随机平均支付对策的条件数及其对非阿基米德半定规划的意义
Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming
摘要：
半定规划可以考虑在任何实封闭的域上，包括具有非阿基米德赋值的Puixix级数的域。非阿基米德半定程序编码的经典半定程序的参数家庭，足够大的参数值。最近，非阿基米德半定规划与完全信息随机平均支付对策之间建立了对应关系。这种对应关系依赖于热带几何学。它允许一个解决一般的非阿基米德半定可行性问题，大规模的，通过随机对策算法。在本文中，我们表明，这些游戏的平均支付可以解释为相应的非阿基米德可行性问题的条件数。这个数字衡量一个可行实例与不可行实例的距离，反之亦然。我们证明了它与希尔伯特射影度量中的球的最大半径一致，即包含在可行集内。几何解释的条件数特别依赖于对偶定理的热带半定可行性方案。然后，我们根据条件数限制可行性问题的复杂性。根据随机对策的特点，给出了该条件数的明确界。因此，我们表明，最简单的算法来决定是否随机平均支付游戏是赢的，即值迭代，具有伪多项式的复杂性时，随机位置的数量是固定的。
英文摘要：
Semidefinite programming can be considered over any real closed field, including fields of Puiseux series equipped with their nonarchimedean valuation. Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter. Recently, a correspondence has been established between nonarchimedean semidefinite programs and stochastic mean payoff games with perfect information. This correspondence relies on tropical geometry. It allows one to solve generic nonarchimedean semidefinite feasibility problems, of large scale, by means of stochastic game algorithms. In this paper, we show that the mean payoff of these games can be interpreted as a condition number for the corresponding nonarchimedean feasibility problems. This number measures how close a feasible instance is from being infeasible, and vice versa. We show that it coincides with the maximal radius of a ball in Hilbert's projective metric, that is included in the feasible set. The geometric interpretation of the condition number relies in particular on a duality theorem for tropical semidefinite feasibility programs. Then, we bound the complexity of the feasibility problem in terms of the condition number. We finally give explicit bounds for this condition number, in terms of the characteristics of the stochastic game. As a consequence, we show that the simplest algorithm to decide whether a stochastic mean payoff game is winning, namely value iteration, has a pseudopolynomial complexity when the number of random positions is fixed.
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