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.
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results, we give an algorithm for the following problem: given a homogeneous polynomial f (x 1 , . . . , x n ) of degree 3, decide whether it can be written as a sum of cubes of linearly independent linear forms with complex coefficients. Compared to previous algorithms for the same problem, the two main novel features of this algorithm are:(i) It is an algebraic algorithm, i.e., it performs only arithmetic operations and equality tests on the coefficients of the input polynomial f . In particular, it does not make any appeal to polynomial factorization.
We introduce tropical spectrahedra, defined as the images by the nonarchimedean valuation of spectrahedra over the field of real Puiseux series. We provide an explicit polyhedral characterization of generic tropical spectrahedra, involving principal tropical minors of size at most 2. One of the key ingredients is Denef-Pas quantifier elimination result over valued fields. We obtain from this that the nonarchimedean valuation maps semialgebraic sets to semilinear sets that are closed. We also prove that, under a regularity assumption, the image by the valuation of a basic semialgebraic set is obtained by tropicalizing the inequalities that define it.
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