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We present two algebraic methods to solve the parametric optimization problem that arises in nonlinear model predictive control. We consider constrained discrete-time polynomial systems and the corresponding constrained finite-time optimal control problem. The first method is based on cylindrical algebraic decomposition. The second uses Gröbner bases and… (More)

We provide a real algebraic symbolic-numeric algorithm for computing the real variety V R (I) of an ideal I ⊆ R[x], assuming V R (I) is finite (while V C (I) could be infinite). Our approach uses sets of linear functionals on R[x], vanishing on a given set of polynomials generating I and their prolongations up to a given degree, as well as on polynomials of… (More)

A new technique for solving polynomial nonlinear constrained optimal control problems is presented. The problem is reformulated into a parametric optimization problem, which in turn is solved in a two step procedure. First, in a pre-computation step, the equation part of the corresponding first order optimality conditions is solved for a generic value of… (More)

For an ideal I ⊆ R[x] given by a set of generators, a new semidefi-nite characterization of its real radical I(V R (I)) is presented, provided it is zero-dimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V R (I)… (More)

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finte. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of [17] are efficient… (More)

Solving systems of polynomial equations is a classical problem of mathematics with an emerging number of modern applications, such as coding theory, robotics, computational statistics, etc. Its importance is reflected by the broad literature that deals with algorithms, ranging from numerical continuation methods to exact methods based e.g. on Gröbner… (More)

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