Abstract

For systems of relations φt(x)≤pt,  t∈T\varphi_t(x)\le p_t,\; t\in T, Ax=yAx=y, where TT is an arbitrary set, φt\varphi_t is a convex l.s.c. function on a Banach space XX for every tt and AA is a linear bounded operator from XX into another Banach space YY, we discuss the following three problems:
(a) stability of solutions with respect to variations of the right hand side;
(b) effect of linear perturbations of functions φt\varphi_t and mapping AA;
(c) distance to infeasibility (the minimal norm of linear perturbations that make the system infeasible.)

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References

  1. D. Azè: A unified theory for metric regularity of multifunctions, J. Convex Analysis 13 (2006) 225–252.https://doi.org/10.68381/jca13023Google Scholar
  2. D. Azé, J.-N. Corvellec: On the sensitivity analysis of Hoffman constants for systems of linear inequalities, SIAM J. Optim. 12 (2002) 912–927.https://doi.org/10.1137/S1052623400375853Google Scholar
  3. M. J. Cánovas, A. L. Dontchev, M. A. López, J. Parra: Metric regularity of semi-infinite constraint systems, Math. Program. 104 (2005) 329–346.https://doi.org/10.1007/s10107-005-0618-zGoogle Scholar
  4. M. J. Cánovas, F. J. Gómez-Senent, J. Parra: Stability of systems of linear equations and inequalities: distance to ill-posedness and metric regularity, Optimization 56 (2007) 1–24.https://doi.org/10.1080/02331930600815926Google Scholar
  5. M. J. Cánovas, F. J. Gómez-Senent, J. Parra: Linear regularity, equi-regularity and intersection mappings for convex semi-infinite inequality systems, Math. Program. 123 (2010) 33–60.https://doi.org/10.1007/s10107-009-0312-7Google Scholar
  6. M. J. Cánovas, M. A. Lopez, M. S. Mordukhovich, J. Parra: Variational analysis in linear semi-infinite and infinite programming. I: Stability of linear inequality system of feasible solutions, SIAM J. Optim. 20 (2009) 1504–1526.https://doi.org/10.1137/090765948Google Scholar
  7. A. L. Dontchev, A. S. Lewis, R. T. Rockafellar: The radius of metric regularity, Trans. Amer. Math. Soc. 355 (2003) 493–517.https://doi.org/10.1090/S0002-9947-02-03088-XGoogle Scholar
  8. A. D. Ioffe: Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979) 61–69.https://doi.org/10.1090/S0002-9947-1979-0531969-6Google Scholar
  9. A. D. Ioffe: Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 28 (1984) 389–416.https://doi.org/10.1090/S0002-9947-1984-0719677-1Google Scholar
  10. A. D. Ioffe: Metric regularity and subdifferential calculus, Usp. Mat. Nauk 55 (2000) 103–162 (in Russian); Russ. Math. Surv. 55 (2000) 501–558 (in English).https://doi.org/10.4213/rm292https://doi.org/10.1070/RM2000v055n03ABEH000292Google Scholar
  11. A. D. Ioffe, Y. Sekiguchi: Exact formulae for regularity estimates, in: Nonlinear Analysis and Convex Analysis (Okinawa, 2005), W. Takahasi, T. Tanaka (eds.), Yokohama Publ., Yokohama (2007) 185–198.Google Scholar
  12. A. D. Ioffe, Y. Sekiguchi: Regularity estimates for convex multifunctions, Math. Program. 117 (2009) 255–270.https://doi.org/10.1007/s10107-007-0157-xGoogle Scholar
  13. A. S. Lewis: Ill-conditioned convex processes and conic linear systems, Math. Oper. Res. 24 (1999) 829–834.https://doi.org/10.1287/moor.24.4.829Google Scholar
  14. B. S. Mordukhovich: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR 254 (1980) 1072–1076.Google Scholar
  15. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  16. B. N. Pshenichnyi: Necessary Conditions for an Extremum, Marcel Dekker, New York (1971).Google Scholar
  17. J. Renegar: Linear programming, complexity theory and elementary functional analysis, Math. Program. 70 (1995) 279–351.https://doi.org/10.1007/BF01585941Google Scholar
  18. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  19. M. Sion: On general minimax theorems, Pac. J. Math. 8 (1958) 171–176.https://doi.org/10.2140/pjm.1958.8.171Google Scholar

Suggested citation

A. Ioffe. “On Stability of Solutions to Systems of Convex Inequalities.” Journal of Convex Analysis 19 (2012), No. 4, 1017–1032. https://doi.org/10.68381/jca19054

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.