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Section: New Results

Tropical methods applied to optimization, perturbation theory and matrix analysis

Majorization inequalities for valuations of eigenvalues using tropical algebra

Participants : Marianne Akian, Stéphane Gaubert.

In [14], with Meisam Sharify (IPM, Tehran, Iran), we establish log-majorization inequalities of the eigenvalues of matrix polynomials using the tropical roots of some scalar polynomials depending only on the norms of the matrix coefficients. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and Pólya for the roots of scalar polynomials.

These works have been presented in  [46].

Tropicalization of the central path and application to the complexity of interior point methods

Participants : Xavier Allamigeon, Stéphane Gaubert.

This work is in collaboration with Pascal Benchimol (EDF Labs) and Michael Joswig (TU Berlin).

In optimization, path-following interior point methods are driven to an optimal solution along a trajectory called the central path. The central path of a linear program $\text{LP}(A,b,c)\equiv min\{c·x\mid Ax\le b,x\ge 0\}$ is defined as the set of the optimal solutions $(x^{\mu },w^{\mu })$ of the barrier problems:

While the complexity of interior point methods is known to be polynomial, an important question is to study the number of iterations which are performed by interior point methods, in particular whether it can be bounded by a polynomial in the dimension ($mn$) of the problem. This is motivated by Smale 9th problem  [113], on the existence of a strongly polynomial complexity algorithm for linear programming. So far, this question has been essentially addressed though the study of the curvature of the central path, which measures how far a path differs from a straight line, see  [71], [70], [73], [72]. In particular, by analogy with the classical Hirsch conjecture, Deza, Terlaky and Zinchencko  [72] proposed the “continuous analogue of the Hirsch conjecture”, which says that the total curvature of the central path is linearly bounded in the number $m$ of constraints.

In a work of X. Allamigeon, P. Benchimol, S. Gaubert, and M. Joswig [41], we prove that primal-dual log-barrier interior point methods are not strongly polynomial, by constructing a family of linear programs with $3r+1$ inequalities in dimension $2r$ for which the number of iterations performed is in $\Omega \left(2^r\right)$. The total curvature of the central path of these linear programs is also exponential in $r$, disproving the continuous analogue of the Hirsch conjecture.

Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. We give an explicit geometric characterization of the tropical central path, as a tropical analogue of the barycenter of a sublevel set of the feasible set induced by the duality gap. We study the convergence properties of the classical central path to the tropical one. This allows us to show that that the number of iterations performed by interior point methods is bounded from below by the number of tropical segments constituting the tropical central path.

Tropical approach to semidefinite programming

Participants : Xavier Allamigeon, Stéphane Gaubert, Mateusz Skomra.

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

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.

We address 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 first study 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 gathered in the preprint  [59].

Then, we show in [17] 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, [110] disproved this conjecture. In [26], 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 applied to our previous results  [59] and [17].