Abstract
We consider vector optimization problems on Banach spaces without convexity assumptions. Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential and the approximate subdifferential by Ioffe using a non-convex scalarization scheme. Finally, we apply the results for deriving necessary conditions for weakly efficient solutions of non-convex location problems.
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References
Amahroq T, Taa A (1997) On Lagrange–Kuhn–Tucker multipliers for multiobjective optimization problems. Optimization 41:159–172
Chankong V, Haimes YY (1983) Multiobjective decision making : theory and methodology. North-Holland, Amsterdam
Clarke FH (1983) Optimization and nonsmooth analysis. Wiley, New York
Clarke FH, Ledyaev YuS, Stern RJ, Wolenski PR (1998) Nonsmooth analysis and control theory. Springer, Berlin Heidelberg New York
Chandra S, Dutta J, Lalitha CS (2004) Regularity cconditions and optimality in nonsmooth vector optimization. Numer Funct Anal Appl 25:479–501
Craven BD (1989) Nonsmooth multiobjective programming. Numer Funct Anal Optim 10(1–2): 65–76
Demyanov VF, Rubinov A (1995) Constructive nonsmooth analysis. Peter-Verlag, Frankfurt
El Abdouni B, Thibault L (1992) Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces. Optimization 26:277–285
Gerth(Tammer) C, Weidner PW (1990) Nonconvex separation theorems and some applications in vector optimization. J Optim Theory Appl 67:297–320
Ioffe AD (1986) Approximate subdifferentials and applications II. Mathematika 33:111–128
Ioffe AD (1989) Approximate subdifferentials and applications III. The metric theory. Mathematika 36:1–38
Ioffe AD (2000) Metric regularity and subdifferential calculus. Russ Math Surveys 55:501–558
Jahn J (1986) Mathematical vector optimization in partially ordered spaces. Peter Lang, Frankfurt Bern New York
Jahn J (2004) Vector optimization: theory applications and extensions. Springer, Berlin Heidelberg New York
Jourani A, Thibault L (1993) The approximate subdifferential of composite functions. Bull Aust Math Soc 47:443–455
Li XF (2000) Constraint qualification in nonsmooth multiobjective optimization. J Optimi Theory Appl 106:373–398
Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer, Boston
Miettinen K, Mäkelä MM (2000) Tangent and normal cones in nonconvex multiobjective optimization. In: Haimes YY, Steuer RE (eds). Research and practice in multiple criteria decision making. Lecture notes in economils and mathematical systems. Springer, Berlin Heidelberg New York, pp 114–124
Minami M (1983) Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space. J Optim Theory Appl 41:451–461
Mordukhovich BS (1976) Maximum principle in problems of time optimal control with nonsmooth constraints. J Appl Math Mech 40:960–969
Mordukhovich BS (1985) On necessary conditions for an extremum in non-smooth optimization. Sov Math Dokl 32:215–220
Mordukhovich BS (1994) Generalized differential calculus for nonsmooth and set-valued mappings. J Math Anal Appl 183:250–288
Mordukhovich BS (2001) The extremal principle and its application to optimization and economics. In: Optimization and related topic, Ballarat 1999, applied optimization vol 47. Kluwer, Dodrecht
Mordukhovich BS (2005) Variational analysis and generalized differentiation, I: basic theory, II: applications (series: fundamental principles of mathematics), vol 330 and 331. Springer, Berlin Heidelberg New York
Mordukhovich BS, Shao Y (1996) Nonsmooth sequential analysis in Asplund spaces. Transa Am Math Soc 348:1235–1280
Rockafellar RT, Wets RJB (1998) Variational analysis. Springer, Berlin Heidelberg New York
Zălinescu C (2002) Convex analysis in general vector spaces. World Scientific, New Jersey
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Dutta, J., Tammer, C. Lagrangian conditions for vector optimization in Banach spaces. Math Meth Oper Res 64, 521–540 (2006). https://doi.org/10.1007/s00186-006-0079-z
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DOI: https://doi.org/10.1007/s00186-006-0079-z