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Volumn 48, Issue 5, 2009, Pages 3403-3432

Nonsmooth optimization using Mordukhovich's subdifferential

Author keywords

(Semi ) infinite programming; (Weak proper) Paretooptimality; Asplund space; Convex analysis; Functional analysis; Mordukhovich's subdifferential; Nonsmooth optimization; Penalization; Variational function (inequality)

Indexed keywords

BANACH SPACES; FUNCTIONAL ANALYSIS; FUNCTIONAL PROGRAMMING; MATHEMATICAL PROGRAMMING; NONLINEAR PROGRAMMING; OPTIMIZATION; PARETO PRINCIPLE; SITE SELECTION; VARIATIONAL TECHNIQUES; VECTOR SPACES;

EID: 77958007218     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/070710664     Document Type: Article
Times cited : (49)

References (59)
  • 1
    • 0038821453 scopus 로고    scopus 로고
    • Generalized invexity and duality in multiobjective programming problems
    • B. AGHEZZAF and M. HACHIMI, Generalized invexity and duality in multiobjective programming problems, J. Global Optim., 18(2000), pp. 91-101.
    • (2000) J. Global Optim. , vol.18 , pp. 91-101
    • Aghezzaf, B.1    Hachimi, M.2
  • 2
    • 0035891482 scopus 로고    scopus 로고
    • (p, r)-invex sets and functions
    • T. ANTCZAK, (p, r)-invex sets and functions, J. Math. Anal. Appl., 263(2001), pp. 355-379.
    • (2001) J. Math. Anal. Appl. , vol.263 , pp. 355-379
    • Antczak, T.1
  • 3
    • 13444253970 scopus 로고    scopus 로고
    • Mean value in invexity analysis
    • T. ANTCZAK, Mean value in invexity analysis, Nonlinear Anal., 60(2005), pp. 1473-1484.
    • (2005) Nonlinear Anal. , vol.60 , pp. 1473-1484
    • Antczak, T.1
  • 5
    • 35148820393 scopus 로고    scopus 로고
    • Necessary conditions in multiobjective optimization with equilibrium constraints
    • T. Q. BAO, P. GUPTA, and B. S. MORDUKHOVICH, Necessary conditions in multiobjective optimization with equilibrium constraints, J. Optim. Theory Appl., 134(2007), pp. 179-203.
    • (2007) J. Optim. Theory Appl. , vol.134 , pp. 179-203
    • Bao, T.Q.1    Gupta, P.2    Mordukhovich, B.S.3
  • 6
    • 69949085190 scopus 로고    scopus 로고
    • Relative Pareto minimizers for multiobjective problems: Existence and optimality conditions
    • T. Q. BAO and B. S. MORDUKHOVICH, Relative Pareto minimizers for multiobjective problems: Existence and optimality conditions, Math. Program. 122(2010), pp. 301-347.
    • (2010) Math. Program , vol.122 , pp. 301-347
    • Bao, T.Q.1    Mordukhovich, B.S.2
  • 7
    • 14644415575 scopus 로고    scopus 로고
    • Preinvex functions and weak efficient solutions for some vectorial optimization problem in Banach spaces
    • L. BATISTA DOS SANTOS, R. OSUNA-GÓMEZ, M. A. ROJAS-MEDAR, and A. RUFIÁN-LIZANA, Preinvex functions and weak efficient solutions for some vectorial optimization problem in Banach spaces, Comput. Math. Appl., 48 (2004) pp. 885-895.
    • (2004) Comput. Math. Appl. , vol.48 , pp. 885-895
    • Batista Dos Santos, L.1    Osuna-Gómez, R.2    Rojas-Medar, M.A.3    Rufián-Lizana, A.4
  • 9
    • 0027683985 scopus 로고
    • Optimality conditions and subdifferentiable multiobjective fractional programming
    • C. R. BECTOR, S. CHANDRA, and I. HUSAIN, Optimality conditions and subdifferentiable multiobjective fractional programming, J. Optim. Theory Appl., 79 (1993) pp. 105-125.
    • (1993) J. Optim. Theory Appl. , vol.79 , pp. 105-125
    • Bector, C.R.1    Chandra, S.2    Husain, I.3
  • 12
    • 0033246201 scopus 로고    scopus 로고
    • Optimality conditions for Pareto nonsmooth nonconvex programming in Banach spaces
    • A. J. V. BRANDÃO, M. A. ROJAS-MEDAR, and G. N. SILVA, Optimality conditions for Pareto nonsmooth nonconvex programming in Banach spaces, J. Optim. Theory Appl., 103(1999), pp. 65-73.
    • (1999) J. Optim. Theory Appl. , vol.103 , pp. 65-73
    • Brandão, A.J.V.1    Rojas-Medar, M.A.2    Silva, G.N.3
  • 13
    • 0043266457 scopus 로고    scopus 로고
    • The gap function of a convex multicriteria optimization problem
    • G. Y. CHEN, C. J. GOH, and X. Q. YANG, The gap function of a convex multicriteria optimization problem, European J. Oper. Res., 111(1998), pp. 142-151.
    • (1998) European J. Oper. Res. , vol.111 , pp. 142-151
    • Chen, G.Y.1    Goh, C.J.2    Yang, X.Q.3
  • 17
    • 84971124250 scopus 로고
    • Invex functions and constrained local minima
    • B. D. CRAVEN, Invex functions and constrained local minima, Bull. Austral. Math. Soc., 24(1981), pp. 357-366.
    • (1981) Bull. Austral. Math. Soc. , vol.24 , pp. 357-366
    • Craven, B.D.1
  • 19
    • 0002729923 scopus 로고
    • Sufficient conditions for global minima of suitably convex functionals from variational and control theory
    • G. M. EWING, Sufficient conditions for global minima of suitably convex functionals from variational and control theory, SIAM Rev., 19(1977), pp. 202-220.
    • (1977) SIAM Rev. , vol.19 , pp. 202-220
    • Ewing, G.M.1
  • 21
    • 0002097095 scopus 로고    scopus 로고
    • The notion of invexity in vector optimization: Smooth and nonsmooth case
    • J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic
    • G. GIORGI and A. GUERRAGGIO, The notion of invexity in vector optimization: Smooth and nonsmooth case, in Generalized Convexity, Generalized Monotonicity, J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic, 1998, pp. 389-405.
    • (1998) Generalized Convexity, Generalized Monotonicity , pp. 389-405
    • Giorgi, G.1    Guerraggio, A.2
  • 22
    • 34047265721 scopus 로고    scopus 로고
    • Various types of nonsmooth invexity
    • G. GIORGI and A. GUERRAGGIO, Various types of nonsmooth invexity, J. Inf. Optim. Sci., 29(2004), pp. 415-424.
    • (2004) J. Inf. Optim. Sci. , vol.29 , pp. 415-424
    • Giorgi, G.1    Guerraggio, A.2
  • 24
    • 49149139478 scopus 로고
    • On sufficiency of the Kuhn-Tucker conditions
    • M. A. HANSON, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80(1981), pp. 545-550.
    • (1981) J. Math. Anal. Appl. , vol.80 , pp. 545-550
    • Hanson, M.A.1
  • 25
    • 0023295307 scopus 로고
    • Necessary and sufficiency conditions in constrained optimization
    • M. A. HANSON and B. MOND, Necessary and sufficiency conditions in constrained optimization, Math. Program., 37(1987), pp. 51-58.
    • (1987) Math. Program. , vol.37 , pp. 51-58
    • Hanson, M.A.1    Mond, B.2
  • 26
    • 0035885645 scopus 로고    scopus 로고
    • Multiobjective programming under generalized type I invexity
    • M. A. HANSON, R. PINI, and C. SINGH, Multiobjective programming under generalized type I invexity, J. Math. Anal. Appl., 261(2001), pp. 562-577.
    • (2001) J. Math. Anal. Appl. , vol.261 , pp. 562-577
    • Hanson, M.A.1    Pini, R.2    Singh, C.3
  • 27
    • 0020116651 scopus 로고
    • The gap function of a convex program
    • D. W. HEARN, The gap function of a convex program, Oper. Res. Lett., 1(1982), pp. 67-71.
    • (1982) Oper. Res. Lett. , vol.1 , pp. 67-71
    • Hearn, D.W.1
  • 28
    • 0002818016 scopus 로고
    • On generalized convex mathematical programming
    • V. JEYAKUMAR and B. MOND, On generalized convex mathematical programming, J. Austral. Math. Soc. Ser. B, 34(1992), pp. 43-53.
    • (1992) J. Austral. Math. Soc. Ser. B , vol.34 , pp. 43-53
    • Jeyakumar, V.1    Mond, B.2
  • 29
    • 0028393255 scopus 로고
    • Optimality criteria and duality in multipleobjective optimization involving generalized invexity
    • R. N. KAUL, S. K. SUNEJA, and N. K. SRIVASTAVA, Optimality criteria and duality in multipleobjective optimization involving generalized invexity, J. Optim. Theory Appl., 80(1994), pp. 465-482.
    • (1994) J. Optim. Theory Appl. , vol.80 , pp. 465-482
    • Kaul, R.N.1    Suneja, S.K.2    Srivastava, N.K.3
  • 30
    • 28044466214 scopus 로고    scopus 로고
    • On optimality and duality for nonsmooth multiobjective fractional optimization problems
    • G. S. KIM, M. H. KIM, and G. M. LEE, On optimality and duality for nonsmooth multiobjective fractional optimization problems, Nonlinear Anal., 63(2005), pp. 1867-1876.
    • (2005) Nonlinear Anal. , vol.63 , pp. 1867-1876
    • Kim, G.S.1    Kim, M.H.2    Lee, G.M.3
  • 34
    • 0031461725 scopus 로고    scopus 로고
    • Optimality and duality in fractional programming involving semilocally convex and related functions
    • V. LYALL, S. SUNEJA, and S. AGGARWAL, Optimality and duality in fractional programming involving semilocally convex and related functions, Optimization, 41(1997), pp. 237-255.
    • (1997) Optimization , vol.41 , pp. 237-255
    • Lyall, V.1    Suneja, S.2    Aggarwal, S.3
  • 36
    • 0022131568 scopus 로고
    • The essence of invexity
    • D. H. MARTIN, The essence of invexity, J. Optim. Theory Appl., 42(1985), pp. 65-76.
    • (1985) J. Optim. Theory Appl. , vol.42 , pp. 65-76
    • Martin, D.H.1
  • 37
    • 9244265520 scopus 로고    scopus 로고
    • Duality in vector optimization in Banach spaces with generalized convexity
    • S. K. MISHRA, G. GIORGI, and S. Y. WANG, Duality in vector optimization in Banach spaces with generalized convexity, J. Global Optim., 29(2004), pp. 415-424.
    • (2004) J. Global Optim. , vol.29 , pp. 415-424
    • Mishra, S.K.1    Giorgi, G.2    Wang, S.Y.3
  • 38
    • 29844458709 scopus 로고    scopus 로고
    • Some nondifferentiable multiobjective programming problems
    • S. K. MISHRA and M. A. NOOR, Some nondifferentiable multiobjective programming problems, J. Math. Anal. Appl., 316(2006), pp. 472-482.
    • (2006) J. Math. Anal. Appl. , vol.316 , pp. 472-482
    • Mishra, S.K.1    Noor, M.A.2
  • 39
    • 33344471259 scopus 로고    scopus 로고
    • Vector variational-like inequalities and nonsmooth vector optimization problems
    • S. K. MLSHRA and S. Y. WANG, Vector variational-like inequalities and nonsmooth vector optimization problems, Nonlinear Anal., 64(2006), pp. 1939-1945.
    • (2006) Nonlinear Anal. , vol.64 , pp. 1939-1945
    • Mlshra, S.K.1    Wang, S.Y.2
  • 40
    • 24044533857 scopus 로고    scopus 로고
    • Multiple objective fractional programming involving semilocally type l-preinvex and related functions
    • S. K. MLSHRA, S.-Y. WANG, and K. K. LAI, Multiple objective fractional programming involving semilocally type l-preinvex and related functions, J. Math. Anal. Appl., 310(2005), pp. 626-640.
    • (2005) J. Math. Anal. Appl. , vol.310 , pp. 626-640
    • Mlshra, S.K.1    Wang, S.-Y.2    Lai, K.K.3
  • 41
    • 0000512702 scopus 로고
    • On invex sets and preinvex functions
    • S. R. MOHAN, S. K. NEOGY, On invex sets and preinvex functions, J. Math. Anal. Appl., 189(1995), pp. 901-908.
    • (1995) J. Math. Anal. Appl. , vol.189 , pp. 901-908
    • Mohan, S.R.1    Neogy, S.K.2
  • 42
    • 33646940121 scopus 로고
    • Maximum principle in problems oftime optimal control with nonsmooth constraints
    • B. S. MORDUKHOVICH, Maximum principle in problems oftime optimal control with nonsmooth constraints, J. Appl. Math. Mech., 40(1976), pp. 960-969.
    • (1976) J. Appl. Math. Mech. , vol.40 , pp. 960-969
    • Mordukhovich, B.S.1
  • 45
    • 0000838472 scopus 로고
    • Variational-like inequalities
    • M. A. NOOR, Variational-like inequalities, Optimization, 30(1994), pp. 323-330.
    • (1994) Optimization , vol.30 , pp. 323-330
    • Noor, M.A.1
  • 46
    • 0011263263 scopus 로고    scopus 로고
    • Preinvex functions and variational inequalities
    • M. A. NOOR, Preinvex functions and variational inequalities, J. Nat. Geom., 9(1996), pp. 63-76.
    • (1996) J. Nat. Geom. , vol.9 , pp. 63-76
    • Noor, M.A.1
  • 47
    • 12344308381 scopus 로고    scopus 로고
    • On generalized preinvex functions and monotonicities
    • M. A. NOOR, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math., 5(2004), pp. 1-9.
    • (2004) J. Inequal. Pure Appl. Math. , vol.5 , pp. 1-9
    • Noor, M.A.1
  • 49
    • 0346973099 scopus 로고    scopus 로고
    • Optimality and duality in fractional multip le objective programming involving semilocally preinvex and related functions
    • V. PREDA, Optimality and duality in fractional multip le objective programming involving semilocally preinvex and related functions, J. Math. A nal. Appl., 288(2003), pp. 365-382.
    • (2003) J. Math. A Nal. Appl. , vol.288 , pp. 365-382
    • Preda, V.1
  • 50
    • 0001506393 scopus 로고
    • Directional Lipschitzian functions and subdifferential calculus, Proc
    • R. T. ROCKAFELLAR, Directional Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. (3), 39(1979), pp. 331-335.
    • (1979) London Math. Soc. (3) , vol.39 , pp. 331-335
    • Rockafellar, R.T.1
  • 54
    • 0000873824 scopus 로고
    • R. Horst and P. M. Pardalos, eds., Kluwer Academic, Dordrecht, The Net herlands
    • S. SCHAIBLE, Fractional programming. Handbook of global optimization, R. Horst and P. M. Pardalos, eds., Kluwer Academic, Dordrecht, The Net herlands, 1995, pp. 495-608.
    • (1995) Fractional Programming. Handbook of Global Optimization , pp. 495-608
    • Schaible, S.1
  • 55
    • 67949101783 scopus 로고    scopus 로고
    • A proof for Antczak's mean value theorem in invexity analysis
    • M. SOLEIMANI-DAMANEH, A proof for Antczak's mean value theorem in invexity analysis, Nonlinear Anal., 68(2008), pp. 1073-1074.
    • (2008) Nonlinear Anal. , vol.68 , pp. 1073-1074
    • Soleimani-Damaneh, M.1
  • 56
    • 33847671907 scopus 로고    scopus 로고
    • Characterization, of nonsmooth quasiconvex and pseudoconvex functions
    • M. SOLEIMANI-DAMANEH, Characterization, of nonsmooth quasiconvex and pseudoconvex functions, J. Math. Anal. Appl., 330(2007), pp. 2168-217 6.
    • (2007) J. Math. Anal. Appl. , vol.330 , pp. 2168-2176
    • Soleimani-Damaneh, M.1
  • 57
    • 40749086100 scopus 로고    scopus 로고
    • A mean value theorem in Asplund spaces
    • M. SOLEIMANI-DAMANEH, A mean value theorem in Asplund spaces, Nonlinear Anal., 68(2008), pp. 3103-3106.
    • (2008) Nonlinear Anal. , vol.68 , pp. 3103-3106
    • Soleimani-Damaneh, M.1
  • 58
    • 67651108860 scopus 로고    scopus 로고
    • Nonsmooth multiple-objective optimization in separable Hilbert spaces
    • M. SOLEIMANI-DAMANEH and J. J. NIETO, Nonsmooth multiple-objective optimization in separable Hilbert spaces, Nonlinear Anal., 71(2009), pp. 4553-4558.
    • (2009) Nonlinear Anal. , vol.71 , pp. 4553-4558
    • Soleimani-Damaneh, M.1    Nieto, J.J.2
  • 59
    • 24044539991 scopus 로고    scopus 로고
    • Optimality and duality in fractional programming involving semilocally preinvex and related functions
    • I. M. STANCU-MINASIAN, Optimality and duality in fractional programming involving semilocally preinvex and related functions, J. Inf. Optim. Sci., 23(2002), pp. 185-201.
    • (2002) J. Inf. Optim. Sci. , vol.23 , pp. 185-201
    • Stancu-Minasian, I.M.1


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