Conditioning for optimization problems under general perturbations
Given a function f ∈ C 1 , 1 ( B ( 0 , r ) ) , where B ( 0 , r ) denotes a ball of radius r in a real Banach space E , we provide the definition of a positive extended real number c ˆ ( f ) defined through the function, that plays a role in the study of the sensitivity of the Argmin map of the pertu...
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| Vydané v: | Nonlinear analysis Ročník 75; číslo 1; s. 37 - 45 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Amsterdam
Elsevier Ltd
2012
Elsevier |
| Predmet: | |
| ISSN: | 0362-546X, 1873-5215 |
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| Abstract | Given a function
f
∈
C
1
,
1
(
B
(
0
,
r
)
)
, where
B
(
0
,
r
)
denotes a ball of radius
r
in a real Banach space
E
, we provide the definition of a positive extended real number
c
ˆ
(
f
)
defined through the function, that plays a role in the study of the sensitivity of the Argmin map of the perturbed function
F
g
(
p
,
u
)
=
f
(
u
)
−
g
(
p
,
u
)
. This number coincides with the number
c
2
(
f
)
introduced by Zolezzi (2003) if linear perturbations
g
(
p
,
u
)
=
〈
p
,
u
〉
are considered. |
|---|---|
| AbstractList | Given a function
f
∈
C
1
,
1
(
B
(
0
,
r
)
)
, where
B
(
0
,
r
)
denotes a ball of radius
r
in a real Banach space
E
, we provide the definition of a positive extended real number
c
ˆ
(
f
)
defined through the function, that plays a role in the study of the sensitivity of the Argmin map of the perturbed function
F
g
(
p
,
u
)
=
f
(
u
)
−
g
(
p
,
u
)
. This number coincides with the number
c
2
(
f
)
introduced by Zolezzi (2003) if linear perturbations
g
(
p
,
u
)
=
〈
p
,
u
〉
are considered. Given a function f[isin]C super(1,1(B(0,r)), where B(0,r) denotes a ball of radius r in a real Banach space E, we provide the definition of a positive extended real number [inline image] defined through the function, that plays a role in the study of the sensitivity of the Argmin map of the perturbed function F) sub(g)(p,u)=f(u)-g(p,u). This number coincides with the number c sub(2(f) introduced by Zolezzi (2003) if linear perturbations g(p,u)=a pi ,ua[control] are considered.) |
| Author | Kassay, G. Bianchi, M. Pini, R. |
| Author_xml | – sequence: 1 givenname: M. surname: Bianchi fullname: Bianchi, M. email: monica.bianchi@unicatt.it organization: Università Cattolica del Sacro Cuore di Milano, Italy – sequence: 2 givenname: G. surname: Kassay fullname: Kassay, G. email: kassay@math.ubbcluj.ro organization: Babes Bolyai University Cluj, Romania – sequence: 3 givenname: R. surname: Pini fullname: Pini, R. email: rita.pini@unimib.it organization: Università degli Studi di Milano Bicocca, Italy |
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| Cites_doi | 10.1023/A:1004600631797 10.1007/BF02288367 10.1137/S1052623402411885 10.1007/BF01585941 10.1137/S1052623496309296 |
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| Keywords | 90C31 Condition number theorem Sensitivity 49K40 Convex optimization Nonlinear analysis Optimization method Condition number Mathematical model |
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| References | Zolezzi (br000035) 2003; 14 Zolezzi (br000030) 2002; 9 Poliquin, Rockafellar (br000025) 1998; 8 Renegar (br000020) 1995; 70 Lucchetti (br000005) 2006 Dontchev, Rockafellar (br000040) 2009 Kassay, Kolumban (br000050) 2000; 107 Donthchev, Zolezzi (br000010) 1993; vol. 1543 Hu, Papageorgiou (br000045) 1997 Eckart, Young (br000015) 1936; 1 Poliquin (10.1016/j.na.2011.07.061_br000025) 1998; 8 Hu (10.1016/j.na.2011.07.061_br000045) 1997 Kassay (10.1016/j.na.2011.07.061_br000050) 2000; 107 Eckart (10.1016/j.na.2011.07.061_br000015) 1936; 1 Dontchev (10.1016/j.na.2011.07.061_br000040) 2009 Zolezzi (10.1016/j.na.2011.07.061_br000035) 2003; 14 Renegar (10.1016/j.na.2011.07.061_br000020) 1995; 70 Zolezzi (10.1016/j.na.2011.07.061_br000030) 2002; 9 Lucchetti (10.1016/j.na.2011.07.061_br000005) 2006 Donthchev (10.1016/j.na.2011.07.061_br000010) 1993; vol. 1543 |
| References_xml | – volume: 14 start-page: 507 year: 2003 end-page: 516 ident: br000035 article-title: Condition number theorems in optimization publication-title: SIAM J. Optim. – volume: vol. 1543 year: 1993 ident: br000010 publication-title: Well-Posed Optimization Problems – year: 2006 ident: br000005 publication-title: Convexity and Well-Posed Problems – year: 2009 ident: br000040 article-title: Implicit Functions and Solution Mappings – volume: 107 start-page: 35 year: 2000 end-page: 50 ident: br000050 article-title: Multivalued parametric variational inequalities with publication-title: J. Optim. Theory Appl. – volume: 1 start-page: 211 year: 1936 end-page: 218 ident: br000015 article-title: The approximation of one matrix by another of lower rank publication-title: Psychometrika – volume: 9 start-page: 693 year: 2002 end-page: 700 ident: br000030 article-title: On the distance theorem in quadratic optimization publication-title: J. Convex Anal. – volume: 70 start-page: 279 year: 1995 end-page: 351 ident: br000020 article-title: Linear programming, complexity theory and elementary functional analysis publication-title: Math. Programming – volume: 8 start-page: 287 year: 1998 end-page: 299 ident: br000025 article-title: Tilt stability of a local minimum publication-title: SIAM J. Optim. – year: 1997 ident: br000045 article-title: Handbook of Multivalued Analysis-Volume I: Theory – year: 2009 ident: 10.1016/j.na.2011.07.061_br000040 – year: 2006 ident: 10.1016/j.na.2011.07.061_br000005 – year: 1997 ident: 10.1016/j.na.2011.07.061_br000045 – volume: 9 start-page: 693 year: 2002 ident: 10.1016/j.na.2011.07.061_br000030 article-title: On the distance theorem in quadratic optimization publication-title: J. Convex Anal. – volume: 107 start-page: 35 year: 2000 ident: 10.1016/j.na.2011.07.061_br000050 article-title: Multivalued parametric variational inequalities with α-pseudomonotone maps publication-title: J. Optim. Theory Appl. doi: 10.1023/A:1004600631797 – volume: 1 start-page: 211 year: 1936 ident: 10.1016/j.na.2011.07.061_br000015 article-title: The approximation of one matrix by another of lower rank publication-title: Psychometrika doi: 10.1007/BF02288367 – volume: vol. 1543 year: 1993 ident: 10.1016/j.na.2011.07.061_br000010 – volume: 14 start-page: 507 year: 2003 ident: 10.1016/j.na.2011.07.061_br000035 article-title: Condition number theorems in optimization publication-title: SIAM J. Optim. doi: 10.1137/S1052623402411885 – volume: 70 start-page: 279 year: 1995 ident: 10.1016/j.na.2011.07.061_br000020 article-title: Linear programming, complexity theory and elementary functional analysis publication-title: Math. Programming doi: 10.1007/BF01585941 – volume: 8 start-page: 287 year: 1998 ident: 10.1016/j.na.2011.07.061_br000025 article-title: Tilt stability of a local minimum publication-title: SIAM J. Optim. doi: 10.1137/S1052623496309296 |
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| Snippet | Given a function
f
∈
C
1
,
1
(
B
(
0
,
r
)
)
, where
B
(
0
,
r
)
denotes a ball of radius
r
in a real Banach space
E
, we provide the definition of a positive... Given a function f[isin]C super(1,1(B(0,r)), where B(0,r) denotes a ball of radius r in a real Banach space E, we provide the definition of a positive extended... |
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| SubjectTerms | Algebra Banach space Calculus of variations and optimal control Condition number theorem Conditioning Convex optimization Exact sciences and technology Linear and multilinear algebra, matrix theory Mathematical analysis Mathematics Nonlinearity Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Numerical methods in mathematical programming, optimization and calculus of variations Numerical methods in optimization and calculus of variations Optimization Perturbation methods Real numbers Sciences and techniques of general use Sensitivity |
| Title | Conditioning for optimization problems under general perturbations |
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