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Volumn 20, Issue 3, 2009, Pages 1311-1332

Constraint qualifications for extended Farkas's Lemmas and Lagrangian dualities in convex infinite programming

Author keywords

Conic programming; Convex inequality system; Farkas lemma; Strong lagrangian duality

Indexed keywords

CONIC PROGRAMMING; CONSTRAINED MINIMIZATION PROBLEM; CONSTRAINED SYSTEMS; CONSTRAINT QUALIFICATION; FARKAS LEMMA; INEQUALITY SYSTEMS; LAGRANGIAN DUALITY; SEMI-CONTINUOUS; SUFFICIENT CONDITIONS;

EID: 73249145584     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/080739124     Document Type: Article
Times cited : (81)

References (52)
  • 1
    • 33745437485 scopus 로고    scopus 로고
    • Lower semicontinuous regularization for vector-valued mappings
    • M. AÏT MANSOOUR, A. METRANE, AND M. THÉRA, Lower semicontinuous regularization for vector-valued mappings, J. Global Optim., 35 (2006), pp. 283-309.
    • (2006) J. Global Optim , vol.35 , pp. 283-309
    • AÏT MANSOOUR, M.1    METRANE, A.2    THÉRA, M.3
  • 3
    • 61349122119 scopus 로고    scopus 로고
    • Regularity conditions via quasi-relative interior in convex programming
    • R. I. BOŢ, E. R. CSETNEK, AND G. WANKA, Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim., 19 (2008), pp. 217-233.
    • (2008) SIAM J. Optim , vol.19 , pp. 217-233
    • BOŢ, R.I.1    CSETNEK, E.R.2    WANKA, G.3
  • 4
    • 55449133662 scopus 로고    scopus 로고
    • A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces
    • R. I. BOŢ, S. M. GRAD, AND G. WANKA, A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr., 281 (2008), pp. 1088-1107.
    • (2008) Math. Nachr , vol.281 , pp. 1088-1107
    • BOŢ, R.I.1    GRAD, S.M.2    WANKA, G.3
  • 5
    • 34548654625 scopus 로고    scopus 로고
    • On strong and total Lagrange duality for convex optimization problems
    • R. I. BOŢ, S. M. GRAD, AND G. WANKA, On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337 (2008), pp. 1315-1325.
    • (2008) J. Math. Anal. Appl , vol.337 , pp. 1315-1325
    • BOŢ, R.I.1    GRAD, S.M.2    WANKA, G.3
  • 6
    • 43049179419 scopus 로고    scopus 로고
    • New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces
    • R. I. BO̧Ţ, S. M. GRAD, AND G. WANKA, New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces, Nonlinear Anal., 69 (2008), pp. 323-336.
    • (2008) Nonlinear Anal , vol.69 , pp. 323-336
    • BO̧Ţ, R.I.1    GRAD, S.M.2    WANKA, G.3
  • 7
    • 35748973861 scopus 로고    scopus 로고
    • Some new Farkas-type results for inequality systems with DC functions
    • R. I. BOŢ, I. B. HODREA, AND G. WANKA, Some new Farkas-type results for inequality systems with DC functions, J. Global Optim., 39 (2007), pp. 595-608.
    • (2007) J. Global Optim , vol.39 , pp. 595-608
    • BOŢ, R.I.1    HODREA, I.B.2    WANKA, G.3
  • 8
    • 18744402742 scopus 로고    scopus 로고
    • Farkas-type results with conjugate functions
    • R. I. BO̧T AND G. WANKA, Farkas-type results with conjugate functions, SIAM J. Optim., 15 (2005), pp. 540-554.
    • (2005) SIAM J. Optim , vol.15 , pp. 540-554
    • BO̧T, R.I.1    WANKA, G.2
  • 9
    • 30944465215 scopus 로고    scopus 로고
    • An alternative formulation for a new closed cone constraint qualification
    • R. I. BOŢ AND G. WANKA, An alternative formulation for a new closed cone constraint qualification, Nonlinear Anal., 64 (2006), pp. 1367-1381.
    • (2006) Nonlinear Anal , vol.64 , pp. 1367-1381
    • BOŢ, R.I.1    WANKA, G.2
  • 10
    • 33645961296 scopus 로고    scopus 로고
    • A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces
    • R. I. BOŢ AND G. WANKA, A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 64 (2006), pp. 2787-2804.
    • (2006) Nonlinear Anal , vol.64 , pp. 2787-2804
    • BOŢ, R.I.1    WANKA, G.2
  • 11
    • 20144382882 scopus 로고    scopus 로고
    • A simple closure condition for the normal cone intersection formula
    • R. S. BURACHIK AND V. JEYAKUMAR, A simple closure condition for the normal cone intersection formula, Proc. Amer. Math. Soc, 133 (2005), pp. 1741-1748.
    • (2005) Proc. Amer. Math. Soc , vol.133 , pp. 1741-1748
    • BURACHIK, R.S.1    JEYAKUMAR, V.2
  • 12
    • 27244447811 scopus 로고    scopus 로고
    • A new geometric condition for Fenchel's duality in infinite dimensional spaces
    • R. S. BURACHIK AND V. JEYAKUMAR, A new geometric condition for Fenchel's duality in infinite dimensional spaces, Math. Program. Ser. B, 104 (2005), pp. 229-233.
    • (2005) Math. Program. Ser. B , vol.104 , pp. 229-233
    • BURACHIK, R.S.1    JEYAKUMAR, V.2
  • 13
    • 23744457391 scopus 로고    scopus 로고
    • A dual condition for the convex subdifferential sum formula with applications
    • R. S. BURACHIK AND V. JEYAKUMAR, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), pp. 279-290.
    • (2005) J. Convex Anal , vol.12 , pp. 279-290
    • BURACHIK, R.S.1    JEYAKUMAR, V.2
  • 14
    • 0029205133 scopus 로고
    • A Gauss-Newton method for convex composite optimization
    • J. V. BURKE AND M. C. FERRIS, A Gauss-Newton method for convex composite optimization, Math. Program., 71 (1995), pp. 179-194.
    • (1995) Math. Program , vol.71 , pp. 179-194
    • BURKE, J.V.1    FERRIS, M.C.2
  • 15
    • 0000881030 scopus 로고
    • Sous-différentiels de foncitions convexes composées
    • C. COMBARI, M. LAGHDIR, AND L. THIBAULT, Sous-différentiels de foncitions convexes composées, Ann. Sci. Math. Québec, 18 (1994), pp. 119-148.
    • (1994) Ann. Sci. Math. Québec , vol.18 , pp. 119-148
    • COMBARI, C.1    LAGHDIR, M.2    THIBAULT, L.3
  • 16
    • 33645146580 scopus 로고    scopus 로고
    • From linear to convex systems: Consistency, Farkas lemma and applications
    • N. DINH, M. A. GOBERNA, AND M. A. LÓ PEZ, From linear to convex systems: Consistency, Farkas lemma and applications, J. Convex Anal., 13 (2006), pp. 279-290.
    • (2006) J. Convex Anal , vol.13 , pp. 279-290
    • DINH, N.1    GOBERNA, M.A.2    LÓ PEZ, M.A.3
  • 17
    • 34648820239 scopus 로고    scopus 로고
    • New Farkas-type constraint qualifications in convex infinite programming
    • N. DINH, M. A. GOBERNA, M. A. LÓPEZ, AND T. Q. SON, New Farkas-type constraint qualifications in convex infinite programming, ESAIM Control Optim. Calc. Var., 13 (2007), pp. 580-597.
    • (2007) ESAIM Control Optim. Calc. Var , vol.13 , pp. 580-597
    • DINH, N.1    GOBERNA, M.A.2    LÓPEZ, M.A.3    SON, T.Q.4
  • 18
    • 17444364855 scopus 로고    scopus 로고
    • Sequential Lagrangian conditions for convex programs with applications to semidefinite programming
    • N. DINH, V. JEYAKUMAR, AND G. M. LEE, Sequential Lagrangian conditions for convex programs with applications to semidefinite programming, J. Optim. Theory Appl., 125 (2005), pp. 85-112.
    • (2005) J. Optim. Theory Appl , vol.125 , pp. 85-112
    • DINH, N.1    JEYAKUMAR, V.2    LEE, G.M.3
  • 19
    • 73549104284 scopus 로고    scopus 로고
    • Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs
    • to appear
    • N. DINH, B. MORDUKHOVICH, AND T. T. A. NGHIA, Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs, Math. Program., to appear.
    • Math. Program
    • DINH, N.1    MORDUKHOVICH, B.2    NGHIA, T.T.A.3
  • 20
    • 77952203383 scopus 로고    scopus 로고
    • Qualification and optimality conditions for convex and DC programs with infinite constraints
    • to appear
    • N. DINH, B. MORDUKHOVICH, AND T. T. A. NGHIA, Qualification and optimality conditions for convex and DC programs with infinite constraints, Acta Math. Vietnam., to appear.
    • Acta Math. Vietnam
    • DINH, N.1    MORDUKHOVICH, B.2    NGHIA, T.T.A.3
  • 21
    • 77952524496 scopus 로고    scopus 로고
    • A closedness condition and its applications to DC programs with convex constraints
    • to appear
    • N. DINH, T. T. A. NGHIA, AND G. VALLET, A closedness condition and its applications to DC programs with convex constraints, Optimization, to appear.
    • Optimization
    • DINH, N.1    NGHIA, T.T.A.2    VALLET, G.3
  • 22
    • 45849105598 scopus 로고    scopus 로고
    • Farkas-type results and duality for DC programs with convex constraints
    • N. DINH, G. VALLET, AND T. T. A. NGHIA, Farkas-type results and duality for DC programs with convex constraints, J. Convex Anal., 15 (2008), pp. 235-262.
    • (2008) J. Convex Anal , vol.15 , pp. 235-262
    • DINH, N.1    VALLET, G.2    NGHIA, T.T.A.3
  • 23
    • 67349154226 scopus 로고    scopus 로고
    • Necessary and sufficient conditions for solvability of systems of infinite convex inequalities
    • M. A. GOBERNA, V. JEYAKUMAR, AND M. A. LÓPEZ, Necessary and sufficient conditions for solvability of systems of infinite convex inequalities, Nonlinear Anal., 68 (2008), pp. 1184-1194.
    • (2008) Nonlinear Anal , vol.68 , pp. 1184-1194
    • GOBERNA, M.A.1    JEYAKUMAR, V.2    LÓPEZ, M.A.3
  • 25
    • 67649560531 scopus 로고    scopus 로고
    • Subdifferential calculus rules in convex analysis: A unifying approach via pointwise supremum functions
    • A. HANTOUTE, M. A. LÓPEZ, AND C. ZǍLINESCU, Subdifferential calculus rules in convex analysis: A unifying approach via pointwise supremum functions, SIAM J. Optim., 19 (2008), pp. 863-882.
    • (2008) SIAM J. Optim , vol.19 , pp. 863-882
    • HANTOUTE, A.1    LÓPEZ, M.A.2    ZǍLINESCU, C.3
  • 26
    • 0242587976 scopus 로고    scopus 로고
    • Farkas lemma: Generalizations
    • C. A. Floudas and P. Pardalos, eds, Kluwer, Dordrecht
    • V. JEYAKUMAR, Farkas lemma: Generalizations, in Encyclopedia of Optimization II, C. A. Floudas and P. Pardalos, eds., Kluwer, Dordrecht, 2001, pp. 87-91.
    • (2001) Encyclopedia of Optimization II , pp. 87-91
    • JEYAKUMAR, V.1
  • 27
    • 0242679729 scopus 로고    scopus 로고
    • Characterizing set containments involving infinite convex constraints and reverse-convex constraints
    • V. JEYAKUMAR, Characterizing set containments involving infinite convex constraints and reverse-convex constraints, SIAM J. Optim., 13 (2003), pp. 947-959.
    • (2003) SIAM J. Optim , vol.13 , pp. 947-959
    • JEYAKUMAR, V.1
  • 28
    • 29144527037 scopus 로고    scopus 로고
    • The strong conical hull intersection property for convex programming
    • V. JEYAKUMAR, The strong conical hull intersection property for convex programming, Math. Program. Ser. A, 106 (2006), pp. 81-92.
    • (2006) Math. Program. Ser. A , vol.106 , pp. 81-92
    • JEYAKUMAR, V.1
  • 29
    • 38049050343 scopus 로고    scopus 로고
    • Constraint qualifications characterizing Lagrangian duality in convex optimization
    • V. JEYAKUMAR, Constraint qualifications characterizing Lagrangian duality in convex optimization, J. Optim. Theory Appl., 136 (2008), pp. 31-41.
    • (2008) J. Optim. Theory Appl , vol.136 , pp. 31-41
    • JEYAKUMAR, V.1
  • 30
    • 2442552101 scopus 로고    scopus 로고
    • New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs
    • V. JEYAKUMAR, G. M. LEE, and N. DINH, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM J. Optim., 14 (2003), pp. 534-547.
    • (2003) SIAM J. Optim , vol.14 , pp. 534-547
    • JEYAKUMAR, V.1    LEE, G.M.2    DINH, N.3
  • 31
    • 42149160025 scopus 로고    scopus 로고
    • Complete characterization of stable Farkas' lemma and coneconvex programming duality
    • V. JEYAKUMAR AND G. M. LEE, Complete characterization of stable Farkas' lemma and coneconvex programming duality, Math. Program. Ser. A, 114 (2008), pp. 335-347.
    • (2008) Math. Program. Ser. A , vol.114 , pp. 335-347
    • JEYAKUMAR, V.1    LEE, G.M.2
  • 32
    • 5444255683 scopus 로고    scopus 로고
    • Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs
    • V. JEYAKUMAR, G. M. LEE, and N. DINH, Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs, J. Optim. Theory Appl., 123 (2004), pp. 83-103.
    • (2004) J. Optim. Theory Appl , vol.123 , pp. 83-103
    • JEYAKUMAR, V.1    LEE, G.M.2    DINH, N.3
  • 33
    • 23744470880 scopus 로고    scopus 로고
    • Limiting ∈-subgradient characterizations of constrained best approximation
    • V. JEYAKUMAR AND H. MOHEBI, Limiting ∈-subgradient characterizations of constrained best approximation, J. Approx. Theory, 135 (2005), pp. 145-159.
    • (2005) J. Approx. Theory , vol.135 , pp. 145-159
    • JEYAKUMAR, V.1    MOHEBI, H.2
  • 34
    • 0037272070 scopus 로고    scopus 로고
    • On best uniform restricted range approximation in complex-valued continuous function spaces
    • C. LI, On best uniform restricted range approximation in complex-valued continuous function spaces, J. Approx. Theory, 120 (2003), pp. 71-84.
    • (2003) J. Approx. Theory , vol.120 , pp. 71-84
    • LI, C.1
  • 35
    • 2442583687 scopus 로고    scopus 로고
    • Constraint qualification, the strong CHIP, and best approximation with convex constraints in Banach spaces
    • C. LI and K. F. NG, Constraint qualification, the strong CHIP, and best approximation with convex constraints in Banach spaces, SIAM J. Optim., 14 (2003), pp. 584-607.
    • (2003) SIAM J. Optim , vol.14 , pp. 584-607
    • LI, C.1    NG, K.F.2
  • 36
    • 18744368765 scopus 로고    scopus 로고
    • On constraint qualification for an infinite system of convex inequalities in a Banach space
    • C. LI AND K. F. NG, On constraint qualification for an infinite system of convex inequalities in a Banach space, SIAM J. Optim., 15 (2005), pp. 488-512.
    • (2005) SIAM J. Optim , vol.15 , pp. 488-512
    • LI, C.1    NG, K.F.2
  • 37
    • 26844494126 scopus 로고    scopus 로고
    • On best restricted range approximation in continuous complex-valued function spaces
    • C. LI AND K. F. NG, On best restricted range approximation in continuous complex-valued function spaces, J Approx. Theory, 136 (2005), pp. 159-181.
    • (2005) J Approx. Theory , vol.136 , pp. 159-181
    • LI, C.1    NG, K.F.2
  • 38
    • 33646734955 scopus 로고    scopus 로고
    • Strong CHIP for infinite system of closed convex sets in normed linear spaces
    • C. LI AND K. F. NG, Strong CHIP for infinite system of closed convex sets in normed linear spaces, SIAM J. Optim., 16 (2005), pp. 311-340.
    • (2005) SIAM J. Optim , vol.16 , pp. 311-340
    • LI, C.1    NG, K.F.2
  • 39
    • 44649127428 scopus 로고    scopus 로고
    • Majorizing functions and convergence of the Gauss-Newton method for convex composite optimization
    • C. LI AND K. F. NG, Majorizing functions and convergence of the Gauss-Newton method for convex composite optimization, SIAM J. Optim., 18 (2007), pp. 613-642.
    • (2007) SIAM J. Optim , vol.18 , pp. 613-642
    • LI, C.1    NG, K.F.2
  • 40
    • 44649108980 scopus 로고    scopus 로고
    • The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces
    • C. LI, K. F. NG, and T. K. PONG, The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces, SIAM J. Optim., 18 (2007), pp. 643-665.
    • (2007) SIAM J. Optim , vol.18 , pp. 643-665
    • LI, C.1    NG, K.F.2    PONG, T.K.3
  • 41
    • 61349182011 scopus 로고    scopus 로고
    • Constraint qualifications for convex inequality systems with applications in constrained optimization
    • C. LI, K. F. NG, and T. K. PONG, Constraint qualifications for convex inequality systems with applications in constrained optimization, SIAM J. Optim., 19 (2008), pp. 163-187.
    • (2008) SIAM J. Optim , vol.19 , pp. 163-187
    • LI, C.1    NG, K.F.2    PONG, T.K.3
  • 42
    • 0034550220 scopus 로고    scopus 로고
    • Constraint qualifications for semi-infinite systems of convex inequalities
    • W. LI, C. NAHAK, AND I. SINGER, Constraint qualifications for semi-infinite systems of convex inequalities, SIAM J. Optim., 11 (2000), pp. 31-52.
    • (2000) SIAM J. Optim , vol.11 , pp. 31-52
    • LI, W.1    NAHAK, C.2    SINGER, I.3
  • 44
    • 0009119348 scopus 로고
    • Semi-continuous mappings in general topology
    • J. P. PENOT AND M. THÉRA, Semi-continuous mappings in general topology, Arch. Math. (Basel), 38 (1982), pp. 158-166.
    • (1982) Arch. Math. (Basel) , vol.38 , pp. 158-166
    • PENOT, J.P.1    THÉRA, M.2
  • 45
    • 84972539430 scopus 로고
    • Extension of Fenchel's duality theorem for convex functions
    • R. T. ROCKAFELLAR, Extension of Fenchel's duality theorem for convex functions, Duke Math. J., 33 (1966), pp. 81-89.
    • (1966) Duke Math. J , vol.33 , pp. 81-89
    • ROCKAFELLAR, R.T.1
  • 46
    • 84966228167 scopus 로고
    • First and second-order epidifferentiability in nonlinear programming
    • R. T. ROCKAFELLAR, First and second-order epidifferentiability in nonlinear programming, Trans. Amer. Math. Soc., 307 (1988), pp. 75-108.
    • (1988) Trans. Amer. Math. Soc , vol.307 , pp. 75-108
    • ROCKAFELLAR, R.T.1
  • 48
    • 73249149124 scopus 로고    scopus 로고
    • G. S. SMIRNOV AND R. G. SMIRNOV, Best uniform restricted range approximation of complexvalued functions, C. R. Math. Acad. Sci. Canada, 19 (1997), pp. 58-63.
    • G. S. SMIRNOV AND R. G. SMIRNOV, Best uniform restricted range approximation of complexvalued functions, C. R. Math. Acad. Sci. Canada, 19 (1997), pp. 58-63.
  • 49
    • 0000232172 scopus 로고    scopus 로고
    • Best uniform approximation of complex-valued functions by generalized polynomials having restricted range
    • G. S. SMIRNOV AND R. G. SMIRNOV, Best uniform approximation of complex-valued functions by generalized polynomials having restricted range, J. Approx. Theory, 100 (1999), pp. 284-303.
    • (1999) J. Approx. Theory , vol.100 , pp. 284-303
    • SMIRNOV, G.S.1    SMIRNOV, R.G.2
  • 52
    • 78651460024 scopus 로고    scopus 로고
    • Strong KKT conditions and weak sharp solutions in convex composite optimization
    • to appear
    • X. Y. ZHENG AND K. F. NG, Strong KKT conditions and weak sharp solutions in convex composite optimization, Math. Program., to appear.
    • Math. Program
    • ZHENG, X.Y.1    NG, K.F.2


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