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Error bounds for inequality systems defining convex sets

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Abstract

The main goal in this paper is to devise an approach to explicitly calculate the constant in the Hoffman’s error bound for (not necessarily convex) inequality systems defining convex sets. We give a constructive proof of the Hoffman’s error bound and show that we can use our method to calculate the constant at least in simple cases.

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References

  1. Azé, D.: A survey on error bounds for lower semicontinuous functions. ESAIM Proc. 13, 1–17 (2003)

    Article  MathSciNet  Google Scholar 

  2. Azé, D., Corvellec, J.-N.: On the sensitivity analysis of Hoffman constants for systems of linear inequalities. SIAM J. Optim. 12, 913–927 (2002)

    Article  MathSciNet  Google Scholar 

  3. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. Springer, Amsterdam (2012)

    Book  Google Scholar 

  4. Bergthaller, C., Singer, I.: The distance to a polyhedron. Linear Algebra Appl. 169, 111–129 (1992)

    Article  MathSciNet  Google Scholar 

  5. Demyanov, V.F., Rubinov, A.M.: Quasidifferentiable functionals. Soviet Math. Dokl. 21, 14–18 (1980)

    MATH  Google Scholar 

  6. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity problems. Springer, New York (2003)

    MATH  Google Scholar 

  7. Güler, O., Hoffman, A.J., Rothblum, U.G.: Approximations to solution to systems of linear inequalities. SIAM J. Matrix Anal. Appl. 16, 686–696 (1995)

    Article  MathSciNet  Google Scholar 

  8. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Stand. 49, 263–265 (1952)

    Article  MathSciNet  Google Scholar 

  9. Ioffe, A.D.: Metric regularity–a survey Part II. Appl. J. Aust. Math. Soc. 101(3), 376–417 (2016)

    Article  MathSciNet  Google Scholar 

  10. Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Co., Amsterdam-New York (1979)

    MATH  Google Scholar 

  11. Klatte, D.: Hoffman’s error bound for systems of convex inequalities. Mathematical programming with data perturbations, 185-199, Lecture Notes in Pure and Appl. Math., 195, Dekker, New York, (1998)

  12. Klatte, D., Li, W.: Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Program. 84, 137–160 (1999)

    Article  MathSciNet  Google Scholar 

  13. Klatte, D., Thiere, G.: Error bounds for solutions of linear equations and inequalities. ZOR - Math. Methods Oper. Res. 41, 191–214 (1995)

    Article  MathSciNet  Google Scholar 

  14. Klatte, D., Thiere, G.: A note on Lipschitz constants for solutions of linear inequalities and equations. Linear Algebra Appl. 244, 365–374 (1996)

    Article  MathSciNet  Google Scholar 

  15. Lemaréchal, C.: An introduction to the theory of nonsmooth optimization. Optimization 17, 827–858 (1986)

    Article  MathSciNet  Google Scholar 

  16. Lewis, A.S., Pang, J.-S.: Error bounds for convex inequality systems. Nonconvex Optim. Appl. 27, 75–110 (1998)

    Article  MathSciNet  Google Scholar 

  17. Li, W.: The sharp Lipschitz constant for feasible and optimal solutions of a perturbed linear program. Linear Algebra Appl. 187, 15–40 (1993)

    Article  MathSciNet  Google Scholar 

  18. Luo, Z.-Q., Tseng, P.: Perturbation analysis of a condition number for linear systems. SIAM J. Matrix Anal. Appl. 15, 636–659 (1994)

    Article  MathSciNet  Google Scholar 

  19. Mangasarian, O.L.: Simple computable bounds for solutions of linear complementarity problems and linear programs. In: Cottle, R.W. (ed.) Mathematical Programming, II. Mathematical Programming Studies, vol. 25, pp. 1–12. Springer, Berlin (1985)

    Google Scholar 

  20. Mangasarian, O.L.: Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification. Math. Program. 83, 187–194 (1998)

    MathSciNet  MATH  Google Scholar 

  21. Martínez-Legaz, J.E.: Optimality conditions for pseudoconvex minimization over convex sets defined by tangentially convex constraints. Optim. Lett. 9, 1017–1023 (2015)

    Article  MathSciNet  Google Scholar 

  22. Ngai, H.V., Kruger, A., Théra, M.: Stability of error bounds for semi-infinite convex constraint systems. SIAM J. Optim. 20, 2080–2096 (2010)

    Article  MathSciNet  Google Scholar 

  23. Pshenichnyi, B.N.: Necessary Conditions for an Extremum. Marcel Dekker Inc., New York (1971)

    Google Scholar 

  24. Robinson, S.M.: Bounds for error in the solution set of a perturbed linear problem. Linear Algebra Appl. 6, 69–81 (1973)

    Article  MathSciNet  Google Scholar 

  25. Robinson, S.M.: An application of error bounds for convex programming in a linear space. SIAM J. Control 13, 271–273 (1975)

    Article  MathSciNet  Google Scholar 

  26. Rosenbloom, P.C.: Quelques classes de problèmes extré maux. Bull. Soc. Math. France 79, 1–58 (1951)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are grateful to Dominique Azé for helpful comments. We are also grateful to two anonymous referees, whose constructive suggestions allowed us to improve the paper.

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Authors and Affiliations

  1. Department of Economic Sciences, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Joydeep Dutta

  2. Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, Bellaterra, Spain

    Juan Enrique Martínez-Legaz

  3. BGSMath, Barcelona, Spain

    Juan Enrique Martínez-Legaz

Authors
  1. Joydeep Dutta
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  2. Juan Enrique Martínez-Legaz
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Correspondence to Juan Enrique Martínez-Legaz.

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Dedicated to Prof. Marco A. López on the occasion of his 70th birthday.

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Juan Enrique Martínez-Legaz acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities, through Grant PGC2018-097960-B-C21 and the Severo Ochoa Program for Centers of Excellence in R&D (CEX2019-000915-S). He is affiliated with MOVE (Markets, Organizations and Votes in Economics).

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Dutta, J., Martínez-Legaz, J.E. Error bounds for inequality systems defining convex sets. Math. Program. 189, 299–314 (2021). https://doi.org/10.1007/s10107-020-01575-1

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