# Computation of the distance to semi-algebraic sets

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 5, page 139-156
- ISSN: 1292-8119

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topFerrier, Christophe. "Computation of the distance to semi-algebraic sets." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 139-156. <http://eudml.org/doc/197274>.

@article{Ferrier2010,

abstract = {
This paper is devoted to the computation of distance to set, called S, defined by polynomial 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 minimization problem: $ d(u,S) = \inf_\{x \in S\} \| x-u\|^2$, where u is in $\mathcal\{R\}^\{n\}$. To have, at least, lower approximation of distance, we consider the dual bound m(u) associated with the dual problem and give sufficient conditions to guarantee m(u) = d(u,S). The second part deal with numerical computation of m(u) using an interior point method in semidefinite programming. Last, various examples, namely from chemistry and robotic, are given.
},

author = {Ferrier, Christophe},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Distance; dual bond; optimality conditions; polynomial systems; interior point
methods; semidefinite programming; location of zeros.; interior point methods; location of zeros},

language = {eng},

month = {3},

pages = {139-156},

publisher = {EDP Sciences},

title = {Computation of the distance to semi-algebraic sets},

url = {http://eudml.org/doc/197274},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Ferrier, Christophe

TI - Computation of the distance to semi-algebraic sets

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 139

EP - 156

AB -
This paper is devoted to the computation of distance to set, called S, defined by polynomial 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 minimization problem: $ d(u,S) = \inf_{x \in S} \| x-u\|^2$, where u is in $\mathcal{R}^{n}$. To have, at least, lower approximation of distance, we consider the dual bound m(u) associated with the dual problem and give sufficient conditions to guarantee m(u) = d(u,S). The second part deal with numerical computation of m(u) using an interior point method in semidefinite programming. Last, various examples, namely from chemistry and robotic, are given.

LA - eng

KW - Distance; dual bond; optimality conditions; polynomial systems; interior point
methods; semidefinite programming; location of zeros.; interior point methods; location of zeros

UR - http://eudml.org/doc/197274

ER -

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