Abstract
The paper is devoted to studying the Hoffman global error bound for convex quadratic/affine inequality/equality systems in the context of Banach spaces. We prove that the global error bound holds if the Hoffman local error bound is satisfied for each subsystem at some point of the solution set of the system under consideration. This result is applied to establishing the equivalence between the Hoffman error bound and the Abadie qualification condition, as well as a general version of Wang & Pang's result [30], on error bound of Hölderian type. The results in the present paper generalize and unify recent works by Luo & Luo in [17], Li in [16] and Wang & Pang in [30].
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References
Azé D., Corvellec J-N.: On the sensitivity analysis of Hofman constants for systems of linear inequalities. SIAM J. Otim. 12, 91–927 (2002)
Azé D., Corvellec J-N.: Characterization of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optiim. Calc. Var. 10, 409–425 (2004)
Auslender A., Crouzeix J.P.: Global regularily theorems. Math. Oper. Res. 13, 243–253 (1998)
Bartelt M., Li, W.: Exact order of Hoffman's error bounds for eliptic quadratic inequalities derived from vector-valued Chebyshev approximation. Math. Program. Ser. B (2000) pp. 223–253
Burke J.V., Tseng P.: A unified analysis of Hoffman's bound via Fenchel duality. SIAM J. Optim. 6, 265–282 (1996)
Borwein J.M., Fitzpatrick S.: Existence of nearest points in Banach spaces. Can. J. Math. (XLI)(4), 702–720 (1989)
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York, 1983
Chou C.C., Liu C.G., Ng K. F.: On error bounds for systems. Preprint, 2001
Cornejo O., Jourani A., Zalinescu C.: Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems. J. Optim. Theory Appl. 95, 127–148 (1997)
Hoffman A.J.: On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Standards 49, 263–265 (1952)
Ioffe A.: Regular Points of Lipschitz functions, Trans. Am. Soc. 251, 61–69 (1979)
Klatte D., Li W.: Asymptotic Constraint qualifications and error bounds for convex inequalities. Math. Program. 84, 137–160 (1999)
Jourani A.: Hoffman's error bound, local controlability and sensivity analysis. SIAM J. Control Optim. 38, 947–970 (2000)
Jourani A., Thibault L.: Metric inequality and subdifferential calculus in Banach spaces. Set-Valued Anal. 3, 87–100 (1995)
Lewis A.S., Pang J.S.: Error bound for convex inequality systems. In: Generalized convexity and generalized monotonicity, Crouzeix J.P. et al. (eds.), Kluwer Academic Pub., Dordrecht, 1997, pp. 75–100
Li W.: Abadie's constraint qualification, metric regularity. and error bound for differentiable convex inequalities, SIAM J.Optim. 7(4), 966–978 (1997)
Luo X.D., Luo Z.Q.: Extension of Hoffman's error bound to polynominal systems. SIAM J. Optim. 4, 383–392 (1994)
Luo Z.Q, Sturn J.F.: Error bounds for quadratic systems. In: High Perpormance Optimization, H. Frenk et al. (eds.), Kluwer Dordrecht, Netherlands, 2000, pp. 383–404
Luo Z.Q., Pang J.S.: Error bounds for the analytic systems and their applications. Math. Program. 67, 1–28 (1994)
Mangasarian O.L.: A condition number for differentiable convex inequalities. Math. of Oper. Res. 10, 175–179 (1985)
Mangasarian O.L., Shiau T.H.: Error bounds for monotone linear complementarity problems. Math Program. 36, 81–89 (1986)
Ngai H., Théra M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9, 187–216 (2001)
Ngai H., Théra M.: Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization. Set-Valued Anal. 12, 195–223 (2004)
Ng K.F., Zheng X.Y.: Error bound for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12, 1–17 (2001)
Pang J.S.: Error bounds in Mathematical programming. Math. Program. Ser. B 79, 299–232 (1997)
Phelps R.R.: Convex functions, Monotone operators and Differentiablility. (2nd edn), Lecture Notes in Math. 1364, Springer-Verlag, Berlin, 1993
Rockafellar R.T.: Convex Analysis. Princeton University Press, 1970
Robinson S.M.: An application of error bound for convex programming in a linear space. SIAM J. Control. Optim. 13, 271–273 (1975)
Tiba D., Zalinescu C.: On the Necessity of some Constraint Qualification Conditions in Convex Programming. J. Convex Analysis, 95–110 (2004)
Wang T., Pang J.S.: Global error bound for convex quadratic inequality systems. Optimization, 31, 1–12 (1994)
Wu Z., Ye J.: On error bounds for lower semicontinuous functions. Mat. Program 92, 301–314 (2002)
Zalinescu C.: Weak sharp minimum, well behaving functions and global error bounds for convex inequalities in Banach spaces. Proc. 12-th Baikal Int. Conf. on Optimization Methods and their Applications, V. Bulatov and V. Baturin (eds.), Institute of System Dynamics and Control Theory of SB RAS, Irkutsk, 2001, pp. 272–284
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Dedicated to Terry Rockafellar in honor of his 70 th birthday
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Ngai, H., Théra, M. Error bounds for convex differentiable inequality systems in Banach spaces. Math. Program. 104, 465–482 (2005). https://doi.org/10.1007/s10107-005-0624-1
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DOI: https://doi.org/10.1007/s10107-005-0624-1