zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Weak sharp minima revisited. III: Error bounds for differentiable convex inclusions. (English) Zbl 1163.90016
The term weak sharp minima is coined by Ferris in the late 1980’s to describe an extension of the notion of sharp minima to include the possibility of a non-unique solution set. It unifies a number of important ideas in optimization and many authors have studied this notion extensively. The notion of weak sharp minima is also an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. This is the third paper in this series. Part I of this work [Control Cybern. 31, No. 3, 439–469 (2002; Zbl 1105.90356)] provides the foundation for the theory of weak sharp minima. The basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II [Math. Program. 104, No. 2–3 (B), 235–261 (2005; Zbl 1124.90349)], the applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming are studied. In Part III, the authors continue their study of weak sharp minima by focusing on applications to error bounds for differentiable convex inclusions. A number of standard constraint qualifications for such inclusions are also examined.
90C25Convex programming
90C31Sensitivity, stability, parametric optimization
49J52Nonsmooth analysis (other weak concepts of optimality)
[1]Abadie J.: On the Kuhn–Tucker Theorem. In: Abadie J. (ed) Nonlinear Programming, pp. 21–36. North Holland (1967)
[2]Aubin J.-P. and Ekeland I. (1984). Applied Nonlinear Analysis. Wiley–Interscience, New York
[3]Auslender A. and Crouzeix J.-P. (1988). Global regularity theorem. Math. Oper. Res. 13: 243–253 · Zbl 0655.90059 · doi:10.1287/moor.13.2.243
[4]Auslender A. and Teboulle M. (2003). Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Heidelberg
[5]Bauschke, H.: Projection algorithms and monotone operators. Ph.D. Thesis, Simon Fraser University, Department of Mathematics, Burnaby, British Columbia, V5A 1S6, Canada (1996)
[6]Bauschke H., Borwein J. and Li W. (1999). Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Prog. 86: 135–160 · Zbl 0998.90088 · doi:10.1007/s101070050083
[7]Borwein J. and Lewis A. (2000). Convex Analysis and Nonlinear Optimization, Theory and Examples CMS Books in Mathematics. Springer, New York
[8]Burke J.V. (1991). An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29: 968–998 · Zbl 0737.90060 · doi:10.1137/0329054
[9]Burke J.V. (1991). Calmness and exact penalization. SIAM J. Control Optim. 29: 493–497 · Zbl 0734.90090 · doi:10.1137/0329027
[10]Burke J.V. and Deng S. (2002). Weak sharp minima revisited, part I: Basic theory. Control Cybernetics 31: 439–469
[11]Burke J.V. and Deng S. (2005). Weak sharp minima revisited, part II: Application to linear regularity and error bounds. Math. Program. Ser. B 104: 235–261 · Zbl 1124.90349 · doi:10.1007/s10107-005-0615-2
[12]Burke J.V. and Ferris M.C. (1993). Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31: 1340–1359 · Zbl 0791.90040 · doi:10.1137/0331063
[13]Burke J.V. and Ferris M.C. (1995). A Gauss–Newton method for convex composite optimization. Math. Prog. 71: 179–194
[14]Burke J.V. and Moré J.J. (1988). On the identification of active constraints. SIAM J. Numer. Anal. 25: 1197–1211 · Zbl 0662.65052 · doi:10.1137/0725068
[15]Burke J.V. and Tseng P. (1996). A unified analysis of Hoffman’s bound via Fenchel duality. SIAM J. Optim. 6: 265–282 · Zbl 0849.90093 · doi:10.1137/0806015
[16]Clarke F.H. (1976). A new approach to Lagrange multipliers. Math. Oper. Res. 2: 165–174 · Zbl 0404.90100 · doi:10.1287/moor.1.2.165
[17]Deng S. (1998). Global error bounds for convex inequality systems in Banach spaces. SIAM J. Control Optim. 36: 1240–1249 · Zbl 0909.90250 · doi:10.1137/S0363012995293645
[18]Dontchev A.L. and Rockafellar R.T. (2004). Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12: 79–109 · Zbl 1046.49021 · doi:10.1023/B:SVAN.0000023394.19482.30
[19]Ekeland, I., Temam, R.: Convex analysis and variational problems. North Holland (1976)
[20]Ferris, M.C.: Weak sharp minima and penalty functions in mathematical programming. Tech. Report 779, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin (1988)
[21]Henrion R and Jourani A. (2002). Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13: 520–534 · Zbl 1028.49017 · doi:10.1137/S1052623401386071
[22]Henrion R. and Outrata J. (2001). A subdifferential criterion for calmness of multifunctions. J. Math. Anal. Appl. 258: 110–130 · Zbl 0983.49010 · doi:10.1006/jmaa.2000.7363
[23]Henrion R. and Outrata J. (2005). Calmness of constraint systems with applications. Math. Prog. Ser. B. 104: 437–464 · Zbl 1093.90058 · doi:10.1007/s10107-005-0623-2
[24]Hiriart-Urruty J.-B. and Lemaréchal C. (1993). Convex Analysis and Minimization Algorithms I, volume 306 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg
[25]Hoffman A.J. (1952). On approximate solutions to systems of linear inequalities. J. Res. Nat. Bur. Stand. 49: 263–265
[26]Jourani A. (2000). Hoffman’s error bounds, local controllability and sensitivity analysis. SIAM J. Control Optim. 38: 947–970 · Zbl 0945.46023 · doi:10.1137/S0363012998339216
[27]Klatte D. (1997). Hoffman’s error bound for systems of convex inequalities. In: Fiacco, A.V. (eds) Mathematical Programming with Data Perturbations, pp 185–199. Marcel Dekker Publ., Moscow
[28]Klatte D. and Li W. (1999). Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Prog. 84: 137–160
[29]Lewis A.S. and Pang J.-S. (1998). Error bounds for convex inequality systems. In: Crouzeix, J.P., Martinez- Legaz, J.-E. and Volle, M. (eds) Proceedings of the Fifth International Symposium on Generalized Convexity held in Luminy June 17-21, 1996, pp 75–101. Kluwer Academic Publishers, Dordrecht
[30]Li W. (1997). Abadie’s constraint qualification, metric regularity and error bounds for differentiable convex inequalities. SIAM J. Optim. 7: 966–978 · Zbl 0891.90132 · doi:10.1137/S1052623495287927
[31]Luo X.D. and Luo Z.Q. (1994). Extension of hoffman’s error bound to polynomial systems. SIAM J. Optim. 4: 383–392 · Zbl 0821.90113 · doi:10.1137/0804021
[32]Maguregui, J.: Regular multivalued functions and algorithmic applications. Ph.D. Thesis, University of Wisconsin at Madison, Madison, WI (1977)
[33]Ng K.F. and Yang W.H. (2004). Regularities and their relations to error bounds. Math. Prog. Ser. A 99: 521–538 · Zbl 1077.90050 · doi:10.1007/s10107-003-0464-9
[34]Ngai H.V. and Thera M. (2005). Error bounds for convex differentiable inequality systems in Banach spaces. Math. Program. Ser. B 104: 465–482 · Zbl 1089.49028 · doi:10.1007/s10107-005-0624-1
[35]Pang J.-S. (1997). Error bounds in mathematical programming. Math. Prog. 79: 299–333
[36]Polyak, B.T.: Sharp minima. In: Proceedings of the IIASA Workshop on Generalized Lagrangians and Their Applications: Laxenburg, Austria. Institute of Control Sciences Lecture Notes, Moscow (1979)
[37]Robinson S.M. (1972). Normed convex processes. Trans. Amer. Math. Soc. 174: 127–140 · doi:10.1090/S0002-9947-1972-0313769-9
[38]Robinson S.M. (1975). An application of error bounds for convex programming in a linear space. SIAM J. Control 13: 271–273 · Zbl 0297.90072 · doi:10.1137/0313015
[39]Robinson S.M. (1976). Regularity and stability for convex multivalued functions. Math. Oper. Res. 1: 130–143 · Zbl 0418.52005 · doi:10.1287/moor.1.2.130
[40]Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton
[41]Rockafellar R.T. (1974). Conjugate Duality and Optimization. SIAM, Philadelphia
[42]Rockafellar R.T. and Wets R.J.-B. (1998). Variational Analysis. Springer, Heidelberg
[43]Ursescu C. (1975). Multifunctions with closed convex graph. Czech. Math. J. 25: 438–441
[44]Yosida K. (1980). Functional Analysis. Springer, Heidelberg
[45]Zălinescu, C.: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces. In: Bulatov, V., Baturin, V. (ed.) Proceedings of the 12th Baikal International Conference on Optimization Methods and their Applications, pp. 272–284, Institute of System Dynamics and Control Theory of SB RAS, Irkutsk (2001)
[46]Zheng X.Y. and Ng K.F. (2004). Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14: 757–772 · Zbl 1079.90103 · doi:10.1137/S1052623403423102
[47]Zheng X.Y. and Ng K.F. (2004). Error moduli for conic convex systems on Banach spaces. Math. Oper. Res. 29: 213–228 · Zbl 1082.90120 · doi:10.1287/moor.1030.0088