Computation of the distance to semi-algebraic sets

Christophe Ferrier

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

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

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 ( u , S ) = inf x ∈ S ∥ x - u ∥ 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 -

References

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  1. F. Alizadeth, Interior point methods in semidefinite programming with application to combinatorial optimisation. SIAM J. Optim.5 (1995) 13-51.  
  2. A. Bellido, Construction de fonctions d'itération pour le calcul simultané des solutions d'équations et de systèmes d'équations algébriques. Thèse de doctorat de l'Universté Paul Sabatier, Toulouse (1992).  
  3. S. Boyd et al., Linear Matrix Inequalities Problem in Control Theory. SIAM, Philadelphia, Stud. Appl. Math.15 (1995).  
  4. J.-P. Dedieu and J.-C. Yakoubsohn, Localization of an algebraic hypersurface by the exclusion algorithm. Appl. Algebra Engrg. Comm. Comput.2 (1992) 239-256.  
  5. Ch. Ferrier, Hilbert's 17th problem and best dual bounds in quadratic minimization. Cybernetics and System Analysis5 (1998) 76-91.  
  6. A.V. Fiacco and G.P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley (1968). Reprinted SIAM, 1990.  
  7. R. Fletcher, Semi-definite matrix constraints in optimization. SIAM J. Control Optim.23 (1985) 493-513.  
  8. C. Lemarechal and J.-B. Hiriart-Urruty, Convex Analysis and Minimization Algorithms II. Springer Verlag, Comprehensive Studies in Mathematics306 (1991).  
  9. F. Jarre, Interior-point methods for convex programming. Appl. Math. Optim.26 (1992) 287-391.  
  10. F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM J. Control Optim.31 (1993) 1360-1377.  
  11. N. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica4 (1984) 373-395.  
  12. R.B. Kearfott, Some tests of generalized bisection. ACM Trans. Math. Software13 (1987) 197-200.  
  13. Yu. Nesterov and A. Nemirovsky, Interior-point polynomial methods in convex programming. SIAM, Philadelphia, Stud. Appl. Math.13 (1994).  
  14. N.Z. Shor, Dual estimate in multi-extremal problems. J. Global Optim.2 (1992) 411-418.  
  15. G. Sonnevend, An ``analytical centre'' for polyhedrons and a new classe of global algorithms for linear (smooth, convex) programming. Springer Verlag, Lecture Notes in Control and Inform. Sci.84, System Modeling and Optimisation. 12th IFIP Conference on system optimisation 1984 (1986) 866-878.  
  16. G. Sonnevend and J. Stoer, Global ellipsoidal approximation and homotopy methods for solving convex analitic programs. Appl. Math. Optim.21 (1990) 139-165.  
  17. D.E. Stewart, Matrix Computation in C. University of Queensland, Australia (1992). ftp site: des@thrain.anu.edu.au. directory: pub/meschach  
  18. L. Vandenberghe and S. Boyd, Semidefinite programming. SIAM Rev.1 (1996) 49-95.  
  19. J. Verschelde, P. Verlinden and R. Cools, Homotopy exploiting newton polytopes for solving sparse polynomials systems. SIAM J. Numer. Anal.31 (1994) 915-930.  
  20. A. Wright, Finding solutions to a system of polynomial equations. Math. Comp.44 (1985) 125-133.  

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