Computation of the distance to semi-algebraic sets

This paper is devoted to the computation of distance to set, called S, dened by polyno- mial equations. First we consider the case of quadratic systems. Then, application of results stated for quadratic systems to the quadratic equivalent of polynomial systems (see (5)), allows us to compute distance to semi-algebraic sets. Problem of computing distance can be viewed as non convex mini- mization problem: d(u;S )=i nfx2Skx uk 2 ,w hereu is in n . To have, at least, lower approximation of distance, we consider the dual bound m(u) associated with the dual problem and give sucient conditions to guarantee m(u )= d(u;S). The second part deal with numerical computation of m(u) using an interior point method in semidenite programming. Last, various examples, namely from chemistry and robotic, are given.