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Real algebraic geometry and infinite-dimensional conic optimization
Intervenant(s) : Didier Henrion (LAAS-CNRS Toulouse, FR and Czech Tech. Univ. in Prague, CZ) We show how some problems of real algebraic geometry can be approached via infinite-dimensional conic optimization (linear programming in Banach spaces), and how this results in a readily implementable floating-point numerical solution with hierarchies of semidefinite-programming relaxations. More specifically, we address the problem of numerically approximating the volume, and more generally, the moments of the uniform measure on a given real semi-algebraic set. We show how this generates, as a by-product, a hierarchy of outer semi-algebraic approximations of the set with increasing degree and complexity, and with guarantees of almost uniform convergence. If time allows, we extend these ideas to semi-algebraic outer approximations of transcendental subsets of the Euclidean space such as basins of attraction for ordinary differential equations with polynomial vector fields and semi-algebraic constraints.
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