10 research outputs found
Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
A general issue in computational optimization is to develop combinatorial
algorithms for semidefinite programming. We address this issue when the base
field is nonarchimedean. We provide a solution for a class of semidefinite
feasibility problems given by generic matrices. Our approach is based on
tropical geometry. It relies on tropical spectrahedra, which are defined as the
images by the valuation of nonarchimedean spectrahedra. We establish a
correspondence between generic tropical spectrahedra and zero-sum stochastic
games with perfect information. The latter have been well studied in
algorithmic game theory. This allows us to solve nonarchimedean semidefinite
feasibility problems using algorithms for stochastic games. These algorithms
are of a combinatorial nature and work for large instances.Comment: v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions +
benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign
patterns + some results have been replaced by references to the companion
work arXiv:1610.0674
Tropical totally positive matrices
We investigate the tropical analogues of totally positive and totally
nonnegative matrices. These arise when considering the images by the
nonarchimedean valuation of the corresponding classes of matrices over a real
nonarchimedean valued field, like the field of real Puiseux series. We show
that the nonarchimedean valuation sends the totally positive matrices precisely
to the Monge matrices. This leads to explicit polyhedral representations of the
tropical analogues of totally positive and totally nonnegative matrices. We
also show that tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of
FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author
is sported by the French Chateaubriand grant and INRIA postdoctoral
fellowshi
Smoothed analysis of deterministic discounted and mean-payoff games
We devise a policy-iteration algorithm for deterministic two-player
discounted and mean-payoff games, that runs in polynomial time with high
probability, on any input where each payoff is chosen independently from a
sufficiently random distribution.
This includes the case where an arbitrary set of payoffs has been perturbed
by a Gaussian, showing for the first time that deterministic two-player games
can be solved efficiently, in the sense of smoothed analysis.
More generally, we devise a condition number for deterministic discounted and
mean-payoff games, and show that our algorithm runs in time polynomial in this
condition number.
Our result confirms a previous conjecture of Boros et al., which was claimed
as a theorem and later retracted. It stands in contrast with a recent
counter-example by Christ and Yannakakis, showing that Howard's
policy-iteration algorithm does not run in smoothed polynomial time on
stochastic single-player mean-payoff games.
Our approach is inspired by the analysis of random optimal assignment
instances by Frieze and Sorkin, and the analysis of bias-induced policies for
mean-payoff games by Akian, Gaubert and Hochart
On the convex formulations of robust Markov decision processes
Robust Markov decision processes (MDPs) are used for applications of dynamic
optimization in uncertain environments and have been studied extensively. Many
of the main properties and algorithms of MDPs, such as value iteration and
policy iteration, extend directly to RMDPs. Surprisingly, there is no known
analog of the MDP convex optimization formulation for solving RMDPs. This work
describes the first convex optimization formulation of RMDPs under the
classical sa-rectangularity and s-rectangularity assumptions. By using entropic
regularization and exponential change of variables, we derive a convex
formulation with a number of variables and constraints polynomial in the number
of states and actions, but with large coefficients in the constraints. We
further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or
entropy-based uncertainty sets, showing that, in these cases, RMDPs can be
reformulated as conic programs based on exponential cones, quadratic cones, and
non-negative orthants. Our work opens a new research direction for RMDPs and
can serve as a first step toward obtaining a tractable convex formulation of
RMDPs
Tropical complementarity problems and Nash equilibria
Linear complementarity programming is a generalization of linear programming
which encompasses the computation of Nash equilibria for bimatrix games. While
the latter problem is PPAD-complete, we show that the tropical analogue of the
complementarity problem associated with Nash equilibria can be solved in
polynomial time. Moreover, we prove that the Lemke--Howson algorithm carries
over the tropical setting and performs a linear number of pivots in the worst
case. A consequence of this result is a new class of (classical) bimatrix games
for which Nash equilibria computation can be done in polynomial time
Signed tropical halfspaces and convexity
We extend the fundamentals for tropical convexity beyond the tropically
positive orthant expanding the theory developed by Loho and V\'egh (ITCS 2020).
We study two notions of convexity for signed tropical numbers called
'TO-convexity' (formerly 'signed tropical convexity') and the novel notion
'TC-convexity'. We derive several separation results for TO-convexity and
TC-convexity. A key ingredient is a thorough understanding of TC-hemispaces -
those TC-convex sets whose complement is also TC-convex. Furthermore, we use
new insights in the interplay between convexity over Puiseux series and its
signed valuation. Remarkably, TC-convexity can be seen as a natural convexity
notion for representing oriented matroids as it arises from a generalization of
the composition operation of vectors in an oriented matroid. We make this
explicit by giving representations of linear spaces over the real tropical
hyperfield in terms of TC-convexity.Comment: v1: 48 pages, 8 figures; v2: 58 pages, 10 figures, new section on
oriented matroids + minor improvement
International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022
Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022.
Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress.
The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library