Skip to main content
Top

2015 | OriginalPaper | Chapter

7. Monotone Operators Approached via Convex Analysis

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The monotone operators started being intensively investigated during the 1960’s by authors like Browder, Brézis or Minty, and it did not take much time until their connections with convex analysis were noticed by Rockafellar, Gossez and others. The fact that the (convex) subdifferential of a proper, convex and lower semicontinuous function is a maximally monotone operator was one of the reasons for connecting these at a first sight maybe unrelated research fields.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Aliprantis, C.D., Florenzano, M., Martins-da-Rocha, V.F., Tourky, R.: Equilibrium analysis in financial markets with countably many securities. J. Math. Econ. 40, 683–699 (2004) Aliprantis, C.D., Florenzano, M., Martins-da-Rocha, V.F., Tourky, R.: Equilibrium analysis in financial markets with countably many securities. J. Math. Econ. 40, 683–699 (2004)
2.
go back to reference Alizadeh, M.H., Hadjisavvas, N.: On the Fitzpatrick transform of a monotone bifunction. Optimization 62, 693–701 (2013) Alizadeh, M.H., Hadjisavvas, N.: On the Fitzpatrick transform of a monotone bifunction. Optimization 62, 693–701 (2013)
3.
go back to reference Anbo, Y.: Nonstandard arguments and the characterization of independence in generic structures. RIMS Kôkyûroku 1646, 4–17 (2009) Anbo, Y.: Nonstandard arguments and the characterization of independence in generic structures. RIMS Kôkyûroku 1646, 4–17 (2009)
4.
go back to reference Attouch, H., Baillon, J.-B., Théra, M.: Variational sum of monotone operators. J. Convex Anal. 1, 1–29 (1994) Attouch, H., Baillon, J.-B., Théra, M.: Variational sum of monotone operators. J. Convex Anal. 1, 1–29 (1994)
5.
go back to reference Attouch, H., Théra, M.: A general duality principle for the sum of two operators. J. Convex Anal. 3, 1–24 (1996) Attouch, H., Théra, M.: A general duality principle for the sum of two operators. J. Convex Anal. 3, 1–24 (1996)
6.
go back to reference Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. Ser. A. 122, 301–347 (2010) Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. Ser. A. 122, 301–347 (2010)
7.
go back to reference Bao, T.Q., Mordukhovich, B.S.: Extended Pareto optimality in multiobjective problems. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, pp. 467–515. Springer, Berlin/Heidelberg (2012) Bao, T.Q., Mordukhovich, B.S.: Extended Pareto optimality in multiobjective problems. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, pp. 467–515. Springer, Berlin/Heidelberg (2012)
8.
go back to reference Bartz, S., Bauschke, H.H., Borwein, J.M., Reich, S., Wang, X.: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative. Nonlinear Anal. Theory Methods Appl. 66, 1198–1223 (2007) Bartz, S., Bauschke, H.H., Borwein, J.M., Reich, S., Wang, X.: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative. Nonlinear Anal. Theory Methods Appl. 66, 1198–1223 (2007)
9.
go back to reference Bauschke, H.H., Moffat, S.M., Wang, X.: Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappings. Math. Program. 139, 55–70 (2013) Bauschke, H.H., Moffat, S.M., Wang, X.: Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappings. Math. Program. 139, 55–70 (2013)
10.
go back to reference Bauschke, H.H., Wang, X., Yao, L.: Rectangularity and paramonotonicity of maximally monotone operators. Optimization 63, 487–504 (2014) Bauschke, H.H., Wang, X., Yao, L.: Rectangularity and paramonotonicity of maximally monotone operators. Optimization 63, 487–504 (2014)
11.
go back to reference Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979) Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)
12.
go back to reference Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
13.
go back to reference Boncea, H.V., Grad, S.-M.: Characterizations of ɛ-duality gap statements for composed optimization problems. Nonlinear Anal. Theory Methods Appl. 92, 96–107 (2013) Boncea, H.V., Grad, S.-M.: Characterizations of ɛ-duality gap statements for composed optimization problems. Nonlinear Anal. Theory Methods Appl. 92, 96–107 (2013)
14.
go back to reference Boncea, H.V., Grad, S.-M.: Characterizations of ɛ-duality gap statements for constrained optimization problems. Cent. Eur. J. Math. 11, 2020–2033, (2013) Boncea, H.V., Grad, S.-M.: Characterizations of ɛ-duality gap statements for constrained optimization problems. Cent. Eur. J. Math. 11, 2020–2033, (2013)
15.
go back to reference Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15, 953–970 (2005) Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15, 953–970 (2005)
16.
go back to reference Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977) Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)
17.
go back to reference Borwein, J.M.: The geometry of Pareto efficiency over cones. Math. Operationsforsch. Stat. Ser. Optim. 11, 235–248 (1980) Borwein, J.M.: The geometry of Pareto efficiency over cones. Math. Operationsforsch. Stat. Ser. Optim. 11, 235–248 (1980)
18.
go back to reference Borwein, J.M.: On the existence of Pareto efficient points. Math. Oper. Res. 8, 64–73 (1983) Borwein, J.M.: On the existence of Pareto efficient points. Math. Oper. Res. 8, 64–73 (1983)
19.
go back to reference Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13, 561–586 (2006) Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13, 561–586 (2006)
20.
go back to reference Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Program. Ser. B 57, 15–48 (1992) Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Program. Ser. B 57, 15–48 (1992)
21.
go back to reference Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin/Heidelberg (2010) Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin/Heidelberg (2010)
22.
go back to reference Boţ, R.I., Csetnek, E.R.: Error bound results for convex inequality systems via conjugate duality. Top 20, 296–309 (2012) Boţ, R.I., Csetnek, E.R.: Error bound results for convex inequality systems via conjugate duality. Top 20, 296–309 (2012)
23.
go back to reference Boţ, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61, 35–65 (2012) Boţ, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61, 35–65 (2012)
24.
go back to reference Boţ, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008) Boţ, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008)
25.
go back to reference Boţ, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008) Boţ, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)
26.
go back to reference Boţ, R.I., Dumitru, A., Wanka, G.: A new Fenchel dual problem in vector optimization. Proc. Indian Acad. Sci. Math. Sci. 119, 251–265 (2009) Boţ, R.I., Dumitru, A., Wanka, G.: A new Fenchel dual problem in vector optimization. Proc. Indian Acad. Sci. Math. Sci. 119, 251–265 (2009)
27.
go back to reference Boţ, R.I., Grad, S.-M.: Regularity conditions for formulae of biconjugate functions. Taiwan. J. Math. 12, 1921–1942 (2008) Boţ, R.I., Grad, S.-M.: Regularity conditions for formulae of biconjugate functions. Taiwan. J. Math. 12, 1921–1942 (2008)
28.
go back to reference Boţ, R.I., Grad, S.-M.: Lower semicontinuous type regularity conditions for subdifferential calculus. Optim. Methods Softw. 25, 37–48 (2010) Boţ, R.I., Grad, S.-M.: Lower semicontinuous type regularity conditions for subdifferential calculus. Optim. Methods Softw. 25, 37–48 (2010)
29.
go back to reference Boţ, R.I., Grad, S.-M.: Wolfe duality and Mond-Weir duality via perturbations. Nonlinear Anal. Theory Methods Appl. 73, 374–384 (2010) Boţ, R.I., Grad, S.-M.: Wolfe duality and Mond-Weir duality via perturbations. Nonlinear Anal. Theory Methods Appl. 73, 374–384 (2010)
30.
go back to reference Boţ, R.I., Grad, S.-M.: Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators. Cent. Eur. J. Math. 9, 162–172 (2011) Boţ, R.I., Grad, S.-M.: Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators. Cent. Eur. J. Math. 9, 162–172 (2011)
31.
go back to reference Boţ, R.I., Grad, S.-M.: Duality for vector optimization problems via a general scalarization. Optimization 60, 1269–1290 (2011) Boţ, R.I., Grad, S.-M.: Duality for vector optimization problems via a general scalarization. Optimization 60, 1269–1290 (2011)
32.
go back to reference Boţ, R.I., Grad, S.-M.: Extending the classical vector Wolfe and Mond-Weir duality concepts via perturbations. J. Nonlinear Convex Anal. 12, 81–101 (2011) Boţ, R.I., Grad, S.-M.: Extending the classical vector Wolfe and Mond-Weir duality concepts via perturbations. J. Nonlinear Convex Anal. 12, 81–101 (2011)
33.
go back to reference Boţ, R.I., Grad, S.-M.: Approaching the maximal monotonicity of bifunctions via representative functions. J. Convex Anal. 19, 713–724 (2012) Boţ, R.I., Grad, S.-M.: Approaching the maximal monotonicity of bifunctions via representative functions. J. Convex Anal. 19, 713–724 (2012)
34.
go back to reference Boţ, R.I., Grad, S.-M.: On linear vector optimization duality in infinite-dimensional spaces. Numer. Algebra Control Optim. 1, 407–415 (2011) Boţ, R.I., Grad, S.-M.: On linear vector optimization duality in infinite-dimensional spaces. Numer. Algebra Control Optim. 1, 407–415 (2011)
35.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Brézis-Haraux-type approximation of the range of a monotone operator composed with a linear mapping. In: Kása, Z., Kassay, G., Kolumbán, J. (eds.) Proceedings of the International Conference in Memoriam Gyula Farkas, Cluj-Napoca, pp. 36–49. Cluj University Press, Cluj-Napoca (2006) Boţ, R.I., Grad, S.-M., Wanka, G.: Brézis-Haraux-type approximation of the range of a monotone operator composed with a linear mapping. In: Kása, Z., Kassay, G., Kolumbán, J. (eds.) Proceedings of the International Conference in Memoriam Gyula Farkas, Cluj-Napoca, pp. 36–49. Cluj University Press, Cluj-Napoca (2006)
36.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Fenchel-Lagrange duality versus geometric duality in convex optimization. J. Optim. Theory Appl. 129, 33–54 (2006) Boţ, R.I., Grad, S.-M., Wanka, G.: Fenchel-Lagrange duality versus geometric duality in convex optimization. J. Optim. Theory Appl. 129, 33–54 (2006)
37.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: A general approach for studying duality in multiobjective optimization. Math. Methods Oper. Res. 65, 417–444 (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: A general approach for studying duality in multiobjective optimization. Math. Methods Oper. Res. 65, 417–444 (2007)
38.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: A new regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Applications for maximal monotone operators. In: Castellani, G. (ed.) Seminario Mario Volpato, vol. 3, pp. 16–30. Ca’Foscari University of Venice, Venice (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: A new regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Applications for maximal monotone operators. In: Castellani, G. (ed.) Seminario Mario Volpato, vol. 3, pp. 16–30. Ca’Foscari University of Venice, Venice (2007)
39.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Almost convex functions: conjugacy and duality. In: Konnov, I.V., Luc, D.T., Rubinov, A.M. (eds.) Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 101–114. Springer, Berlin (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: Almost convex functions: conjugacy and duality. In: Konnov, I.V., Luc, D.T., Rubinov, A.M. (eds.) Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 101–114. Springer, Berlin (2007)
40.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Brézis-Haraux-type approximation in nonreflexive Banach spaces. In: Allevi, E., Bertocchi, M., Gnudi, A., Konnov, I.V. (eds.) Nonlinear Analysis with Applications in Economics, Energy and Transportation, pp. 155–170. Bergamo University Press, Bergamo (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: Brézis-Haraux-type approximation in nonreflexive Banach spaces. In: Allevi, E., Bertocchi, M., Gnudi, A., Konnov, I.V. (eds.) Nonlinear Analysis with Applications in Economics, Energy and Transportation, pp. 155–170. Bergamo University Press, Bergamo (2007)
41.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Fenchel’s duality theorem for nearly convex functions. J. Optim. Theory Appl. 132, 509–515 (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: Fenchel’s duality theorem for nearly convex functions. J. Optim. Theory Appl. 132, 509–515 (2007)
42.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Maximal monotonicity for the precomposition with a linear operator. SIAM J. Optim. 17, 1239–1252 (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: Maximal monotonicity for the precomposition with a linear operator. SIAM J. Optim. 17, 1239–1252 (2007)
43.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: New constraint qualification and conjugate duality for composed convex optimization problems. J. Optim. Theory Appl. 135, 241–255 (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: New constraint qualification and conjugate duality for composed convex optimization problems. J. Optim. Theory Appl. 135, 241–255 (2007)
44.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Weaker constraint qualifications in maximal monotonicity. Numer. Funct. Anal. Optim. 28, 27–41 (2007) Boţ, R.I., Grad, S.-M., Wanka, G.: Weaker constraint qualifications in maximal monotonicity. Numer. Funct. Anal. Optim. 28, 27–41 (2007)
45.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces. Math. Nachr. 281, 1088–1107 (2008) Boţ, R.I., Grad, S.-M., Wanka, G.: A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces. Math. Nachr. 281, 1088–1107 (2008)
46.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces. Nonlinear Anal. Theory Methods Appl. 69, 323–336 (2008) Boţ, R.I., Grad, S.-M., Wanka, G.: New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces. Nonlinear Anal. Theory Methods Appl. 69, 323–336 (2008)
47.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337, 1315–1325 (2008) Boţ, R.I., Grad, S.-M., Wanka, G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337, 1315–1325 (2008)
48.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin/Heidelberg (2009) Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin/Heidelberg (2009)
49.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Generalized Moreau-Rockafellar results for composed convex functions. Optimization 58, 917–933 (2009) Boţ, R.I., Grad, S.-M., Wanka, G.: Generalized Moreau-Rockafellar results for composed convex functions. Optimization 58, 917–933 (2009)
50.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: New regularity conditions for Lagrange and Fenchel-Lagrange duality in infinite dimensional spaces. Math. Inequal. Appl. 12, 171–189 (2009) Boţ, R.I., Grad, S.-M., Wanka, G.: New regularity conditions for Lagrange and Fenchel-Lagrange duality in infinite dimensional spaces. Math. Inequal. Appl. 12, 171–189 (2009)
51.
go back to reference Boţ, R.I., Grad, S.-M., Wanka, G.: Classical linear vector optimization duality revisited. Optim. Lett. 6, 199–210 (2012) Boţ, R.I., Grad, S.-M., Wanka, G.: Classical linear vector optimization duality revisited. Optim. Lett. 6, 199–210 (2012)
52.
go back to reference Boţ, R.I., Kassay, G., Wanka, G.: Strong duality for generalized convex optimization problems. J. Optim. Theory Appl. 127, 45–70 (2005) Boţ, R.I., Kassay, G., Wanka, G.: Strong duality for generalized convex optimization problems. J. Optim. Theory Appl. 127, 45–70 (2005)
53.
go back to reference Boţ, R.I., Kassay, G., Wanka, G.: Duality for almost convex optimization problems via the perturbation approach. J. Glob. Optim. 42, 385–399 (2009) Boţ, R.I., Kassay, G., Wanka, G.: Duality for almost convex optimization problems via the perturbation approach. J. Glob. Optim. 42, 385–399 (2009)
54.
go back to reference Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (I). Optimization 53, 281–300 (2004) Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (I). Optimization 53, 281–300 (2004)
55.
go back to reference Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (II). Optimization 53, 301–324 (2004) Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (II). Optimization 53, 301–324 (2004)
56.
go back to reference Boţ, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal. Theory Methods Appl. 64, 2787–2804 (2006) Boţ, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal. Theory Methods Appl. 64, 2787–2804 (2006)
57.
go back to reference Boţ, R.I., Wanka, G.: An alternative formulation for a new closed cone constraint qualification. Nonlinear Anal. Theory Methods Appl. 64, 1367–1381 (2006) Boţ, R.I., Wanka, G.: An alternative formulation for a new closed cone constraint qualification. Nonlinear Anal. Theory Methods Appl. 64, 1367–1381 (2006)
58.
go back to reference Breckner, W., Kolumbán, J.: Dualität bei Optimierungsaugaben in Topologischen Vektorräumen. Mathematica 10, 229–244 (1968) Breckner, W., Kolumbán, J.: Dualität bei Optimierungsaugaben in Topologischen Vektorräumen. Mathematica 10, 229–244 (1968)
59.
go back to reference Breckner, W., Kolumbán, J.: Konvexe Optimierungsaufgaben in Topologischen Vektorräumen. Math. Scand. 25, 227–247 (1969) Breckner, W., Kolumbán, J.: Konvexe Optimierungsaufgaben in Topologischen Vektorräumen. Math. Scand. 25, 227–247 (1969)
60.
go back to reference Brézis, H., Haraux, A.: Image d’une somme d’opérateurs monotones et applications. Isr. J. Math. 23, 165–186 (1976) Brézis, H., Haraux, A.: Image d’une somme d’opérateurs monotones et applications. Isr. J. Math. 23, 165–186 (1976)
61.
go back to reference Brumelle, S.: Duality for multiple objective convex programs. Math. Oper. Res. 6, 159–172 (1981) Brumelle, S.: Duality for multiple objective convex programs. Math. Oper. Res. 6, 159–172 (1981)
62.
go back to reference Bueno, O., Martínez-Legaz, J.-E., Svaiter, B.F.: On the monotone polar and representable closures of monotone operators. J. Convex Anal. 21, 495–505 (2014) Bueno, O., Martínez-Legaz, J.-E., Svaiter, B.F.: On the monotone polar and representable closures of monotone operators. J. Convex Anal. 21, 495–505 (2014)
63.
go back to reference Burachik, R.S., Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12, 279–290 (2005) Burachik, R.S., Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12, 279–290 (2005)
64.
go back to reference Burachik, R.S., Jeyakumar, V., Wu, Z.-Y.: Necessary and sufficient conditions for stable conjugate duality. Nonlinear Anal. Theory Methods Appl. 64, 1998–2006 (2006) Burachik, R.S., Jeyakumar, V., Wu, Z.-Y.: Necessary and sufficient conditions for stable conjugate duality. Nonlinear Anal. Theory Methods Appl. 64, 1998–2006 (2006)
65.
go back to reference Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10, 297–316 (2002) Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10, 297–316 (2002)
66.
go back to reference Cambini, R., Carosi, L.: Duality in multiobjective optimization problems with set constraints. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized Convexity, Generalized Monotonicity and Applications, Proceedings of the 7th International Symposium on Generalized Convexity and Generalized Monotonicity held in Hanoi, 27–31 Aug 2002. Nonconvex Optimization and Its Applications, vol. 77, pp. 131–146. Springer, New York (2005) Cambini, R., Carosi, L.: Duality in multiobjective optimization problems with set constraints. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized Convexity, Generalized Monotonicity and Applications, Proceedings of the 7th International Symposium on Generalized Convexity and Generalized Monotonicity held in Hanoi, 27–31 Aug 2002. Nonconvex Optimization and Its Applications, vol. 77, pp. 131–146. Springer, New York (2005)
67.
go back to reference Cammaroto, F., Di Bella, B.: Separation theorem based on the quasirelative interior and application to duality theory. J. Optim. Theory Appl. 125, 223–229 (2005) Cammaroto, F., Di Bella, B.: Separation theorem based on the quasirelative interior and application to duality theory. J. Optim. Theory Appl. 125, 223–229 (2005)
68.
go back to reference Cammaroto, F., Di Bella, B.: On a separation theorem involving the quasi-relative interior. Proc. Edinb. Math. Soc. II. Ser. 50, 605–610 (2007) Cammaroto, F., Di Bella, B.: On a separation theorem involving the quasi-relative interior. Proc. Edinb. Math. Soc. II. Ser. 50, 605–610 (2007)
69.
go back to reference Carrizosa, E., Fliege, J.: Generalized goal programming: polynomial methods and applications. Math. Program. 93, 281–303 (2002) Carrizosa, E., Fliege, J.: Generalized goal programming: polynomial methods and applications. Math. Program. 93, 281–303 (2002)
70.
go back to reference Chbani, Z., Riahi, H.: The range of sums of monotone operators and applications to Hammerstein inclusions and nonlinear complementarity problems. In: Benkirane, A., Grossez, P. (eds.) Nonlinear Partial Differential Equations, Fés, 1994. Pitman Research Notes in Mathematics Series, vol. 343, pp. 61–72. Longman, Harlow (1996) Chbani, Z., Riahi, H.: The range of sums of monotone operators and applications to Hammerstein inclusions and nonlinear complementarity problems. In: Benkirane, A., Grossez, P. (eds.) Nonlinear Partial Differential Equations, Fés, 1994. Pitman Research Notes in Mathematics Series, vol. 343, pp. 61–72. Longman, Harlow (1996)
71.
go back to reference Chien, T.Q.: Nondifferentiable and quasidifferentiable duality in vector optimization theory. Kybernetika 21, 298–312 (1985) Chien, T.Q.: Nondifferentiable and quasidifferentiable duality in vector optimization theory. Kybernetika 21, 298–312 (1985)
72.
go back to reference Chu, L.-J.: On Brézis-Haraux approximation with applications. Far East J. Math. Sci. 4, 425–442 (1996) Chu, L.-J.: On Brézis-Haraux approximation with applications. Far East J. Math. Sci. 4, 425–442 (1996)
73.
go back to reference Chu, L.-J.: On the sum of monotone operators. Mich. Math. J. 43, 273–289 (1996) Chu, L.-J.: On the sum of monotone operators. Mich. Math. J. 43, 273–289 (1996)
74.
go back to reference Coulibaly, A., Crouzeix, J.-P.: Condition numbers and error bounds in convex programming. Math. Program. Ser. B 116, 79–113 (2009) Coulibaly, A., Crouzeix, J.-P.: Condition numbers and error bounds in convex programming. Math. Program. Ser. B 116, 79–113 (2009)
75.
go back to reference Crouzeix, J.-P., Ocaña Anaya, E.: Maximality is nothing but continuity. J. Convex Anal. 17, 521–534 (2010) Crouzeix, J.-P., Ocaña Anaya, E.: Maximality is nothing but continuity. J. Convex Anal. 17, 521–534 (2010)
76.
go back to reference Daniele, P., Giuffré, S.: General infinite dimensional duality theory and applications to evolutionary network equilibrium problems. Optim. Lett. 1, 227–243 (2007) Daniele, P., Giuffré, S.: General infinite dimensional duality theory and applications to evolutionary network equilibrium problems. Optim. Lett. 1, 227–243 (2007)
77.
go back to reference Daniele, P., Giuffré, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007) Daniele, P., Giuffré, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007)
78.
go back to reference Dauer, J.P., Saleh, O.A.: A characterization of proper minimal points as solutions of sublinear optimization problems. J. Math. Anal. Appl. 178, 227–246 (1993) Dauer, J.P., Saleh, O.A.: A characterization of proper minimal points as solutions of sublinear optimization problems. J. Math. Anal. Appl. 178, 227–246 (1993)
79.
go back to reference Debreu, G.: Theory of Value. Wiley, New York (1959) Debreu, G.: Theory of Value. Wiley, New York (1959)
80.
go back to reference Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for ɛ-Pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010) Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for ɛ-Pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010)
81.
go back to reference Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009) Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)
82.
go back to reference Egudo, R.R.: Proper efficiency and multiobjective duality in nonlinear programming. J. Inf. Optim. Sci. 8, 155–166 (1987) Egudo, R.R.: Proper efficiency and multiobjective duality in nonlinear programming. J. Inf. Optim. Sci. 8, 155–166 (1987)
83.
go back to reference Egudo, R.R.: Efficiency and generalized convex duality for multiobjective programs. J. Math. Anal. Appl. 138, 84–94 (1989) Egudo, R.R.: Efficiency and generalized convex duality for multiobjective programs. J. Math. Anal. Appl. 138, 84–94 (1989)
84.
go back to reference Egudo, R.R., Weir, T., Mond, B.: Duality without constraint qualification for multiobjective programming. J. Aust. Math. Soc. Ser. B 33, 531–544 (1992) Egudo, R.R., Weir, T., Mond, B.: Duality without constraint qualification for multiobjective programming. J. Aust. Math. Soc. Ser. B 33, 531–544 (1992)
85.
go back to reference Eichfelder, G., Jahn, J.: Set-semidefinite optimization. J. Convex Anal. 15, 767–801 (2008) Eichfelder, G., Jahn, J.: Set-semidefinite optimization. J. Convex Anal. 15, 767–801 (2008)
86.
go back to reference Fitzpatrick, S.P.: Representing monotone operators by convex functions. In: Fitzpatrick, S.P., Giles, J.R. (eds.) Workshop/Miniconference on Functional Analysis and Optimization. Proceedings of the Centre for Mathematical Analysis, vol. 20, pp. 59–65. Australian National University, Canberra (1988) Fitzpatrick, S.P.: Representing monotone operators by convex functions. In: Fitzpatrick, S.P., Giles, J.R. (eds.) Workshop/Miniconference on Functional Analysis and Optimization. Proceedings of the Centre for Mathematical Analysis, vol. 20, pp. 59–65. Australian National University, Canberra (1988)
87.
go back to reference Fliege, J.: Approximation Techniques for the Set of Efficient Points. Habilitation Thesis, Faculty of Mathematics, University of Dortmund (2001) Fliege, J.: Approximation Techniques for the Set of Efficient Points. Habilitation Thesis, Faculty of Mathematics, University of Dortmund (2001)
88.
go back to reference Fliege, J., Heseler, A.: Constructing approximations to the efficient set of convex quadratic multiobjective problems. Reports on applied mathematics, 211, Faculty of Mathematics, University of Dortmund (2002) Fliege, J., Heseler, A.: Constructing approximations to the efficient set of convex quadratic multiobjective problems. Reports on applied mathematics, 211, Faculty of Mathematics, University of Dortmund (2002)
89.
go back to reference Flores-Bazán, F., Flores-Bazán, F., Laengle, S.: Characterizing efficiency on infinite-dimensional commodity spaces with ordering cones having possibly empty interior. J. Optim. Theory Appl. (2014). doi:10.1007/s10957-014-0558-y Flores-Bazán, F., Flores-Bazán, F., Laengle, S.: Characterizing efficiency on infinite-dimensional commodity spaces with ordering cones having possibly empty interior. J. Optim. Theory Appl. (2014). doi:10.1007/s10957-014-0558-y
90.
go back to reference Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Gordan-type alternative theorems and vector optimization revisited. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, pp. 29–59. Springer, Berlin/Heidelberg (2012) Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Gordan-type alternative theorems and vector optimization revisited. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, pp. 29–59. Springer, Berlin/Heidelberg (2012)
91.
go back to reference Flores-Bazán, F., Hadjisavvas, N., Vera, C.: An optimal alternative theorem and applications. J. Glob. Optim. 37, 229–243 (2007) Flores-Bazán, F., Hadjisavvas, N., Vera, C.: An optimal alternative theorem and applications. J. Glob. Optim. 37, 229–243 (2007)
92.
go back to reference Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60, 1399–1419 (2011) Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60, 1399–1419 (2011)
93.
go back to reference Flores-Bazán, F., Hernández, E.: Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. J. Glob. Optim. 56, 299–315 (2013) Flores-Bazán, F., Hernández, E.: Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. J. Glob. Optim. 56, 299–315 (2013)
94.
go back to reference Focke, J: Vektormaximumproblem und parametrische Optimierung. Math. Operationsforsch. Stat. 4, 365–369 (1973) Focke, J: Vektormaximumproblem und parametrische Optimierung. Math. Operationsforsch. Stat. 4, 365–369 (1973)
95.
go back to reference Friedman, H.M.: A way out. In: Link, G. (ed.) One Hundred Years of Russell’s Paradox, pp. 49–86. de Gruyter, Berlin/New York (2004) Friedman, H.M.: A way out. In: Link, G. (ed.) One Hundred Years of Russell’s Paradox, pp. 49–86. de Gruyter, Berlin/New York (2004)
96.
go back to reference Gale, D., Kuhn, H.W., Tucker, A.W.: Linear programming and the theory of games. In: Koopmans, T.C. (ed.) Activity Analysis of Production and Allocation, pp. 317–329. Wiley, New York (1951) Gale, D., Kuhn, H.W., Tucker, A.W.: Linear programming and the theory of games. In: Koopmans, T.C. (ed.) Activity Analysis of Production and Allocation, pp. 317–329. Wiley, New York (1951)
97.
go back to reference Gerstewitz, C.: Nichtkonvexe Dualität in der Vektoroptimierung. Wiss. Z. Tech. Hochsch. Carl Schorlemmer Leuna-Merseburg 25, 357–364 (1983) Gerstewitz, C.: Nichtkonvexe Dualität in der Vektoroptimierung. Wiss. Z. Tech. Hochsch. Carl Schorlemmer Leuna-Merseburg 25, 357–364 (1983)
98.
go back to reference Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 31, 61–81 (1985) Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 31, 61–81 (1985)
99.
go back to reference Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990) Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)
100.
go back to reference Gong, X.-H.: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 12–31 (2005) Gong, X.-H.: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 12–31 (2005)
101.
go back to reference Göpfert, A.: Multicriterial duality, examples and advances. In: Fandel, G., Grauer, M., Kurzhanski, A., Wierzbicki, A.P. (eds.) Large-Scale Modelling and Interactive Decision Analysis, Proceedings of the International Workshop Held in Eisenach, 18–21 Nov 1985. Lecture Notes in Economics and Mathematical Systems, vol. 273, pp. 52–58. Springer, Berlin (1986) Göpfert, A.: Multicriterial duality, examples and advances. In: Fandel, G., Grauer, M., Kurzhanski, A., Wierzbicki, A.P. (eds.) Large-Scale Modelling and Interactive Decision Analysis, Proceedings of the International Workshop Held in Eisenach, 18–21 Nov 1985. Lecture Notes in Economics and Mathematical Systems, vol. 273, pp. 52–58. Springer, Berlin (1986)
102.
go back to reference Göpfert, A., Gerth, C.: Über die Skalarisierung und Dualisierung von Vektoroptimierungsproblemen. Z. Anal. Anwend. 5, 377–384 (1986) Göpfert, A., Gerth, C.: Über die Skalarisierung und Dualisierung von Vektoroptimierungsproblemen. Z. Anal. Anwend. 5, 377–384 (1986)
103.
go back to reference Göpfert, A., Nehse, R.: Vektoroptimierung: Theorie, Verfahren und Anwendungen. Teubner, Leipzig (1990) Göpfert, A., Nehse, R.: Vektoroptimierung: Theorie, Verfahren und Anwendungen. Teubner, Leipzig (1990)
104.
go back to reference Gossez, J.-P.: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs. J. Math. Anal. Appl. 34, 371–395 (1971) Gossez, J.-P.: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs. J. Math. Anal. Appl. 34, 371–395 (1971)
105.
go back to reference Grad, A.: Generalized Duality and Optimality Conditions. Editura Mega, Cluj-Napoca (2010) Grad, A.: Generalized Duality and Optimality Conditions. Editura Mega, Cluj-Napoca (2010)
106.
go back to reference Grad, A.: Quasi interior-type optimality conditions in set-valued duality. J. Nonlinear Convex Anal. 14, 301–317 (2013) Grad, A.: Quasi interior-type optimality conditions in set-valued duality. J. Nonlinear Convex Anal. 14, 301–317 (2013)
107.
go back to reference Grad, S.-M.: Recent Advances in Vector Optimization and Set-Valued Analysis via Convex Duality. Habilitation Thesis, Faculty of Mathematics, Chemnitz University of Technology (2014) Grad, S.-M.: Recent Advances in Vector Optimization and Set-Valued Analysis via Convex Duality. Habilitation Thesis, Faculty of Mathematics, Chemnitz University of Technology (2014)
108.
go back to reference Grad, S.-M., Pop, E.L.: Alternative generalized Wolfe type and Mond-Weir type vector duality. J. Nonlinear Convex Anal. 15, 867–884 (2014) Grad, S.-M., Pop, E.L.: Alternative generalized Wolfe type and Mond-Weir type vector duality. J. Nonlinear Convex Anal. 15, 867–884 (2014)
109.
go back to reference Grad, S.-M., Pop, E.L.: Characterizing relatively minimal elements via linear scalarization. In: Huisman, D., Louwerse, I., Wagelmans, A.P.M. (eds.) Operations Research Proceedings 2013. Springer, Berlin/Heidelberg, 153–159 (2014) Grad, S.-M., Pop, E.L.: Characterizing relatively minimal elements via linear scalarization. In: Huisman, D., Louwerse, I., Wagelmans, A.P.M. (eds.) Operations Research Proceedings 2013. Springer, Berlin/Heidelberg, 153–159 (2014)
110.
go back to reference Grad, S.-M., Pop, E.L.: Vector duality for convex vector optimization problems by means of the quasi interior of the ordering cone. Optimization 63, 21–37 (2014) Grad, S.-M., Pop, E.L.: Vector duality for convex vector optimization problems by means of the quasi interior of the ordering cone. Optimization 63, 21–37 (2014)
111.
go back to reference Graña Drummond, L.M., Iusem, A.N.: First order conditions for ideal minimization of matrix-valued problems. J. Convex Anal. 10, 1–19 (2003) Graña Drummond, L.M., Iusem, A.N.: First order conditions for ideal minimization of matrix-valued problems. J. Convex Anal. 10, 1–19 (2003)
112.
go back to reference Graña Drummond, L.M., Iusem, A.N., Svaiter, B.F.: On first order optimality conditions for vector optimization. Acta Math. Appl. Sin. Engl. Ser. 19, 371–386 (2003) Graña Drummond, L.M., Iusem, A.N., Svaiter, B.F.: On first order optimality conditions for vector optimization. Acta Math. Appl. Sin. Engl. Ser. 19, 371–386 (2003)
113.
go back to reference Guerraggio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994) Guerraggio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)
114.
go back to reference Gutiérrez, C., Jiménez, B., Novo, V.: On approximate solutions in vector optimization problems via scalarization. Comput. Optim. Appl. 35, 305–324 (2006) Gutiérrez, C., Jiménez, B., Novo, V.: On approximate solutions in vector optimization problems via scalarization. Comput. Optim. Appl. 35, 305–324 (2006)
115.
go back to reference Ha, T.X.D.: Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems. Nonlinear Anal. Theory Methods Appl. 75, 1305–1323 (2012) Ha, T.X.D.: Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems. Nonlinear Anal. Theory Methods Appl. 75, 1305–1323 (2012)
116.
go back to reference Hadjisavvas, N., Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization 59, 147–160 (2010) Hadjisavvas, N., Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization 59, 147–160 (2010)
117.
go back to reference Hamel, A.H., Heyde, F., Löhne, A., Tammer, C., Winkler, K.: Closing the duality gap in linear vector optimization. J. Convex Anal. 11, 163–178 (2004) Hamel, A.H., Heyde, F., Löhne, A., Tammer, C., Winkler, K.: Closing the duality gap in linear vector optimization. J. Convex Anal. 11, 163–178 (2004)
118.
go back to reference Hartley, R.: On cone-efficiency, cone-convexity and cone-compactness. SIAM J. Appl. Math. 34, 211–222 (1978) Hartley, R.: On cone-efficiency, cone-convexity and cone-compactness. SIAM J. Appl. Math. 34, 211–222 (1978)
119.
go back to reference Helbig, S.: Parametric optimization with a bottleneck objective and vector optimization. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization, pp. 146–159. Springer, Berlin (1987) Helbig, S.: Parametric optimization with a bottleneck objective and vector optimization. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization, pp. 146–159. Springer, Berlin (1987)
120.
go back to reference Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982) Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)
121.
go back to reference Heyde, F., Löhne, A.: Geometric duality in multiple objective linear programming. SIAM J. Optim. 19, 836–845 (2008) Heyde, F., Löhne, A.: Geometric duality in multiple objective linear programming. SIAM J. Optim. 19, 836–845 (2008)
122.
go back to reference Heyde, F., Löhne, A., Tammer, C.: Set-valued duality theory for multiple objective linear programs and application to mathematical finance. Math. Methods Oper. Res. 69, 159–179 (2009) Heyde, F., Löhne, A., Tammer, C.: Set-valued duality theory for multiple objective linear programs and application to mathematical finance. Math. Methods Oper. Res. 69, 159–179 (2009)
123.
go back to reference Heyde, F., Löhne, A., Tammer, C.: The attainment of the solution of the dual program in vertices for vectorial linear programs. In: Barichard, V., Ehrgott, M., Gandibleux, X., T’kindt, M.V. (eds.) Multiobjective Programming and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol. 618, pp. 13–24. Springer, Berlin/Heidelberg (2009) Heyde, F., Löhne, A., Tammer, C.: The attainment of the solution of the dual program in vertices for vectorial linear programs. In: Barichard, V., Ehrgott, M., Gandibleux, X., T’kindt, M.V. (eds.) Multiobjective Programming and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol. 618, pp. 13–24. Springer, Berlin/Heidelberg (2009)
124.
go back to reference Hiriart-Urruty, J.-B.: New concepts in nondifferentiable programming. Bull. Soc. Math. Fr. Suppl. Mém. 60, 57–85 (1979) Hiriart-Urruty, J.-B.: New concepts in nondifferentiable programming. Bull. Soc. Math. Fr. Suppl. Mém. 60, 57–85 (1979)
125.
go back to reference Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979) Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)
126.
go back to reference Hiriart-Urruty, J.-B.: ɛ-subdifferential calculus. In: Aubin, J.-P., Vinter, R.B. (eds.) Convex Analysis and Optimization. Pitman Research Notes in Mathematics Series, vol. 57, pp. 43–92. Pitman, Boston (1982) Hiriart-Urruty, J.-B.: ɛ-subdifferential calculus. In: Aubin, J.-P., Vinter, R.B. (eds.) Convex Analysis and Optimization. Pitman Research Notes in Mathematics Series, vol. 57, pp. 43–92. Pitman, Boston (1982)
127.
go back to reference Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals. Grundlehren der Mathematischen Wissenschaften, vol. 305. Springer, Berlin (1993) Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals. Grundlehren der Mathematischen Wissenschaften, vol. 305. Springer, Berlin (1993)
128.
go back to reference Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods. Grundlehren der Mathematischen Wissenschaften, vol. 306. Springer, Berlin (1993) Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods. Grundlehren der Mathematischen Wissenschaften, vol. 306. Springer, Berlin (1993)
129.
go back to reference Huong, N.T.T., Yen, N.D.: The Pascoletti-Serafini scalarization scheme and linear vector optimization. J. Optim. Theory Appl. 162, 559–576 (2014) Huong, N.T.T., Yen, N.D.: The Pascoletti-Serafini scalarization scheme and linear vector optimization. J. Optim. Theory Appl. 162, 559–576 (2014)
130.
go back to reference Hurwicz, L: Programming in linear spaces. In: Arrow, K.J., Hurwicz, L., Uzawa, H. (eds.) Studies in Linear and Non-Linear Programming, pp. 38–102. Stanford University Press, Stanford (1958) Hurwicz, L: Programming in linear spaces. In: Arrow, K.J., Hurwicz, L., Uzawa, H. (eds.) Studies in Linear and Non-Linear Programming, pp. 38–102. Stanford University Press, Stanford (1958)
131.
go back to reference Isermann, H.: Proper efficiency and the linear vector maximum problem. Oper. Res. 22, 189–191 (1974) Isermann, H.: Proper efficiency and the linear vector maximum problem. Oper. Res. 22, 189–191 (1974)
132.
go back to reference Isermann, H.: Duality in multiple objective linear programming. In: Zionts, S. (ed.) Multiple Criteria Problem Solving. Lecture Notes in Economics and Mathematical Systems, vol. 155, pp. 274–285. Springer, Berlin (1978) Isermann, H.: Duality in multiple objective linear programming. In: Zionts, S. (ed.) Multiple Criteria Problem Solving. Lecture Notes in Economics and Mathematical Systems, vol. 155, pp. 274–285. Springer, Berlin (1978)
133.
go back to reference Isermann, H.: On some relations between a dual pair of multiple objective linear programs. Z. Oper. Res. Ser. A 22, A33–A41 (1978) Isermann, H.: On some relations between a dual pair of multiple objective linear programs. Z. Oper. Res. Ser. A 22, A33–A41 (1978)
134.
go back to reference Islam, M.A.: Sufficiency and duality in nondifferentiable multiobjective programming. Pure Appl. Math. Sci. 39, 31–39 (1994) Islam, M.A.: Sufficiency and duality in nondifferentiable multiobjective programming. Pure Appl. Math. Sci. 39, 31–39 (1994)
135.
go back to reference Iusem, A.N.: On the maximal monotonicity of diagonal subdifferential operators. J. Convex Anal. 18, 489–503 (2011) Iusem, A.N.: On the maximal monotonicity of diagonal subdifferential operators. J. Convex Anal. 18, 489–503 (2011)
136.
go back to reference Iusem, A.N., Svaiter, B.F.: On diagonal subdifferential operators in nonreflexive Banach spaces. Set-Valued Var. Anal. 20, 1–14 (2012) Iusem, A.N., Svaiter, B.F.: On diagonal subdifferential operators in nonreflexive Banach spaces. Set-Valued Var. Anal. 20, 1–14 (2012)
137.
go back to reference Iwanow, E.H., Nehse, R.: Some results on dual vector optimization problems. Optimization 16, 505–517 (1985) Iwanow, E.H., Nehse, R.: Some results on dual vector optimization problems. Optimization 16, 505–517 (1985)
138.
go back to reference Jahn, J.: Duality in vector optimization. Math. Program. 25, 343–353 (1983) Jahn, J.: Duality in vector optimization. Math. Program. 25, 343–353 (1983)
139.
go back to reference Jahn, J.: Scalarization in vector optimization. Math. Program. 29, 203–218 (1984) Jahn, J.: Scalarization in vector optimization. Math. Program. 29, 203–218 (1984)
140.
go back to reference Jahn, J.: Vector Optimization – Theory, Applications, and Extensions. Springer, Berlin/Heidelberg (2004) Jahn, J.: Vector Optimization – Theory, Applications, and Extensions. Springer, Berlin/Heidelberg (2004)
141.
go back to reference Jeyakumar, V., Li, G.Y.: New dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs. Nonlinear Anal. Theory Methods Appl. 71, 2239–2249 (2009) Jeyakumar, V., Li, G.Y.: New dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs. Nonlinear Anal. Theory Methods Appl. 71, 2239–2249 (2009)
142.
go back to reference Jeyakumar, V., Li, G.Y.: Stable zero duality gaps in convex programming: complete dual characterizations with applications to semidefinite programs. J. Math. Anal. Appl. 360, 156–167 (2009) Jeyakumar, V., Li, G.Y.: Stable zero duality gaps in convex programming: complete dual characterizations with applications to semidefinite programs. J. Math. Anal. Appl. 360, 156–167 (2009)
143.
go back to reference Kaliszewski, I.: Norm scalarization and proper efficiency in vector optimization. Found. Control Eng. 11, 117–131 (1986) Kaliszewski, I.: Norm scalarization and proper efficiency in vector optimization. Found. Control Eng. 11, 117–131 (1986)
144.
go back to reference Kaliszewski, I.: Generating nested subsets of efficient solutions. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization, pp. 173–182. Springer, Berlin (1987) Kaliszewski, I.: Generating nested subsets of efficient solutions. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization, pp. 173–182. Springer, Berlin (1987)
145.
go back to reference Kanniappan, P.: Duality theorems for convex programming without constraint qualification. J. Aust. Math. Soc. Ser. A 36, 252–266 (1984) Kanniappan, P.: Duality theorems for convex programming without constraint qualification. J. Aust. Math. Soc. Ser. A 36, 252–266 (1984)
146.
go back to reference Khanh, P.Q.: Optimality conditions via norm scalarization in vector optimization. SIAM J. Control Optim. 31, 646–658 (1993) Khanh, P.Q.: Optimality conditions via norm scalarization in vector optimization. SIAM J. Control Optim. 31, 646–658 (1993)
147.
go back to reference Kim, G.S., Lee, G.M.: On ɛ-approximate solutions for convex semidefinite optimization problems. Taiwan. J. Math. 11, 765–784 (2007) Kim, G.S., Lee, G.M.: On ɛ-approximate solutions for convex semidefinite optimization problems. Taiwan. J. Math. 11, 765–784 (2007)
148.
go back to reference Klee, V.L.: Convex sets in linear spaces. Duke Univ. Math. Ser. 16, 443–466 (1948) Klee, V.L.: Convex sets in linear spaces. Duke Univ. Math. Ser. 16, 443–466 (1948)
149.
go back to reference Kolumbán, J.: Dualität bei Optimierungsaufgaben. In: Alexits, G., Stechkin, S.B. (eds.) Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory), Budapest, 1969, pp. 261–265. Akadémiai Kiadó, Budapest (1972) Kolumbán, J.: Dualität bei Optimierungsaufgaben. In: Alexits, G., Stechkin, S.B. (eds.) Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory), Budapest, 1969, pp. 261–265. Akadémiai Kiadó, Budapest (1972)
150.
go back to reference Kornbluth, J.S.H.: Duality, indifference and sensitivity analysis in multiple objective linear programming. Oper. Res. Q. 25, 599–614 (1974) Kornbluth, J.S.H.: Duality, indifference and sensitivity analysis in multiple objective linear programming. Oper. Res. Q. 25, 599–614 (1974)
151.
go back to reference Kusraev, A.G., Kutateladze, S.S.: Subdifferentials: Theory and Applications. Mathematics and Its Applications, vol. 323. Kluwer, Dordrecht (1995) Kusraev, A.G., Kutateladze, S.S.: Subdifferentials: Theory and Applications. Mathematics and Its Applications, vol. 323. Kluwer, Dordrecht (1995)
152.
go back to reference Kuwano, I., Tanaka, T., Yamada, S.: Several nonlinear scalarization methods for set-valued maps. RIMS Kôkyûroku 1643, 75–86 (2009) Kuwano, I., Tanaka, T., Yamada, S.: Several nonlinear scalarization methods for set-valued maps. RIMS Kôkyûroku 1643, 75–86 (2009)
153.
go back to reference Lampe, U.: Dualität und eigentliche Effizienz in der Vektoroptimierung. Humboldt-Univ. Berl., Fachbereich Math., Semin.ber. 37, 45–54 (1981) Lampe, U.: Dualität und eigentliche Effizienz in der Vektoroptimierung. Humboldt-Univ. Berl., Fachbereich Math., Semin.ber. 37, 45–54 (1981)
154.
go back to reference Liu, J., Song, W.: On proper efficiencies in locally convex spaces – a survey. Acta Math. Vietnam. 26, 301–312 (2001) Liu, J., Song, W.: On proper efficiencies in locally convex spaces – a survey. Acta Math. Vietnam. 26, 301–312 (2001)
155.
go back to reference Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin/Heidelberg (1989) Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin/Heidelberg (1989)
156.
go back to reference Luc, D.T.: On duality in multiple objective linear programming. Eur. J. Oper. Res. 210, 158–168 (2011) Luc, D.T.: On duality in multiple objective linear programming. Eur. J. Oper. Res. 210, 158–168 (2011)
157.
go back to reference Luc, D.T., Phong, T.Q., Volle, M.: A new duality approach to solving concave vector maximization problems. J. Glob. Optim. 36, 401–423 (2006) Luc, D.T., Phong, T.Q., Volle, M.: A new duality approach to solving concave vector maximization problems. J. Glob. Optim. 36, 401–423 (2006)
158.
go back to reference Lowe, T.J., Thisse, J.-F., Ward, J.E., Wendell, R.E.: On efficient solutions to multiple objective mathematical problems. Manag. Sci. 30, 13–46 (1984) Lowe, T.J., Thisse, J.-F., Ward, J.E., Wendell, R.E.: On efficient solutions to multiple objective mathematical problems. Manag. Sci. 30, 13–46 (1984)
159.
go back to reference Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969) Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)
160.
go back to reference Mansour, M.A., Chbani, Z., Riahi, H.: Recession bifunction and solvability of noncoercive equilibrium problems. Commun. Appl. Anal. 7, 369–377 (2003) Mansour, M.A., Chbani, Z., Riahi, H.: Recession bifunction and solvability of noncoercive equilibrium problems. Commun. Appl. Anal. 7, 369–377 (2003)
161.
go back to reference Marques Alves, M., Svaiter, B.F.: Maximal monotonicity, conjugation and the duality product in non-reflexive Banach spaces. J. Convex Anal. 17, 553–563 (2010) Marques Alves, M., Svaiter, B.F.: Maximal monotonicity, conjugation and the duality product in non-reflexive Banach spaces. J. Convex Anal. 17, 553–563 (2010)
162.
go back to reference Marques Alves, M., Svaiter, B.F.: On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces. J. Convex Anal. 18, 209–226 (2011) Marques Alves, M., Svaiter, B.F.: On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces. J. Convex Anal. 18, 209–226 (2011)
163.
go back to reference Martínez-Legaz, J.-E.: Some generalizations of Rockafellar’s surjectivity theorem. Pac. J. Optim. 4, 527–535 (2008) Martínez-Legaz, J.-E.: Some generalizations of Rockafellar’s surjectivity theorem. Pac. J. Optim. 4, 527–535 (2008)
164.
go back to reference Mbunga, P.: Structural stability of vector optimization problems. In: Pardalos, P.M., Tseveendorj, I., Enkhbat, R. (eds.) Optimization and Optimal Control, Ulaanbaatar, 2002. Series on Computers and Operations Research, vol. 1, pp. 175–183. World Scientific, River Edge (2003) Mbunga, P.: Structural stability of vector optimization problems. In: Pardalos, P.M., Tseveendorj, I., Enkhbat, R. (eds.) Optimization and Optimal Control, Ulaanbaatar, 2002. Series on Computers and Operations Research, vol. 1, pp. 175–183. World Scientific, River Edge (2003)
165.
go back to reference Miettinen, K., Mäkelä, M.M.: On scalarizing functions in multiobjective optimization. OR Spectr. 24, 193–213 (2002) Miettinen, K., Mäkelä, M.M.: On scalarizing functions in multiobjective optimization. OR Spectr. 24, 193–213 (2002)
166.
go back to reference Miglierina, E., Molho, E.: Scalarization and stability in vector optimization. J. Optim. Theory Appl. 114, 657–670 (2002) Miglierina, E., Molho, E.: Scalarization and stability in vector optimization. J. Optim. Theory Appl. 114, 657–670 (2002)
167.
go back to reference Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2004) Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2004)
168.
go back to reference Mitani, K., Nakayama, H.: A multiobjective diet planning support system using the satisficing trade-off method. J. Multi-Criteria Decis. Anal. 6, 131–139 (1997) Mitani, K., Nakayama, H.: A multiobjective diet planning support system using the satisficing trade-off method. J. Multi-Criteria Decis. Anal. 6, 131–139 (1997)
169.
go back to reference Mond, B., Weir, T.: Generalized concavity and duality. In: Schaible, S., Zemba, W.T. (eds.) Generalized Concavity in Optimization and Economics. Proceedings of the NATO Advanced Study Institute, University of British Columbia, Vancouver, 1980, pp. 263–279. Academic, New York (1981) Mond, B., Weir, T.: Generalized concavity and duality. In: Schaible, S., Zemba, W.T. (eds.) Generalized Concavity in Optimization and Economics. Proceedings of the NATO Advanced Study Institute, University of British Columbia, Vancouver, 1980, pp. 263–279. Academic, New York (1981)
170.
go back to reference Nakayama, H.: Some remarks on dualization in vector optimization. J. Multi-Criteria Decis. Anal. 5, 218–255 (1996) Nakayama, H.: Some remarks on dualization in vector optimization. J. Multi-Criteria Decis. Anal. 5, 218–255 (1996)
171.
go back to reference Pennanen, T.: On the range of monotone composite mappings. J. Nonlinear Convex Anal. 2, 193–202 (2001) Pennanen, T.: On the range of monotone composite mappings. J. Nonlinear Convex Anal. 2, 193–202 (2001)
172.
go back to reference Penot, J.-P., Zălinescu, C.: Some problems about the representation of monotone operators by convex functions. ANZIAM J. 47, 1–20 (2005) Penot, J.-P., Zălinescu, C.: Some problems about the representation of monotone operators by convex functions. ANZIAM J. 47, 1–20 (2005)
173.
go back to reference Phelps, R.R.: Lectures on maximal monotone operators. Extr. Math. 12, 193–230 (1997) Phelps, R.R.: Lectures on maximal monotone operators. Extr. Math. 12, 193–230 (1997)
174.
go back to reference Precupanu, T.: Closedness conditions for the optimality of a family of nonconvex optimization problems. Math. Operationsforsch. Stat. Ser. Optim. 15, 339–346 (1984) Precupanu, T.: Closedness conditions for the optimality of a family of nonconvex optimization problems. Math. Operationsforsch. Stat. Ser. Optim. 15, 339–346 (1984)
175.
go back to reference Qiu, J.H., Hao, Y.: Scalarization of Henig properly efficient points in locally convex spaces. J. Optim. Theory Appl. 147, 71–92 (2010) Qiu, J.H., Hao, Y.: Scalarization of Henig properly efficient points in locally convex spaces. J. Optim. Theory Appl. 147, 71–92 (2010)
176.
go back to reference Riahi, H.: On the range of the sum of monotone operators in general Banach spaces. Proc. Am. Math. Soc. 124, 3333–3338 (1996) Riahi, H.: On the range of the sum of monotone operators in general Banach spaces. Proc. Am. Math. Soc. 124, 3333–3338 (1996)
177.
go back to reference Rocco, M., Martínez-Legaz, J.-E.: On surjectivity results for maximal monotone operators of type (D). J. Convex Anal. 18, 545–576 (2011) Rocco, M., Martínez-Legaz, J.-E.: On surjectivity results for maximal monotone operators of type (D). J. Convex Anal. 18, 545–576 (2011)
178.
go back to reference Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
179.
go back to reference Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970) Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)
180.
go back to reference Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970) Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
181.
go back to reference Rödder, W.: A generalized saddlepoint theory; its application to duality theory for linear vector optimum problems. Eur. J. Oper. Res. 1, 55–59 (1977) Rödder, W.: A generalized saddlepoint theory; its application to duality theory for linear vector optimum problems. Eur. J. Oper. Res. 1, 55–59 (1977)
182.
go back to reference Rubinov, A.M.: Sublinear operator and theirs applications. Russ. Math. Surv. 32, 113–175 (1977) Rubinov, A.M.: Sublinear operator and theirs applications. Russ. Math. Surv. 32, 113–175 (1977)
183.
go back to reference Rubinov, A.M., Gasimov, R.N.: Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation. J. Glob. Optim. 29, 455–477 (2004) Rubinov, A.M., Gasimov, R.N.: Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation. J. Glob. Optim. 29, 455–477 (2004)
184.
go back to reference Rubinov, A.M., Glover, B.M.: Quasiconvexity via two step functions. In: Crouzeix, J.-P., Martínez Legaz, J.E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications, vol. 27, pp. 159–183. Kluwer, Dordrecht (1998) Rubinov, A.M., Glover, B.M.: Quasiconvexity via two step functions. In: Crouzeix, J.-P., Martínez Legaz, J.E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications, vol. 27, pp. 159–183. Kluwer, Dordrecht (1998)
185.
go back to reference Schandl, B., Klamroth, K., Wiecek, M.M.: Norm-based approximation in multicriteria programming. Comput. Math. Appl. 44, 925–942 (2002) Schandl, B., Klamroth, K., Wiecek, M.M.: Norm-based approximation in multicriteria programming. Comput. Math. Appl. 44, 925–942 (2002)
186.
go back to reference Schechter, M.: A subgradient duality theorem. J. Math. Anal. Appl. 61, 850–855 (1977) Schechter, M.: A subgradient duality theorem. J. Math. Anal. Appl. 61, 850–855 (1977)
187.
go back to reference Schönefeld, P.: Some duality theorems for the non-linear vector maximum problem. Unternehmensforsch. 14, 51–63 (1970) Schönefeld, P.: Some duality theorems for the non-linear vector maximum problem. Unternehmensforsch. 14, 51–63 (1970)
188.
go back to reference Shimizu, A., Nishizawa, S., Tanaka, T.: On nonlinear scalarization methods in set-valued optimization. RIMS Kôkyûroku 1415, 20–28 (2005) Shimizu, A., Nishizawa, S., Tanaka, T.: On nonlinear scalarization methods in set-valued optimization. RIMS Kôkyûroku 1415, 20–28 (2005)
189.
go back to reference Simons, S.: The range of a monotone operator. J. Math. Anal. Appl. 199, 176–201 (1996) Simons, S.: The range of a monotone operator. J. Math. Anal. Appl. 199, 176–201 (1996)
190.
go back to reference Simons, S.: From Hahn-Banach to Monotonicity. Lecture Notes in Mathematics, vol. 1693. Springer, Berlin (2008) Simons, S.: From Hahn-Banach to Monotonicity. Lecture Notes in Mathematics, vol. 1693. Springer, Berlin (2008)
191.
go back to reference Tammer, C.: A variational principle and applications for vectorial control approximation problems. Preprint 96–09, Reports on Optimization and Stochastics, Martin-Luther University Halle-Wittenberg (1996) Tammer, C.: A variational principle and applications for vectorial control approximation problems. Preprint 96–09, Reports on Optimization and Stochastics, Martin-Luther University Halle-Wittenberg (1996)
192.
go back to reference Tammer, C., Göpfert, A.: Theory of vector optimization. In: Ehrgott, M., Gandibleux, X. (eds.) Multiple Criteria Optimization: State of the Art – Annotated Bibliographic Surveys. International Series in Operations Research & Management Science, vol. 52, pp. 1–70. Kluwer, Boston (2002) Tammer, C., Göpfert, A.: Theory of vector optimization. In: Ehrgott, M., Gandibleux, X. (eds.) Multiple Criteria Optimization: State of the Art – Annotated Bibliographic Surveys. International Series in Operations Research & Management Science, vol. 52, pp. 1–70. Kluwer, Boston (2002)
193.
go back to reference Tammer, C., Göpfert, A., Riahi, H., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 17. Springer, New York (2003) Tammer, C., Göpfert, A., Riahi, H., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 17. Springer, New York (2003)
194.
go back to reference Tammer, C., Winkler, K.: A new scalarization approach and applications in multicriteria d.c. optimization. J. Nonlinear Convex Anal. 4, 365–380 (2003) Tammer, C., Winkler, K.: A new scalarization approach and applications in multicriteria d.c. optimization. J. Nonlinear Convex Anal. 4, 365–380 (2003)
195.
go back to reference Tanaka, T., Kuroiwa, D.: Some general conditions assuring int A + B = int (A + B). Appl. Math. Lett. 6, 51–53 (1993) Tanaka, T., Kuroiwa, D.: Some general conditions assuring int A + B = int (A + B). Appl. Math. Lett. 6, 51–53 (1993)
196.
go back to reference Tanaka, T., Kuroiwa, D.: The convexity of A and B assures int A + B = int (A + B). Appl. Math. Lett. 6, 83–86 (1993) Tanaka, T., Kuroiwa, D.: The convexity of A and B assures int A + B = int (A + B). Appl. Math. Lett. 6, 83–86 (1993)
197.
go back to reference Tanino, T., Kuk, H.: Nonlinear multiobjective programming. In: Ehrgott, M., Gandibleux, X. (eds.) Multiple Criteria Optimization: State of the Art – Annotated Bibliographic Surveys. International Series in Operations Research & Management Science, vol. 52, pp. 71–128. Kluwer, Boston (2002) Tanino, T., Kuk, H.: Nonlinear multiobjective programming. In: Ehrgott, M., Gandibleux, X. (eds.) Multiple Criteria Optimization: State of the Art – Annotated Bibliographic Surveys. International Series in Operations Research & Management Science, vol. 52, pp. 71–128. Kluwer, Boston (2002)
198.
go back to reference Tasset, T.N.: Lagrange Multipliers for Set-Valued Functions When Ordering Cones Have Empty Interior. PhD Thesis, University of Colorado (2010) Tasset, T.N.: Lagrange Multipliers for Set-Valued Functions When Ordering Cones Have Empty Interior. PhD Thesis, University of Colorado (2010)
199.
go back to reference Tidball, M.M., Pourtallier, O., Altman, E.: Approximations in dynamic zero-sum games. SIAM J. Optim. 35, 2101–2117 (2006) Tidball, M.M., Pourtallier, O., Altman, E.: Approximations in dynamic zero-sum games. SIAM J. Optim. 35, 2101–2117 (2006)
200.
go back to reference Wanka, G., Boţ, R.I.: A new duality approach for multiobjective convex optimization problems. J. Nonlinear Convex Anal. 3, 41–57 (2002) Wanka, G., Boţ, R.I.: A new duality approach for multiobjective convex optimization problems. J. Nonlinear Convex Anal. 3, 41–57 (2002)
201.
go back to reference Wanka, G., Boţ, R.I., Grad, S.-M.: Multiobjective duality for convex semidefinite programming problems. Z. Anal. Anwend. 22, 711–728 (2003) Wanka, G., Boţ, R.I., Grad, S.-M.: Multiobjective duality for convex semidefinite programming problems. Z. Anal. Anwend. 22, 711–728 (2003)
202.
go back to reference Wanka, G., Boţ, R.I., Vargyas, E.T.: Duality for location problems with unbounded unit balls. Eur. J. Oper. Res. 179, 1252–1265 (2007) Wanka, G., Boţ, R.I., Vargyas, E.T.: Duality for location problems with unbounded unit balls. Eur. J. Oper. Res. 179, 1252–1265 (2007)
203.
go back to reference Weidner, P.: An approach to different scalarizations in vector optimization. Wiss. Z. Tech. Hochsch. Ilmenau 36, 103–110 (1990) Weidner, P.: An approach to different scalarizations in vector optimization. Wiss. Z. Tech. Hochsch. Ilmenau 36, 103–110 (1990)
204.
go back to reference Weidner, P.: The influence of proper efficiency on optimal solutions of scalarizing problems in multicriteria optimization. OR Spektrum 16, 255–260 (1994) Weidner, P.: The influence of proper efficiency on optimal solutions of scalarizing problems in multicriteria optimization. OR Spektrum 16, 255–260 (1994)
205.
go back to reference Weir, T.: Proper efficiency and duality for vector valued optimization problems. J. Aust. Math. Soc. Ser. A 43, 21–34 (1987) Weir, T.: Proper efficiency and duality for vector valued optimization problems. J. Aust. Math. Soc. Ser. A 43, 21–34 (1987)
206.
go back to reference Weir, T.: A note on invex functions and duality in multiple objective optimization. Opsearch 25, 98–104 (1988) Weir, T.: A note on invex functions and duality in multiple objective optimization. Opsearch 25, 98–104 (1988)
207.
go back to reference Weir, T.: On efficiency, proper efficiency and duality in multiobjective programming. Asia-Pac. J. Oper. Res. 7, 46–54 (1990) Weir, T.: On efficiency, proper efficiency and duality in multiobjective programming. Asia-Pac. J. Oper. Res. 7, 46–54 (1990)
208.
go back to reference Weir, T., Mond, B.: Duality for generalized convex programming without a constraint qualification. Util. Math. 31, 233–242 (1987) Weir, T., Mond, B.: Duality for generalized convex programming without a constraint qualification. Util. Math. 31, 233–242 (1987)
209.
go back to reference Weir, T., Mond, B.: Generalised convexity and duality in multiple objective programming. Bull. Aust. Math. Soc. 39, 287–299 (1989) Weir, T., Mond, B.: Generalised convexity and duality in multiple objective programming. Bull. Aust. Math. Soc. 39, 287–299 (1989)
210.
go back to reference Weir, T., Mond, B.: Multiple objective programming duality without a constraint qualification. Util. Math. 39, 41–55 (1991) Weir, T., Mond, B.: Multiple objective programming duality without a constraint qualification. Util. Math. 39, 41–55 (1991)
211.
go back to reference Weir, T., Mond, B., Craven, B.D.: On duality for weakly minimized vector valued optimization problems. Optimization 17, 711–721 (1986) Weir, T., Mond, B., Craven, B.D.: On duality for weakly minimized vector valued optimization problems. Optimization 17, 711–721 (1986)
212.
go back to reference Weir, T., Mond, B., Craven, B.D.: Weak minimization and duality. Numer. Funct. Anal. Optim. 9, 181–192 (1987) Weir, T., Mond, B., Craven, B.D.: Weak minimization and duality. Numer. Funct. Anal. Optim. 9, 181–192 (1987)
213.
go back to reference Wierzbicki, A.P.: Basic properties of scalarizing functionals for multiobjective optimization. Math. Operationsforsch. Stat. Ser. Optim. 8, 55–60 (1977) Wierzbicki, A.P.: Basic properties of scalarizing functionals for multiobjective optimization. Math. Operationsforsch. Stat. Ser. Optim. 8, 55–60 (1977)
214.
go back to reference Winkler, K.: Skalarisierung mehrkriterieller Optimierungsprobleme mittels schiefer Normen. In: Habenicht, W., Scheubrein, B., Scheubein, R. (eds.) Multi-Criteria- und Fuzzy-Systeme in Theorie und Praxis, pp. 173–190. Deutscher Universitäts-Verlag, Wiesbaden (2003) Winkler, K.: Skalarisierung mehrkriterieller Optimierungsprobleme mittels schiefer Normen. In: Habenicht, W., Scheubrein, B., Scheubein, R. (eds.) Multi-Criteria- und Fuzzy-Systeme in Theorie und Praxis, pp. 173–190. Deutscher Universitäts-Verlag, Wiesbaden (2003)
215.
go back to reference Wolfe, P.: A duality theorem for non-linear programming. Q. Appl. Math. 19, 239–244 (1961) Wolfe, P.: A duality theorem for non-linear programming. Q. Appl. Math. 19, 239–244 (1961)
216.
go back to reference Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003) Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)
217.
go back to reference Zeidler, E.: Nonlinear Functional Analysis and Applications II-B: Nonlinear Monotone Operators. Springer, New York (1990) Zeidler, E.: Nonlinear Functional Analysis and Applications II-B: Nonlinear Monotone Operators. Springer, New York (1990)
218.
go back to reference Zheng, X.Y.: Scalarization of Henig proper efficient points in a normed space. J. Optim. Theory Appl. 105, 233–247 (2000) Zheng, X.Y.: Scalarization of Henig proper efficient points in a normed space. J. Optim. Theory Appl. 105, 233–247 (2000)
219.
go back to reference Zhou, Z.A., Yang, X.M.: Optimality conditions of generalized subconvexlike set-valued optimization problems based on the quasi-relative interior. J. Optim. Theory Appl. 150, 327–340 (2011) Zhou, Z.A., Yang, X.M.: Optimality conditions of generalized subconvexlike set-valued optimization problems based on the quasi-relative interior. J. Optim. Theory Appl. 150, 327–340 (2011)
220.
go back to reference Zălinescu, C.: Stability for a class of nonlinear optimization problems and applications. In: Clarke, F.H., Demyanov, V.F., Giannessi, F. (eds.) Nonsmooth Optimization and Related Topics, Erice 1988, pp. 437–458. Plenum, New York (1988) Zălinescu, C.: Stability for a class of nonlinear optimization problems and applications. In: Clarke, F.H., Demyanov, V.F., Giannessi, F. (eds.) Nonsmooth Optimization and Related Topics, Erice 1988, pp. 437–458. Plenum, New York (1988)
221.
go back to reference Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002) Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)
222.
go back to reference Zălinescu, C.: A new convexity property for monotone operators. J. Convex Anal. 13, 883–887 (2006) Zălinescu, C.: A new convexity property for monotone operators. J. Convex Anal. 13, 883–887 (2006)
223.
go back to reference Zălinescu, C.: On three open problems related to quasi relative interior J. Convex Anal. 22 (2015, in press) Zălinescu, C.: On three open problems related to quasi relative interior J. Convex Anal. 22 (2015, in press)
Metadata
Title
Monotone Operators Approached via Convex Analysis
Author
Sorin-Mihai Grad
Copyright Year
2015
DOI
https://doi.org/10.1007/978-3-319-08900-3_7