterations and correctly decides which of the two players is winning.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 50 / 58

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SLIDE 141

In the deterministic case, we recover Zwick-Paterson bound. Corollary Let F = A♯ ◦ B be the Shapley operator of a deterministic game, where the finite entries of A, B ∈ Tm×n are integers. If there exists v ∈ Rn such that F(v) = ρ(F) + v with ρ(F) = 0, then Nvi 2n2W .

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 51 / 58

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SLIDE 142

The assumption ρ(F) = 0 can be relaxed, by appealing to the following perturbation and scaling argument. This leads to a bound in which the exponents of M and of n are increased. Corollary Let µ := nMmin{k,n−1}. Then, procedure ValueIteration, applied to the perturbed and rescaled Shapley operator 1 + 2µF, satisfies Nvi 21n4WM3 min{k,n−1} iterations, and this holds unconditionally. If the algorithm reports that Max wins, then Max is winning in the original mean payoff

  • game. If the algorithm reports that Min wins, then Min is strictly

winning in the original mean payoff game. The algorithm can be also adapted to work in finite precision arithmetic.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 52 / 58

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SLIDE 143

Tropical homotopy

The condition number controls the critical temperature t−1

c

such that for t > tc, the archimedean SDP feasibility problem and tropical SDP feasibility problem have the same answer. δ(t) := max

Q(k)

ij

=0

  • |Q(k)

ij | − logt|Q(k) ij (t)|

  • .

Theorem Let m ≥ 2, and v be the value of the stochastic mean payoff game associated with Q(1), . . . , Q(n). Let λ := maxk vk, and suppose that λ = 0. Take any t such that δ(t) < |λ| and t > (2(m − 1)n)1/(2|λ|−2δ(t)) . Then, the spectrahedron S(t) is nontrivial if and only if λ is positive.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 53 / 58

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SLIDE 144

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

slide-145
SLIDE 145

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations. Extends the equivalence between deterministic mean payoff games and tropical linear programming, Akian, SG, Guterman.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

slide-146
SLIDE 146

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations. Extends the equivalence between deterministic mean payoff games and tropical linear programming, Akian, SG, Guterman. Extends the tropicalization of the SDP cone by Yu

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

slide-147
SLIDE 147

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations. Extends the equivalence between deterministic mean payoff games and tropical linear programming, Akian, SG, Guterman. Extends the tropicalization of the SDP cone by Yu This leads to an algorithm for generic semidefinite feasibility problems over Puiseux series.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

slide-148
SLIDE 148

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations. Extends the equivalence between deterministic mean payoff games and tropical linear programming, Akian, SG, Guterman. Extends the tropicalization of the SDP cone by Yu This leads to an algorithm for generic semidefinite feasibility problems over Puiseux series. Metric geometry definition of the condition number, biggest ball in the primal or dual feasible set.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

slide-149
SLIDE 149

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations. Extends the equivalence between deterministic mean payoff games and tropical linear programming, Akian, SG, Guterman. Extends the tropicalization of the SDP cone by Yu This leads to an algorithm for generic semidefinite feasibility problems over Puiseux series. Metric geometry definition of the condition number, biggest ball in the primal or dual feasible set. Controls the number of value iterations to decide the game

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

slide-150
SLIDE 150

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations. Extends the equivalence between deterministic mean payoff games and tropical linear programming, Akian, SG, Guterman. Extends the tropicalization of the SDP cone by Yu This leads to an algorithm for generic semidefinite feasibility problems over Puiseux series. Metric geometry definition of the condition number, biggest ball in the primal or dual feasible set. Controls the number of value iterations to decide the game Recover complexity bound of Boros, Elbassioni, Gurvich, and Makino, with a simpler algorithm.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

slide-151
SLIDE 151

Concluding remarks

Showed: stochastic mean payoff games polynomial time equivalent to feasibility of nonarchimedean semidefinite programs with generic valuations. Extends the equivalence between deterministic mean payoff games and tropical linear programming, Akian, SG, Guterman. Extends the tropicalization of the SDP cone by Yu This leads to an algorithm for generic semidefinite feasibility problems over Puiseux series. Metric geometry definition of the condition number, biggest ball in the primal or dual feasible set. Controls the number of value iterations to decide the game Recover complexity bound of Boros, Elbassioni, Gurvich, and Makino, with a simpler algorithm. Controls the critical temperature under which the SDP feasibility problem “freezes” in its tropical state.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 54 / 58

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SLIDE 152

Thank you !

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 55 / 58

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SLIDE 153

References I

  • X. Allamigeon, S. Gaubert, and M. Skomra. “Solving Generic

Nonarchimedean Semidefinite Programs Using Stochastic Game Algorithms”. In: Journal of Symbolic Computation 85 (2018),

  • pp. 25–54. doi: 10.1016/j.jsc.2017.07.002. eprint:

1603.06916.

  • X. Allamigeon, S. Gaubert, and M. Skomra. “The tropical

analogue of the Helton-Nie conjecture is true”. In: J. Symbolic Computation (2018). doi: 10.1016/j.jsc.2018.06.017. eprint: arXiv:1801.02089.

  • M. Develin and J. Yu. “Tropical polytopes and cellular

resolutions”. In: Experimental Mathematics 16.3 (2007).

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 56 / 58

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References II

  • L. van den Dries and P. Speissegger. “The real field with

convergent generalized power series”. In: Transactions of the AMS 350.11 (1998), pp. 4377–4421.

  • D. Henrion, S. Naldi, and M. Safey El Din. “Exact algorithms

for linear matrix inequalities”. In: SIAM J. Opt. 26.4 (2016),

  • pp. 2512–2539.
  • E. de Klerk and F. Vallentin. “On the Turing model complexity
  • f interior point methods for semidefinite programming”. In:

SIAM J. Opt. 26.3 (2016), pp. 1944–1961. Claus Scheiderer. “Spectrahedral Shadows”. In: SIAM Journal

  • n Applied Algebra and Geometry 2.1 (2018), pp. 26–44. doi:

10.1137/17M1118981.

Allamigeon, Gaubert, Katz, Skomra (Inria-X) Nonarchimedean SDP January 24, 2019, Birmingham 57 / 58

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References III

  • J. Yu. “Tropicalizing the positive semidefinite cone”. In: Proc.
  • Amer. Math. Soc 143.5 (2015), pp. 1891–1895.

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