Duality in set-valued optimization

Song, Wen

Książka - szczegóły

Tytuł książki

Duality in set-valued optimization

Autorzy

Song Wen

Seria

Rozprawy Matematyczne tom/nr w serii: 375 wydano: 1998

Zawartość

Pełne teksty:

Pobierz

Warianty tytułu

Abstrakty

EN

CONTENTS
Introduction...........................................................................................................5
1. Preliminaries on convex and set-valued analysis..............................................6
1.1. Convexity of sets...........................................................................................6
1.2. Convexity of set-valued mappings.................................................................9
1.3. Closed convex processes and invex set-valued mappings..........................12
2. Vector optimization problems...........................................................................14
2.1. Characterization for optimal points of a set..................................................14
2.2. Characterization for optimal solutions of an optimization problem................17
3. Lagrangian multiplier rule................................................................................19
3.1. Lagrangian conditions for weak optimality...................................................19
3.2. Lagrangian conditions for optimality.............................................................21
3.3. Lagrangian conditions for invex set-valued mappings.................................28
4. Lagrangian duality...........................................................................................33
4.1. Duality for weak optimality............................................................................34
4.2. Duality for optimality.....................................................................................35
4.3. Duality for invex set-valued mappings..........................................................36
5. Geometric duality.............................................................................................37
5.1. A general duality principle for sets...............................................................37
5.2. A geometric approach to duality...................................................................39
5.3. Linear optimization problems.......................................................................42
6. Conjugate duality.............................................................................................45
6.1. Conjugate mappings and subdifferentials....................................................45
6.2. A general conjugate duality..........................................................................50
6.3. Duality in vector optimization with constraints...............................................55
6.4. The Fenchel type duality..............................................................................59
References...........................................................................................................62
List of symbols......................................................................................................67
Index.....................................................................................................................68

Słowa kluczowe

Tematy

Kategoryzacja MSC:

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 375

Liczba stron

69

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCCLXXV

Daty

wydano

1998

otrzymano

1997-11-03

poprawiono

1998-01-15

Twórcy

autor

Song Wen

song@impan.gov.pl

Department of Mathematics, Harbin Normal University, 150080 Harbin, China
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

Bibliografia

[1] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Basel, 1990.
[2] H. P. Benson, An improved definition of proper efficiency for vector minimization with respect to cones, J. Math. Anal. Appl. 71 (1979), 232-241.
[3] G. R. Bitran, Duality for nonlinear multiple-criteria optimization problems, J. Optim. Theory Appl. 35 (1981), 367-401.
[4] J. M. Borwein, Proper efficient points for maximizations with respect to cones, SIAM J. Control Optim. 15 (1977), 57-63.
[5] J. M. Borwein, The geometry of Pareto efficiency over cones, Math. Oper. Statist. Ser. Optim. 11 (1980), 235-248.
[6] J. M. Borwein, Convex relations in optimization and analysis, in: Generalized Concavity in Optimization and Economics, Academic Press, 1981, 335-377.
[7] J. M. Borwein, A Lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand. 48 (1981), 189-204.
[8] J. M. Borwein, Automatic continuity and openness of convex relations, Proc. Amer. Math. Soc. 99 (1987), 49-55.
[9] W. W. Breckner, Dualität bei Optimierungsaufgaben in halbgeordneten topologischen Vektorräumen I, Rev. Anal. Numér. Théor. Approx. 1 (1972), 5-35.
[10] W. W. Breckner and G. Kassay, A systematization of convexity concepts for sets and functions, J. Convex Anal. 4 (1997), 109-127.
[11] A. Cambini, Nonlinear separation theorems, duality and optimality conditions, in: Optimization and Related Fields, R. Conti, E. De Giorgi, and F. Giannessi (eds.), Springer, Berlin, 1986, 57-93.
[12] H. W. Corley, Duality theory for maximizations with respect to cones, J. Math. Anal. Appl. 84 (1981), 560-568.
[13] H. W. Corley, Existence and Lagrangian duality for maximizations of set-valued functions, J. Optim. Theory Appl. 54 (1987), 489-501.
[14] B. D. Craven, Mathematical Programming and Control Theory, Chapman and Hall, London, 1978.
[15] B. D. Craven, Nonsmooth multiobjective programming, Numer. Funct. Anal. Optim. 10 (1989), 64-78.
[16] B. D. Craven, A modified Wolfe dual for weak vector minimization, Numer. Funct. Anal. Optim. 10 (1989), 899-907.
[17] B. D. Craven and B. Glover, Invex functions and duality, J. Austral. Math. Soc. Ser. A 39 (1985), 1-20.
[18] B. D. Craven, J. Gwinner and V. Jeyakumar, Nonconvex theorems of the alternative and minimization, Optimization 18 (1987), 151-163.
[19] J. P. Dauer and O. A. Saleh, A characterization of proper minimal points as solutions of sublinear optimization problems, J. Math. Anal. Appl. 178 (1993), 227-246.
[20] J. P. Dauer and W. Stadler, A survey of vector optimization in infinite-dimensional spaces, Part 2, J. Optim. Theory Appl. 51 (1986), 205-241.
[21] P. H. Dien, G. Mastroeni, M. Pappalardo and P. H. Quang, Regularity conditions for constrained extremum problems via image space, J. Optim. Theory Appl. 80 (1994), 19-37.
[22] S. Dolecki and C. Malivert, General duality for vector optimization, Optimization 27 (1993), 97-119.
[23] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.
[24] K. H. Elster and A. Göpert, Recent results on duality in vector optimization, in: Recent Advances and Historical Development of Vector Optimization, J. Jahn and W. Krabs (eds.), Springer, New York, 1987, 129-136.
[25] K. H. Elster and R. Nehse, Optimality Conditions for Some Nonconvex Problems, Lecture Notes in Control and Information Sci. 23, Springer, 1980.
[26] K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47.
[27] W. Fenchel, On conjugate convex functions, Canad. J. Math. 1 (1949), 73-77.
[28] W. Fenchel, Convex Cones, Sets and Functions, Lecture Notes, Princeton University, 1953.
[29] D. Gale, H. W. Kuhn and A. W. Tucker, Linear programming and theory of games, in: Activity Analysis of Production and Allocation, T. C. Koopmann (ed.), Wiley, New York, 1951, 317-329.
[30] F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl. 42 (1984), 331-365.
[31] C. Gros, Generalization of Fenchel's duality theorem for convex vector optimization, European J. Oper. Res. 2 (1978), 368-376.
[32] J. Gwinner, Closed images of convex multivalued mappings in linear topological spaces with applications, J. Math. Anal. Appl. 60 (1977), 75-86.
[33] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545-550.
[34] M. Hayashi and H. Komiya, Perfect duality for convexlike programs, J. Optim. Theory Appl. 38 (1982), 179-189.
[35] M. I. Henig, A cone separation theorem, J. Optim. Theory Appl. 36 (1982), 451-455.
[36] M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl. 36 (1982), 387-407.
[37] R. B. Holmes, Geometric Functional Analysis and its Applications, Springer, New York, 1975.
[38] L. Hurwicz, Programming in linear spaces, in: Studies in Linear and Nonlinear Programming, K. J. Arrow, L. Hurwicz, and H. Uzawa (eds.), Stanford Univ. Press, Stanford, Calif., 1958; 2nd printing, Oxford Univ. Press, London, 1958, 38-102.
[39] G. Isac and V. Postolică, The Best Approximation and Optimization in Locally Convex Spaces, Verlag Peter Lang, 1993.
[40] H. Isermann, On some relations between a dual pair of multiple objective linear programs, Z. Oper. Res. 22 (1978), 33-41.
[41] E. H. Ivanov and R. Nehse, Some results on dual vector optimization, Optimization 16 (1985), 505-517.
[42] J. Jahn, Duality in vector optimization, Math. Programming 25 (1983), 343-353.
[43] J. Jahn, A characterization of properly minimal elements of a set, SIAM J. Control Optim. 23 (1985), 649-656.
[44] J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Verlag Peter Lang, Frankfurt, 1986.
[45] V. Jeyakumar, Convexlike alternative theorems and mathematical programming, Optimization 16 (1985), 643-652.
[46] V. Jeyakumar, A generalization of a minimax theorem of Ky Fan via a theorem of the alternative, J. Optim. Theory Appl. 48 (1986), 525-533.
[47] V. Jeyakumar and J. Gwinner, Inequality systems and optimization, J. Math. Anal. Appl. 159 (1991), 51-71.
[48] V. Jeyakumar, W. Oettli, and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl. 179 (1993), 537-546.
[49] V. Jeyakumar and H. Wolkowicz, Zero duality gaps in infinite-dimensional programming, J. Optim. Theory Appl. 67 (1990), 87-108.
[50] H. Kawasaki, Conjugate relations and weak subdifferentials, Math. Oper. Res. 6 (1981), 593-607.
[51] H. Kawasaki, A duality theorem in multiobjective nonlinear programming, Math. Oper. Res. 7 (1982), 95-110.
[52] H. König, On certain applications of the Hahn-Banach and minimax theorems, Arch. Math. (Basel) 21 (1970), 583-591.
[53] I. Kouada, Upper-semicontinuity and cone-concavity of multi-valued vector functions in a duality theory for vector optimization, Z. Oper. Res. 46 (1997), 169-192.
[54] M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
[55] M. G. Krein and M. A. Rutman, Linear operator leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. no. 26, Providence, R.I., 1950.
[56] H. W. Kuhn and A. W. Tucker, Nonlinear programming, in: Proc. Second Berkeley Sympos. Math. Statist. Probab., J. Neyman (ed.), Berkeley, Calif., 1951, 481-492.
[57] D. Kuroiwa, Convexity for set-valued maps, Appl. Math. Lett. 9 (1996), 97-101.
[58] H. C. Lai and L. S. Yang, Strong duality for infinite-dimensional vector-valued programming problems, J. Optim. Theory Appl. 62 (1989), 449-466.
[59] R. Lehmann and W. Oettli, The theorem of the alternative, the key-theorem, and the vector-maximum problem, Math. Programming 8 (1975), 332-344
[60] L. J. Lin, Optimization of set-valued functions, J. Math. Anal. Appl. 186 (1994), 30-51.
[61] D. T. Luc, On duality theory in multiobjective programming, J. Optim. Theory Appl. 43 (1984), 557-582.
[62] D. T. Luc, Theory of Vector Optimization, Lecture Notes in Econom. and Math. Systems 319, Springer, Berlin, 1989.
[63] D. T. Luc and J. Jahn, Axiomatic approach to duality in optimization, Numer. Funct. Anal. Optim. 13 (1992), 305-326.
[64] D. T. Luc and C. Malivert, Invex optimization problems, Bull. Austral. Math. Soc. 46 (1992), 47-66.
[65] C. Malivert, Fenchel duality in vector optimization, in: Advances in Optimization, Lecture Notes in Econom. and Math. Systems 382, Springer, Berlin, 1992.
[66] O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.
[67] L. Martein, Some results on regularity in vector optimization, Optimization 20 (1989), 787-798.
[68] G. Mastroeni, M. Pappalardo and N. D. Yen, Image of a parametric optimization problem and continuity of the perturbation function, J. Optim. Theory Appl. 81 (1994), 193-202.
[69] H. Nakayama, Geometric consideration of duality in vector optimization, J. Optim. Theory Appl. 44 (1984), 625-655.
[70] H. Nakayama, Some remarks on dualization in vector optimization, J. Multi-Criteria Decision Anal. 5 (1996), 218-225.
[71] J. W. Nieuwenhuis, Supremal points and generalized duality, Math. Oper. Statist. Ser. Optim. 11 (1980), 41-59.
[72] K. Nikodem, K-convex and K-concave set-valued functions, Zeszyty Nauk. Politech. Łódz. 559 (Rozprawy Mat. 114), Łódź, 1989.
[73] K. Nikodem, Remarks on K-midconvex set-valued functions with closed epigraph, Le Matematiche 45 (1990), 277-281.
[74] W. Oettli, A duality theorem for the nonlinear vector-maximum problem, in: Progress in Operations Research (Eger, 1974), A. Prékopa (ed.), Colloq. Math. Soc. János Bolyai 12, North-Holland, 1976, 697-703.
[75] S. Paeck, Convexlike and concavelike conditions in alternative, minimax, and minimization theorems, J. Optim. Theory Appl. 74 (1992), 317-332.
[76] D. Pallaschke and S. Rolewicz, Foundations of Mathematical Optimization, Math. Appl. 388, Kluwer, 1997.
[77] V. Postolică, Vectorial optimization programs with multifunctions and duality, Ann. Sci. Math. Québec 10 (1986), 85-102.
[78] V. Postolică, A generalization of Fenchel's duality theorem, Ann. Sci. Math. Québec 10 (1986),, 199-206.
[79] T. W. Reiland, Generalized invexity for nonsmooth vector-valued mappings, Numer. Funct. Anal. Optim. 10 (1989), 1191-1202.
[80] S. M. Robinson, Stability theory for systems of inequalities in nonlinear programming, part II, differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497-513.
[81] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1970.
[82] R. T. Rockafellar, Conjugate Duality and Optimization, CBMS Lecture Note Ser. 16, SIAM, Philadelphia, 1974.
[83] E. E. Rosinger, Multiobjective duality without convexity, J. Math. Anal. Appl. 66 (1978), 442-450.
[84] P. H. Sach and B. D. Craven, Invexity in multifunction optimization, Numer. Funct. Anal. Optim. 12 (1991), 383-394.
[85] P. H. Sach and B. D. Craven, Invex multifunctions and duality, Numer. Funct. Anal. Optim. 12 (1991), 575-591.
[86] P. H. Sach, N. D. Yen, and B. D. Craven, Generalized invexity and duality theories with multifunctions, Numer. Funct. Anal. Optim. 15 (1994), 131-153.
[87] Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of Multiobjective Optimization, Academic Press, New York, 1985.
[88] P. Schönfeld, Some duality theorems for the non-linear vector maximum problem, Unternehmensforschung 14 (1970), 51-63.
[89] M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171-176.
[90] W. Song, Lagrangian duality for minimization of nonconvex multifunctions, J. Optim. Theory Appl. 93 (1997), 167-182.
[91] W. Song, Duality for vector optimization of set-valued functions, J. Math. Anal. Appl. 201 (1996), 212-225.
[92] W. Song, Optimality conditions and Lagrangian duality for vector optimization of invex set-valued functions, Control and Cybernetics 26 (1997), 68-85.
[93] W. Song, Conjugate duality in set-valued vector optimization, J. Math. Anal. Appl. 216 (1997), 265-283.
[94] W. Song, A generalization of the Fenchel duality in set-valued vector optimization, Z. Oper. Res., to appear.
[95] W. Song, Duality for vector optimization of set-valued mappings, Ph.D. thesis, Inst. Math., Polish Acad. Sci., 1997.
[96] W. Stadler, A survey of multicriteria optimization or the vector maximum problem, Part I: 1776-1960, J. Optim. Theory Appl. 29 (1979), 1-52.
[97] A. Sterna-Karwat, Convexity of the optimal multifunctions and its consequences in vector optimization, Optimization 20 (1989), 799-808.
[98] T. Tanaka, Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions, J. Optim. Theory Appl. 81 (1994), 355-377.
[99] T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl. 167 (1992), 84-97.
[100] T. Tanino and Y. Sawaragi, Duality theory in multiobjective programming, J. Optim. Theory Appl. 27 (1979), 509-529.
[101] T. Tanino and Y. Sawaragi, Conjugate maps and duality in multiobjective optimization, J. Optim. Theory Appl. 31 (1980), 473-499.
[102] F. Tardella, On the image of a constrained extremum problem and some applications to the existence of a minimum, J. Optim. Theory Appl. 60 (1989), 93-104.
[103] R. M. Van Slyke and R. J.-B. Wets, A duality theory for abstract mathematical programs with applications to optimal control theory, J. Math. Anal. Appl. 22 (1968), 679-706.
[104] P. L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), 319-377.
[105] C. Zălinescu, Duality for vectorial nonconvex optimization by convexification and applications, An. Ştiinţ. Univ. "Al. I. Cuza" Iaşi 29 (1983), no. 3, 15-34.
[106] J. Zowe, A duality theorem for a convex programming problem in order complete vector lattices, J. Math. Anal. Appl. 50 (1975), 273-287.

Języki publikacji

EN

Uwagi

1991 Mathematics Subject Classification: 90C29, 90C26, 90C30.

Identyfikator YADDA

bwmeta1.element.zamlynska-acac4a76-3c12-4086-b233-4b0d90e5ce8f

Identyfikatory

ISSN

0012-3862

Kolekcja

DML-PL

Zawartość książki

rozwiń roczniki

Książka - szczegóły

Duality in set-valued optimization

Autorzy

Seria

Zawartość

Warianty tytułu

Abstrakty

Słowa kluczowe

Tematy

Wydawca

Miejsce publikacji

Copyright

Seria

Liczba stron

Liczba rozdzia³ów

Opis fizyczny

Daty

Twórcy

Bibliografia

Języki publikacji

Uwagi

Identyfikator YADDA

Identyfikatory

Kolekcja