Abstract
The aim of this paper is to implement some new techniques, based on conjugate duality in convex optimization, for proving the existence of global error bounds for convex inequality systems. First of all, we deal with systems described via one convex inequality and extend the achieved results, by making use of a celebrated scalarization function, to convex inequality systems expressed by means of a general vector function. We also propose a second approach for guaranteeing the existence of global error bounds of the latter, which meanwhile sharpens the classical result of Robinson.
Similar content being viewed by others
References
Boţ RI (2010) Conjugate duality in convex optimization. Lecture notes in economics and mathematical systems, vol 637. Springer, Berlin
Boţ RI, Wanka G (2008) The conjugate of the pointwise maximum of two convex functions revisited. J Glob Optim 41(4):625–632
Boţ RI, Grad S-M, Wanka G (2009) Duality in vector optimization. Springer, Berlin
Cornejo O, Jourani A, Zălinescu C (1997) Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems. J Optim Theory Appl 95(1):127–148
Csetnek ER (2010) Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operators. Logos, Berlin
Ekeland I, Témam R (1976) Convex analysis and variational problems. North-Holland, Amsterdam
Hiriart-Urruty J-B (1979a) New concepts in nondifferentiable programming. Bull Soc Math Fr, Mém 60:57–85
Hiriart-Urruty J-B (1979b) Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math Oper Res 4(1):79–97
Hiriart-Urruty J-B, Lemaréchal C (1993) Convex analysis and minimization algorithms I, II. Springer, Berlin
Hoffman AJ (1952) On approximate solutions of systems of linear inequalities. J Res Natl Bur Stand 49:263–265
Hu H, Wang Q (2010) Local and global error bounds for proper function. Pac J Optim 6(1):177–186
Klatte D (1998) Hoffman’s error bound for systems of convex inequalities. In: Fiacco AV (ed) Mathematical programming with data perturbations. Dekker, New York, pp 185–199
Lewis AS, Pang J-S (1998) Error bounds for convex inequality systems. In: Generalized convexity, generalized monotonicity: recent results, Luminy, 1996. Nonconvex optimization and its applications, vol 27. Kluwer Academic, Dordrecht, pp 75–110
Li W, Singer I (1998) Global error bounds for convex multifunctions and applications. Math Oper Res 23(2):443–462
Luo X-D, Luo Z-Q (1994) Extension of Hoffman’s error bound to polynomial systems. SIAM J Optim 4(2):383–392
Pang J-S (1997) Error bounds in mathematical programming. Math Program, Ser A 79(1–3):299–332
Robinson SM (1975) An application of error bounds for convex programming in a linear space. SIAM J Control Optim 13:271–273
Rockafellar RT (1966) Level sets and continuity of conjugate convex functions. Trans Am Math Soc 123(1):43–63
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Simons S (2008) From Hahn–Banach to monotonicity. Springer, Berlin
Zaffaroni A (2003) Degrees of efficiency and degrees of minimality. SIAM J Control Optim 42(3):1071–1086
Zălinescu C (2001) Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces. In: Proceedings of the 12th Baikal International conference on optimization methods and their applications, Irkutsk, Russia, pp 272–284
Zălinescu C (2002) Convex analysis in general vector spaces. World Scientific, Singapore
Zălinescu C (2003) A nonlinear extension of Hoffman’s error bound for linear inequalities. Math Oper Res 28(3):524–532
Zheng XY (2003) Error bounds for set inclusions. Sci China Ser A 46:750–763
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Marco A. López on the occasion of his 60th birthday.
Research of R.I. Boţ was partially supported by DFG (German Research Foundation), project WA 922/1-3.
Rights and permissions
About this article
Cite this article
Boţ, R.I., Csetnek, E.R. Error bound results for convex inequality systems via conjugate duality. TOP 20, 296–309 (2012). https://doi.org/10.1007/s11750-011-0187-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-011-0187-7