Section: New Results

Tropical methods applied to optimization, perturbation theory and matrix analysis

Tropicalization of semidefinite programming and its relation with stochastic games

Participants : Xavier Allamigeon, Stéphane Gaubert.

Semidefinite programming consists in optimizing a linear function over a spectrahedron. The latter is a subset of ${ℝ}^n$ defined by linear matrix inequalities, i.e., a set of the form

$\left\{x\in {ℝ}^n:Q^{\left(0\right)}+x_1{Q}^{\left(1\right)}+\cdots +x_n{Q}^{\left(n\right)}⪰0\right\}$

where the $Q^{\left(k\right)}$ are symmetric matrices of order $m$, and $⪰$ denotes the Loewner order on the space of symmetric matrices. By definition, $X⪰Y$ if and only if $X-Y$ is positive semidefinite.

Semidefinite programming is a fundamental tool in convex optimization. It is used to solve various applications from engineering sciences, and also to obtain approximate solutions or bounds for hard problems arising in combinatorial optimization and semialgebraic optimization.

A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. Indeed, semidefinite programs are usually solved via interior point methods. However, the latter provide an approximate solution in a polynomial number of iterations, provided that a strictly feasible initial solution. Semidefinite programming becomes a much harder matter if one requires an exact solution. The feasibility problem belongs to ${\mathrm{𝖭𝖯}}_{ℝ}\cap {\mathrm{𝖼𝗈𝖭𝖯}}_{ℝ}$, where the subscript $ℝ$ refers to the BSS model of computation. It is not known to be in $\mathrm{𝖭𝖯}$ in the bit model.

The PhD thesis of Mateusz Skomra  [118] dealt about semidefinite programming, in the case where the field $ℝ$ is replaced by a nonarchimedean field, like the field of Puiseux series. In this case, methods from tropical geometry can be applied and are expected to allow one, in generic situations, to reduce semialgebraic problems to combinatorial problems, involving only the nonarchimedean valuations (leading exponents) of the coefficients of the input.

To this purpose, we studied tropical spectrahedra, which are defined as the images by the valuation of nonarchimedean spectrahedra. We establish that they are closed semilinear sets, and that, under a genericity condition, they are described by explicit inequalities expressing the nonnegativity of tropical minors of order 1 and 2. These results are presented in the preprint  [52] (now accepted for publication in Disc. Comp. Geom), with further results in the PhD thesis  [118].

We showed in  [53] that the feasibility problem for a generic tropical spectrahedron is equivalent to solving a stochastic mean payoff game (with perfect information). The complexity of these games is a long-standing open problem. They are not known to be polynomial, however they belong to the class $\mathrm{𝖭𝖯}\cap \mathrm{𝖼𝗈𝖭𝖯}$, and they can be solved efficiently in practice. This allows to apply stochastic game algorithms to solve nonarchimedean semidefinite feasibility problems. We obtain in this way both theoretical bounds and a practicable method which solves some large scale instances.

A long-standing problem is to characterize the convex semialgebraic sets that are SDP representable, meaning that they can be represented as the image of a spectrahedron by a (linear) projector. Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers are SDP representable. Recently, [114] disproved this conjecture. In [15], we show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods and on our previous results  [52] and  [53].

In  [50] and  [118], we exploited the tropical geometry approach to introduce a condition number for stochastic mean payoff games (with perfect information). This condition number is defined as the maximal radius of a ball in Hilbert's projective metric, contained in a primal or dual feasible set. We show that the convergence time of value iteration is governed by this condition number, and derive fixed parameter tractability results.

Tropical polynomial systems and colorful interior of convex bodies

Participants : Marianne Akian, Marin Boyet, Xavier Allamigeon, Stéphane Gaubert.

We studied tropical polynomial systems, with motivations from call center performance evaluation (see Section 7.6.1). We introduced a notion of colorful interior of a family of convex bodies, and showed that the solution of such a polynomial system reduces to linear programming if one knows a vector in the colorful interior of an associated family of Newton polytopes. Further properties of colorful interiors were investigated, as well as the relation between tropical colorful interiors and support vector machines. These results were presented by M. Boyet at the SIAM AG conference in Bern.

Universal approximation theorems by log-sum-exp neural networks

Participant : Stéphane Gaubert.

This is a joint work with Giuseppe Calafiore and Corrado Possieri (Torino).

We establish universal properties of functions by neural networks with log-sum-exp activation functions, first for convex functions [19], and then in general [29]. Some consequences, including approximation by subtraction free rational expressions, are derived.