# Computation of the distance to semi-algebraic sets

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

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## Abstract

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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: $d\left(u,S\right)={inf}_{x\in S}{\parallel x-u\parallel }^{2}$, where u is in ${ℛ}^{n}$. 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.

## How to cite

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Ferrier, 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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