References - link.springer.com978-0-387-28395-1/1.pdf · 330 References [12] M. Avriel and I. Zang,...
Transcript of References - link.springer.com978-0-387-28395-1/1.pdf · 330 References [12] M. Avriel and I. Zang,...
References
[1] N.I. Akhiezer, Lectures on Approximation Theory. Gostehizdat, Moscow-Leningrad, 1947 [Russian].
[2] G. Ascoli, Sugli spazi lineari metrici e le loro varieta lineari. Ann. Mat. Pura AppL (4) 10 (1932), 33-81, 203-232.
[3] E. Asplund, Chebyshev sets in Hilbert space. Trans. Amen Math. Soc. 144 (1969), 235-240.
[4] E. Asplund, Differentiability of the metric projection in finite-dimensional Euclidean spaces. Proc. Amen Math. Soc. 38 (1973), 218-219.
[5] M. Atteia, Analyse convexe projective. Comptes Rendus Acad. Sci. Paris 276 (1973), 855-858.
[6] M. Atteia and A. El Qortobi, Quasi-convex duality. In: Optimization and Optimal Control (A. Auslender, W. Oettli, and J. Stoer, eds.). Lecture Notes Control Inf. Sci. 30, Springer-Verlag, Berlin, Heidelberg, 1981, 3-8.
[7] H. Attouch and H. Brezis, Duality for the sum of convex functions in general Banach spaces. In: Aspects of Mathematics and Its Applications (J. A. Barros, ed.), Elsevier, Amsterdam, 1986, 125-133.
[8] H. Attouch and M. Thera, A general duality principle for the sum of two operators. J. Convex Anal. 3 (1996), 1-24.
[9] J.-R Aubin and I. Ekeland, Estimates of the duality gap in nonconvex optimization. Math. Open Res. 1 (1976), 225-245.
[10] G. Auchmuty, Duality for non-convex variational principles. J. Dijf. Eq. 50 (1983), 80-145.
[11] A. Auslender, Optimisation. Methodes numeriques. Masson, Paris, New York, 1976.
330 References
[12] M. Avriel and I. Zang, Generalized arc wise-connected functions and characterizations of local-global minimum properties. J. Optim. Theory AppL 32 (1980), 407-425.
[13] E.J. Balder, An extension of duality-stability relations to nonconvex optimization problems. SIAM J. Control Optim. 15 (1977), 329-343.
[14] V. Barbu and Th. Precupanu, Convexity and Optimization in Banach Spaces. Editura Academiei and Sijthoff & Noordhoff, Bucuresti and Alphen aan de Rijn, 1978; second ed., Editura Academiei and D. Reidel, Bucuresti and Dordrecht-Boston-Lancaster, 1986.
[15] H. Bauer, Minimalstellen von Funktionen und Extremalpunkte. Arch. Math. 9 (1958), 389-393.
[16] H.R Benson, Concave minimization: theory, applications and algorithms. In: [109], 43-148.
[17] V.I. Berdyshev, The stability of the problem of minimization with respect to a perturbation of the set of admissible elements. Matem. Sbornik 103 (145) (1977) [Russian].
[18] B. Bereanu, On the global minimum of a quasi-concave functional. Arch. Math. 25 (1974), 391-393.
[19] C. Bergthaller and I. Singer, The distance to a polyhedron. Linear Alg. Appl. 169(1992), 111-129.
[20] G. Birkhoff, Lattice Theory. Colloquium Publications, vol. 25, American Mathematical Society, 1967.
[21] J.M. Borwein and A.S. Lewis, Convex Analysis and Nonlinear Optimization. Springer-Verlag, New York, Berlin, Heidelberg, 2000.
[22] J.M. Borwein and D. Zhuang, On Fan's minimax theorem. Math. Progr 34 (1986), 232-234.
[23] N. Bourbaki, Topologie Generale. Ch. IV: Nombres reels. Hermann, Paris, 1942.
[24] N. Bourbaki, Espaces Vectoriels Topologiques (2 volumes). Hermann, Paris, 1953, 1955.
[25] D. Braess, Nonlinear Approximation Theory. Springer-Verlag, Berlin, Heidelberg, New York, 1986.
[26] W. Briec and B. Lemaire, Technical efficiency and distance to a reverse convex set. Eur J. Oper Res. 114 (1999), 178-187.
[27] R.C. Buck, Applications of duality in approximation theory. In: Approximation of Functions (H.L. Garabedian, ed.), Elsevier, Amsterdam, London, New York, 1965, 27-42.
[28] R.E. Burkard, H. Hamacher, and J. Tind, On abstract duahty in mathematical programming. Z. Oper Res. 26 (1982), 197-209.
[29] R.E. Burkard and U. Zimmermann, Combinatorial optimization in linearly ordered semi-modules. A survey. In: Modern Applied Mathematics. Optimization and Operations Research (B. Korte, ed.), Amsterdam, 1982, 391-436.
[30] A. Cambini, Non-linear separation theorems, duality and optimality conditions. In: Optimization and Related Fields (R. Conti, E. de Giorgi, and
References 331
F. Giannessi, eds.), Lecture Notes Math. 1190, Springer-Verlag, Berlin, Heidelberg (1986), 57-93.
[31] E.W. Cheney, Introduction to Approximation Theory. McGraw-Hill, New York, St. Louis, 1966.
[32] E.W. Cheney and A.A. Goldstein, Tchebycheff approximation and related extremal problems. J. Math. Mech. 14 (1965), 87-98.
[33] C. Combari, M. Laghdir, and L. Thibault, Sous-differentielles des fonctions convexes composees. An«. Sci. Math. Quebec 18 (1994), 119-148.
[34] J.-P. Crouzeix, Contributions a I'etude des fonctions quasiconvexes. These. Univ. de Clermont, 1977.
[35] J.-P. Crouzeix, Conjugacy in quasiconvex analysis. In: Convex Analysis and Its Applications (A. Auslender, ed.). Lecture Notes Econ. Math. Systems 144, Springer-Verlag, Berlin, Heidelberg, 1977, 66-99.
[36] J.-P. Crouzeix, Continuity and differentiability properties of quasiconvex functions on /?". In: [205], 109-130.
[37] J.-P. Crouzeix, A duality framework in quasiconvex programming. In: [205], 207-225.
[38] J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Voile, eds.. Generalized convexity, generalized monotonicity. Kluwer Acad. Publ., Dordrecht, 1998.
[39] A. Daniilidis and J.-E. Martinez-Legaz, Characterizations of evenly convex sets and evenly quasiconvex functions. J. Math. Anal. Appl. 273 (2002), 58-66.
[40] M.M. Day, Normed Linear Spaces. 3rd ed. Springer-Verlag, New York, Heidelberg, Berlin, 1973.
[41] F. Deutsch, Best Approximation in Inner Product Spaces. Springer-Verlag, New York, Berhn, Heidelberg, 2001.
[42] F. Deutsch, W. Li, and J. Swetits, Fenchel duality and the strong conical hull intersection property. J. Optim. Theory Appl. 102 (1999), 681-695.
[43] F. Deutsch, W. Li, and J. Ward, A dual approach to constrained interpolation from a convex subset of Hilbert space. J. Approx. Theory 90 (1997), 385-414.
[44] F. Deutsch and PH. Maserick, Applications of the Hahn-Banach theorem in approximation theory. SI AM Rev. 9 (1967), 516-530.
[45] PH. Dien, G. Mastroeni, M. Pappalardo, and PH. Quang, Regularity conditions for constrained extremum problems via image space. /. Optim. Theory A/7/7/.,80(1994), 19-37.
[46] F. Di Guglielmo, Estimates of the duality gap for discrete and quasiconvex optimization problems. In: [205], 281-298.
[47] S. Dolecki, Abstract study of optimality conditions. J. Math. Anal Appl. 73 (1980), 24-48; Corrigendum. J. Math. Anal Appl. 82 (1981), 295-296.
[48] S. Dolecki and S. Kurcyusz, On O-convexity in extremal problems. SI AM J. Control Optim. 16 (1978), 277-300.
[49] N. Dunford and J. Schwartz, Linear Operators. Part I: General Theory. In-terscience Publ., New York, London, 1953.
[50] M. Diir, Conditions characterizing minima of differences of functions. Monatsh. Math. 134 (2002), 295-303.
332 References
[51] M. Diir, A parametric characterization of local optimality. Math. Open Res. 57 (2003), 101-109.
[52] M. Diir, R. Horst, and M. Locatelli, Necessary and sufficient global optimality conditions for convex maximization revisited. / Math. Anal. Appl. Ill (1998), 637-649.
[53] M. Eidelheit, Quelques remarques sur les fonctionnelles lineaires. Studia Math. 10(1948), 140-147.
[54] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam, Oxford, 1976.
[55] I. Ekeland and T. Tumbull, Infinite Dimensional Optimization and Convexity. University of Chicago Press, 1983.
[56] R. Ellaia and J.-B. Hiriart-Urruty, The conjugate of the difference of convex functions. J. Optim. Theory Appl. 49 (1986), 493-498.
[57] A. El Qortobi, Contributions a la theorie de la dualite pour les fonctionnelles quasiconvexes. These de 3-eme cycle. Univ. de Toulouse, 1980.
[58] A. El Qortobi, Conjugaison quasi convexe des fonctionnelles positives. Ann. Sci. Math. Quebec 17 (1993), 155-167.
[59] K.-H. Elster and A. Gopfert, Conjugation concepts in optimization. In: Methods of Operations Research, vol. 62 (R Rieder, A. Gessner, A. Peyerimhoff, and F.J. Radermacher, eds.), A. Hain, Frankfurt am Main (1990), 53-65.
[60] K.-H. Elster and R. Nehse, Zur Theorie der Polarfunktionale. Math. Open 5m?. 5(1974), 3-21.
[61] K.-H. Elster, R. Reinhardt, M. Schauble, and G. Donath, EinfUhrung in die Nichtlineare Optimierung. B.G. Teubner Verlagsgesellschaft, Leipzig, 1977.
[62] K.-H. Elster and A. Wolf, Comparison between several conjugation concepts. In: Optimal Control (R. Bulirsch, A. Miele, J. Stoer, and K.H. Well, eds.). Lecture Notes Control Inform. Sci. 95, Springer-Verlag, Berlin, Heidelberg (1987), 79-93.
[63] K.-H. Elster and A. Wolf, On a general concept of conjugate functions as an approach to nonconvex optimization problems. Preprint 149, Univ. of Pisa, 1987.
[64] K.-H. Elster and A. Wolf, Recent results on generalized conjugate functions. In: Trends in Mathematical Optimization (K.-H. Hoffmann, J.-B.Hiriart-Urruty, C. Lemarechal, and J. Zowe, eds.), Birkhauser-Verlag, Basel (1988), 67-78.
[65] J.J.M. Evers and H. van Maaren, Duality principles in mathematics and their relations to conjugate functions. Nieuw. Arch. Wish. 3 (1985), 23-68.
[66] Yu.G. Evtushenko, A.M. Rubinov, and V.G. Zhadan, General Lagrange-type functions in constrained global optimization. Part I: Auxiliary functions and optimality conditions. Optim. Methods and Software 16 (2001), 193-230.
[67] J. Flachs, Global saddle-point duality for quasi-concave programs. Math. ProgK 20 (19SII 321-341.
[68] J. Flachs and M. Pollatschek, Duality theorems for certain programs involving minimum or maximum operations. Math. Progr 16 (1979), 348-370.
References 333
[69] F. Flores-Bazan, On a notion of subdifferentiability for non-convex functions. Optimization 33 (1995), 1-8.
[70] F. Flores-Bazan, On minima of the difference of functions. /. Optim. Theory A/7/?/. 93 (1997), 525-531.
[71] F. Flores-Bazan and J.-E. Martinez-Legaz, Simplified global optimality conditions in generalized conjugation theory. In: [38], 305-329.
[72] F. Flores-Bazan and W. Oettli, Simphfied optimality conditions for minimizing the difference of vector-valued convex mappings. J. Optim. Theory Appl. 108 (2001), 571-586.
[73] C. Franchetti and I. Singer, Deviation and farthest points in normed linear spaces. Rev. Roum. Math. Pures Appl. 24 (1979), 373-381.
[74] C. Franchetti and I. Singer, Best approximation by elements of caverns in normed linear spaces. Boll. Un. Mat. Ital. (5) 17-B (1980), 33-43.
[75] N. Gaffke and R. Mathar, A cyclic projection algorithm via duality. Metrika 36 (1989), 29-54.
[76] D.Y. Gao, Duality Principles in Nonconvex Systems. Theory, Methods and Applications. Kluwer Acad. Publ., Dordrecht, 2000.
[77] D.Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Optim. Theory Appl. 17 (2000), 127-160.
[78] M.R. Garey and D.S. Johnson, Computers and Intractability. Freeman, San Francisco, 1979.
[79] A.L. Garkavi, Duality theorems for the approximation by elements of convex sets. UspekhiMat. Nauk 16, 4 (100) (1961), 141-145 [Russian].
[80] J. Getan, J.-E. Martinez-Legaz, and I. Singer, (*, ^)-dualities. J. Math. Sciences 115 (2003), 2506-2541.
[81] F. Giannessi, Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42 (1984), 331-365.
[82] F. Giannessi, General optimality conditions via a separation scheme. In: Algorithms for Continuous Optimization (E. Spedicato, ed.). Kluwer Acad. Publ., Dordrecht, 1994, 1-23.
[83] E. Giner, Local minimizers of integral functional are global minimizers. Proc. Amen Math. Soc. 123 (1995), 755-767.
[84] K. Glashoff and S.-A. Gustaffson, Linear Approximation and Optimization. Springer-Verlag, New York, Heidelberg, Berlin, 1983.
[85] B.M. Glover, Y. Ishizuka, V. Jeyakumar, and H.D. Tuan, Complete characterization of global optimality for problems involving the pointwise minimum of sublinear functions. SIAM J. Optim. 6 (1996), 362-372.
[86] F. Glover, A multiphase-dual algorithm for the zero-one integer programming problem. Open Res. 13 (1965), 879-919.
[87] F. Glover, Surrogate constraints. Open Res. 16 (1968), 741-769. [88] C.J. Goh and X.Q. Yang, Nonlinear Lagrangian theory for nonconvex opti
mization. J. Optim. Theory Appl. 109 (2001), 99-121. [89] C.J. Goh and X.Q. Yang, Duality in Optimization and Variational Inequali
ties. Taylor&Francis, London, 2002.
334 References
[90] E.G. Golshtein, The Theory of Duality in Mathematical Programming and Its Applications. Nauka, Moscow, 1971 [Russian].
[91] E.G. Golshtein and N.V. Tretyakov, Modified Lagrangians and Monotone Maps in Optimization Theory. Wiley, New York, 1996.
[92] F.J. Gould, Extensions of Lagrange multipliers in nonlinear programming. SIAMJ. Applied Math. 17 (1969), 1280-1297.
[93] F.J. Gould, Nonlinear duality theorems. Cahiers Centre Etudes Rech. Oper 14 (1972), 196-212.
[94] H.J. Greenberg and W.R Pierskalla, Surrogate mathematical programming. Oper Res. 18 (1970), 924-939.
[95] H.J. Greenberg and W.R Rierskalla, Quasi-conjugate functions and surrogate duality. Cahiers Centre Etudes Rech. Oper 15 (1973), 437-448.
[96] R. Hettich und P. Zencke, Numerische Methoden der Approximation und semi-infiniter Optimierung. Teubner, Stuttgart, 1982.
[97] R. Hildenbrandt and R. Nehse, On duality-separability relations. Optimization 16 (19^51 S05-S\S.
[98] J.-B. Hiriart-Urruty, Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and Duality in Optimization (J. Ponstein, ed.). Lecture Notes Econ. Math. Systems 256, Springer-Verlag, Berlin, Heidelberg, 1985, 37-70.
[99] J.-B. Hiriart-Urruty, A general formula on the conjugate of the difference of functions. Canad. Math. Bull. 29 (1986), 482-485.
[100] J.-B. Hiriart-Urruty, From convex optimization to nonconvex optimization. Part 1: Necessary and sufficient conditions for global optimality. In: Nons-mooth Optimization and Related Topics (F.H. Clarke, V.F. Demyanov, and F. Giannessi, eds.), Plenum Press, New York, London, 1989, 219-239.
[101] J.-B. Hiriart-Urruty, Conditions for global optimality. In: [109], 1-26. [102] J.-B. Hiriart-Urruty, Conditions for global optimality 2. / Global Optim. 13
(1998), 349-367. [103] J.-B. Hiriart-Urruty and Yu.S. Ledyaev, A note on the characterization of
the global maxima of a (tangentially) convex function over a convex set. J. Convex Anal. 3 (1996), 55-61.
[104] J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms. Vols. 1, 2. Springer-Verlag, Berhn, Heidelberg, 1993.
[105] A.J. Hoffman, On abstract dual linear programs. Naval Res. Logist. Quarterly 10 (1963), 369-373.
[106] R.B. Holmes, A Course on Optimization and Best Approximation. Lecture Notes Math. 257, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
[107] R.B. Holmes, Geometric Functional Analysis and Its Applications. Springer-Verlag, New York, Heidelberg, Berlin, 1975.
[108] R. Horst, A note on functions whose local minima are global. J. Optim. Theory Appl 36 (1982), 457-463.
[109] R. Horst and P. Pardalos, eds.. Handbook of Global Optimization. Kluwer Acad. Publ., Dordrecht, 1995.
References 335
110] R. Horst and H. Tuy, Global Optimization (Deterministic Approaches). 3rd ed., Springer-Verlag, Berlin, Heidelberg, New York, 1996.
I l l ] A.D. loffe and V.M. Tikhomirov, Theory of Extremal Problems [Russian]. Nauka, Moscow, 1974. English translation: North-Holland, Amsterdam-Oxford, 1979.
112] T.R. Jefferson and C.H. Scott, Duality for quasi-concave programs with explicit constraints. Math. Operationsforsch. Stat. Sen Optim. 11 (1980), 519-530.
113] V. Jeyakumar and B.M. Glover, Characterizing global optimality for DC optimization problems under convex inequality constraints. J. Global Optim. 8 (1996), 171-187.
114] H.Th. Jongen, R Jonker, F. Twilt, Nonlinear Optimization in R". /. Morse Theory, Chebyshev Approximation. Verlag Peter Lang, Frankfurt/Main, 1983.
115] P. Kanniappan, Fenchel-Rockafellar type duality for a non-convex non-differential optimization problem. J. Math. Anal. Appl. 97 (1983), 366-376.
116] S. Karlin, Mathematical Methods and Theory in Games, Programming and Economics. Pergamon Press, London, Paris, 1959.
117] J.L. Kelley and L Namioka, Linear Topological Spaces. Van Nostrand, London, Toronto, 1963.
118] V. Klee, Convexity of Chebyshev sets. Math. Ann. 142 (1961), 292-304. 119] V. Klee, Remarks on nearest points in normed linear spaces. In: Proc. Coll.
on Convexity (Copenhagen, 1965), Univ. of Copenhagen, 1966, 168-176. 120] H. Konno, P.T. Thach, and H. Tuy, Optimization on Low Rank Nonconvex
Structures. Kluwer Acad. Publ., Dordrecht, 1997. 121] G. Kothe, Topological Vector Spaces (two volumes). Springer-Verlag, New
York, Heidelberg, Berlin, 1979. 122] W. Krabs, Optimierung und Approximation. B.G. Teubner, Stuttgart, 1975. 123] M.G. Krein, The L-problem in an abstract normed linear space. In: On Some
Problems of the Theory of Moments (N.I. Akhiezer and M.G. Krein, eds.), Gonti, Kharkov (1937), 171-199 [Russian].
124] C. Kuratowski, Topologie. 4-eme ed., PWN, Warszawa, 1958. 125] S. Kurcyusz, Some remarks on generalized Lagrangians. In: Optimization
Techniques. Modelling and Optimization in the Service of Man. I (J. Cea, ed.). Lecture Notes Computer Sci. 40, Springer-Verlag, Berlin, Heidelberg (1976), 363-388.
126] S. Kurcyusz, On existence and nonexistence of Lagrange multipliers in Ba-nach spaces. J. Optim. Theory Appl. 20 (1976), 81-110.
127] S.S. Kutateladze and A.M. Rubinov, The Minkowski Duality and Its Applications. Nauka, Novosibirsk, 1976 [Russian].
128] M. Laghdir and M. Voile, A general formula for the horizon function of a convex composite function. Arch. Math. 73 (1999), 291-302.
129] P.-J. Laurent, Approximation et Optimisation. Hermann, Paris, 1972. 130] P.-J. Laurent and B. Martinet, Methodes duales pour le calcul de minimum
d'une fonction convexe sur une intersection des convexes. In: Symposium on Optimization. Ill, Nice 1969. Lecture Notes Math. 132 (1970), 159-180.
336 References
[131] B. Lemaire, Duality in reverse convex optimization. SIAMJ. Optim. 8 (1998), 1029-1037.
[132] B. Lemaire and M. Voile, Duality in d.c. programming. In: [38], 331-345. [133] B. Lemaire and M. Voile, A general duality scheme for nonconvex minimiza
tion problems with a strict inequality constraint. J. Global Optim. 13 (1998), 317-327.
[134] P.O. Lindberg, A generalization of Fenchel conjugation giving generahzed Lagrangians and symmetric nonconvex duality. In: Survey of Mathematical Programming. I (A. Prekopa, ed.) North-Holland, Amsterdam (1979), 249-267.
[135] P.O. Lindberg, On quasiconvex duality. Preprint TRITA-MAT 14. Royal Inst. Technology Stockholm (1981).
[136] D.G. Luenberger, Quasi-convex programming. SI AM J. Appl Math. 16 (1968), 1090-1095.
[137] G. Mastroeni and M. Pappalardo, Separation and regularity in the image space. In: New Trends in Mathematical Programming. Kluwer Acad. Publ., Boston (1998), 181-190.
[138] J.-E. Martinez-Legaz, Un concepto generalizado de conjugacion. Appli-cacion a las funciones quasiconvexas. Thesis, Barcelona, 1981.
[139] J.-E. Martinez-Legaz, A generalized concept of conjugation. In: Optimization: Theory and Algorithms (J.-B. Hiriart-Urruty, W. Oettli, and J. Stoer, eds.), Lecture Notes Pure Appl. Math. 86, Marcel Dekker, New York (1983), 45-59.
[140] J.-E. Martinez-Legaz, Quasi-convex duality theory by generalized conjugation methods. Optimization 19 (1988), 603-652.
[141] J.-E. Martinez-Legaz, Generalized conjugation and related topics. In: Generalized Convexity and Fractional Programming with Economic Applications (A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni, and S. Schaible, eds.). Lecture Notes Econ. Math. Systems 345, Springer-Verlag, Berlin, Heidelberg, 1990, 168-197.
[142] J.-E. Martinez-Legaz, Fenchel duality and related properties in general conjugation theory. Southeast Asian Bull. Math. 19 (1995), 99-106.
[143] J.-E. Martmez-Legaz, Generalized convex duality and its economic applications. In: Handbook of Generalized Convexity, Generalized Monotonicity. (N. Hadjisavvas, S. Komlosi, and S. Schaible, eds.). Springer-Verlag, Berlin, Heidelberg (2004), 237-292.
[144] J.-E. Martinez-Legaz, A.M. Rubinov, and I. Singer, Downward sets and their separation and approximation properties. J. Global Optimization 23 (2002), 113-137.
[145] J.-E. Martinez-Legaz and A. Seeger, A formula on the approximate subdif-ferential of the difference of convex functions. Bull. Austral. Math. Soc. 45 (1992), 37-41.
[146] J.-E. Martinez-Legaz and I. Singer, A characterization of Lagrangian dual problems. Note Mat. (G. Kothe Festschrift; VB. MoscatelH, ed.), 10 (1990), Suppl. 2, 389-394.
References 337
[147] J.-E. Martinez-Legaz and I. Singer, v-dualities and ±-dualities. Optimization 22 (1991), 483-511.
[148] J.-E. Martinez-Legaz and I. Singer, Some characterizations of (/?-Lagrangian dual problems. Optimization 22 (1991), 835-843.
[149] J.-E. Martinez-Legaz and I. Singer, Some characterizations of surrogate dual problems. Optimization 24 (1992), 1-11.
[150] J.-E. Martinez-Legaz and I. Singer, Some further characterizations of unper-turbational dual problems. In: Parametric Optimization and Related Topics. Ill (J. Guddat, H.Th. Jongen, B. Kummer, and F. Nozicka, eds.), Peter Lang Publ. House, Frankfurt/Main, 1993, 407-436.
[151] J.-E. Martinez-Legaz and I. Singer, Some characterizations of perturbational dual problems. Optimization 29 (1994), 97-130.
[152] J.-E. Martinez-Legaz and I. Singer, *-dualities. Optimization 30 (1994), 295-315.
[153] J.-E. Martinez-Legaz and I. Singer, Subdifferentials with respect to dualities. Zeitschr Oper Res. ZOR - Math. Methods Open Res. 42 (1995), 109-125.
[154] J.-E. Martinez-Legaz and I. Singer, Dualities associated to binary operations on^ . J. Convex Anal. 2 (1995), 185-209.
[155] J.-E. Martinez-Legaz and I. Singer, On conjugations for functions with values in extensions of ordered groups. Positivity 1 (1997), 193-218.
[156] J.-E. Martinez-Legaz and I. Singer, An extension of d. c. duality theory, with an Appendix on *-subdifferentials. Optimization 42 (1997), 9-37.
[157] J.-E. Martinez-Legaz and M. Voile, Duality in d. c. programming: the case of several d. c. constraints. J. Math. Anal. Appl. 237 (1999), 657-671.
[158] J.-E. Martinez-Legaz and M. Voile, Duality in d. c. programming: the case of several d. c. constraints. New version of [157], presented at the International Conference on Mathematical Programming held in Matrahaza, March 1999 (unpublished).
[159] J.-E. Martinez-Legaz and M. Voile, Duality for d. c. optimization over compact sets. In: Optimization Theory; Recent Developments from Matrahaza (F. Giannessi, P. Pardalos, and T. Rapcsak, eds.). Kluwer Acad. Publ., Dordrecht, 2000, 139-146.
[160] B. Martos, Nonlinear Programming. North-Holland Publ. Co., Amsterdam, 1975.
[161] C. Michelot, Caracterisation des minima locaux des fonctions de la classe d. c. Technical note, Univ. of Dijon, 1987.
[162] J.-J. Moreau, Theoremes "inf-sup." Comptes Rendus Acad. Sci. Paris 258 (1964), 2720-2722.
[163] J.-J. Moreau, Fonctionnelles Convexes. Seminaire sur les equations aux derivees partielles. College de France, 1966.
[164] J.-J. Moreau, Inf-convolution, sous-additivite, convexite des fonctions num-eriques. J. Math. Pares. Appl. 49 (1970), 109-154.
[165] T.S. Motzkin, E.G. Straus, and F.A. Valentine, The number of farthest points. Pacific J. Math. 3 (1953), 221-232.
338 References
166] M. Moussaoui and M. Voile, Sur la quasicontinuite et les fonctions unies en dualite convexe. Comptes Rendus Acad. Sci. Paris Sen I Math. 322 (1996), 839-844.
167] M. Nicolescu, Sur la meilleure approximation d'une fonction donnee par les fonctions d'une famille donnee. Bui Fac. §tL Cemdufi 12 (1938), 120-128.
168] L. Nirenberg, Functional Analysis. Dittoed notes. Courant Inst. Math. Sci. New York Univ., New York, 1961.
169] D. Pallaschke and S. Rolewicz, Foundations of Mathematical Optimization. Convex Analysis Without Linearity. Kluwer Acad. Publ., Dordrecht, Boston, London, 1997.
170] C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Prentice Hall, Englewood Cliffs, New Jersey, 1982.
171] U. Passy and E.Z. Prisman, Conjugacy in quasi-convex programming. Math. ProgK 30 (19S4\ 121-146.
172] U. Passy and E.Z. Prisman, A convex-like duality scheme for quasi-convex programs. Math. Progr 32 (1985), 278-300.
173] T. Pennanen, Graph convex mappings and AT-convex functions. J. Conv. Anal. 6 (1999), 235-266.
174] J.-P. Penot, Duality for radiant and shady programs. Acta Math. Vietnamica 22 (1997), 541-566.
175] J.-P. Penot, Duality for anticonvex programs. J. Global Optim. 19 (2001), 163-182.
176] J.-P. Penot, What is quasiconvex analysis? Optimization 47 (2000), 35-100. 177] J.-P. Penot and M. Voile, On quasi-convex duality. Math. Open Res. 15
(1990), 597-625. 178] J.-P. Penot and C. Zalinescu, Harmonic sum and duality, J. Convex Anal. 1
(2000), 95-113. 179] G. Pickert, Bemerkungen iiber Galois-Verbindungen. Arch. Math. 3 (1952),
285-289. 180] J. Ponstein, Approaches to the Theory of Optimization. Cambridge Univ.
Press, Cambridge, London, 1980. 181] B.N. Pshenichnyi, Lemons sur les jeux differentiels. In: Controle Optimal et
Jeux Differentiels. Cahier de I'lRIA, no. 4 (1971). 182] B.N. Pshenichnyi, Convex Analysis and Extremal Problems. Nauka, Moscow,
1980 [Russian]. 183] R.T. Rockafellar, Duality and stability in extremum problems involving con
vex functions. Pacific J. Math. 21 (1967), 167-187. 184] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton,
1970. 185] R.T. Rockafellar, Conjugate Duality and Optimization. CBMS Reg. Confer.
Series in Applied Math. 16, SIAM, Philadelphia, 1974. 186] R.T. Rockafellar, Augmented Lagrange multiplier functions and duality in
nonconvex programming. SIAM J. Control 12 (1974), 268-283. 187] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag,
Grundlehren Math. Wiss. 317, BerUn, Heidelberg, New York, 1998.
References 339
[188] S. Rolewicz, On general theory of linear systems. Beitrage zur Analysis 8 (1976), 119-127.
[189] S. Rolewicz, Linear systems in Banach spaces. In: Calculus of Variations and Control Theory (D.L. Russell, ed.). Academic Press, New York, San Francisco, London, 1976, 245-256.
[190] S. Rolewicz, On Pontryagin maximum principle for systems with non one-point target set and systems with additional constraints. Math. Operations-forsch. Stat., Ser Optimization 10 (1979), 97-100.
[191] S. Rolewicz, On maximum principle in Banach spaces. In: Methods of Mathematical Programming, Ossolineum, Wroclaw, 1982, 271-276.
[192] S. Rolewicz, Functional Analysis and Control Theory. PWN, Warszawa and Reidel, Dordrecht, 1987.
[193] A.M. Rubinov, Abstract Convexity and Global Optimization. Kluwer Acad. Publ., Boston, Dordrecht, London, 2000.
[194] A.M. Rubinov and B.M. Glover, On generalized quasiconvex conjugation, Contemp. Math. 204 (1997), 199-216.
[195] A.M. Rubinov and B.M. Glover, Duality for increasing positively homogeneous functions and normal sets. Rech. Oper./Oper Res. 32 (1998), 105-123.
[196] A.M. Rubinov and B.M. Glover, Toland-Singer formula cannot distinguish a global minimizer from a choice of stationary points. Numen Funct. Anal. Optim. 20 {\999\ 99-119.
[197] A.M. Rubinov, B.M. Glover, and V. Jeyakumar, A general approach to dual characterizations of solvability of inequality systems with applications. /. Convex Anal. 2 (1995), 309-344.
[198] A.M. Rubinov and B. §im§ek: Dual problems of quasi-convex maximization. Bull. Austral. Math. Soc. 51 (1995), 139-144.
[199] A.M. Rubinov and I. Singer, Best approximation by normal and conormal sets. J. Approx. Theory 107 (2000), 212-243.
[200] A.M. Rubinov and A. Uderzo, On separation functions. J. Optim. Theory Appl. 109 (2001), 345-370.
[201] A. Rubinov and X. Yang, Lagrange-type Functions in Constrained Non-convex Optimization. Kluwer Acad. Publ., Boston, Dordrecht, 2003.
[202] G. Sh. Rubinshtein, Dual extremal problems. Doklady Akad. Nauk SSSR 152 (1963), 288-291 [Russian].
[203] G. Sh. Rubinshtein, Duality in mathematical programming and some problems of convex analysis. Uspekhi Mat. Nauk 25, 5(155), 171-201 [Russian].
[204] H.H. Schaefer, Topological Vector Spaces. Macmillan, New York, 1966. [205] S. Schaible and W.T. Ziemba, eds.. Generalized Concavity in Optimization
and Economics. Acad. Press, New York, 1981. [206] S. Simons, Minimax theorems and their proofs. In: Minimax and Applications
(Ding-Zhu Du and Panos M. Pardalos, eds.), Kluwer Acad. Publ., Dordrecht, Boston, 1995, 1-23.
[207] I. Singer, Properties of the surface of the unit ball and applications to the solution of the problem of uniqueness of the polynomial of best approximation in arbitrary Banach spaces. Studii Cercet. Mat. 1 (1956), 95-145 [Romanian].
340 References
[208] I. Singer, Caracterisation des elements de meilleure approximation dans un espace de Banach quelconque. Acta Sci. Math. 17 (1956), 181-189.
[209] I. Singer, On the uniqueness of the element of best approximation in arbitrary Banach spaces. Studii Cercet. Mat. 8 (1957), 234-244 [Romanian].
[210] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer-Verlag, Grundlehren Math. Wiss. 171, Berlin, Heidelberg, New York, 1970.
[211] I. Singer, The Theory of Best Approximation and Functional Analysis. CBMS Reg. Confer. Series in Applied Math. 13, SIAM, Philadelphia, 1974.
[212] I. Singer, Generalizations of methods of best approximation to convex optimization in locally convex spaces. I: Extension of continuous linear function-als and characterizations of solutions of continuous convex programs. Rev. Roum. Math. Pures Appl. 19 (1974), 65-77.
[213] I. Singer, Generalizations of methods of best approximation to convex optimization in locally convex spaces. II: Hyperplane theorems. /. Math. Anal. A/7/7/. 69 (1979), 571-584.
[214] I. Singer, Some new applications of the Fenchel-Rockafellar duality theorem: Lagrange multiplier theorems and hyperplane theorems for convex optimization and best approximation. Nonlinear Anal. Theory, Meth. Appl. 3 (1979), 239-248.
[215] I. Singer, Maximization of lower semi-continuous convex functional on bounded subsets of locally convex spaces. I: Hyperplane theorems. Appl. Math. Optim. 5 (1979), 349-362.
[216] I. Singer, On the Pontryagin maximum principle for constant-time linear control systems in Banach spaces. J. Optim. Theory Appl. 27 (1979), 315-321.
[217] I. Singer, A Fenchel-Rockafellar type duality theorem for maximization. Bull. Austral. Math. Soc. 20 (1979), 81-89.
[218] I. Singer, Maximization of lower semi-continuous convex functionals on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems. Result. Math. 3 (1980), 235-248.
[219] I. Singer, Extension with larger norm and separation with double support in normed Hnear spaces. Bull. Austral. Math. Soc. 21 (1980), 93-105.
[220] I. Singer, Minimization of continuous convex functionals on complements of convex subsets of locally convex spaces. Math. Open Stat. Ser. Optim. 11 (1980), 235-248.
[221] I. Singer, Duality theorems for linear systems and convex systems. J. Math. Anal. Appl. 76 (1980), 339-368.
[222] I. Singer, Duality theorems for constrained convex optimization. Control and Cybernetics 9 (1980), 37-45.
[223] I. Singer, The norm of a linear functional with respect to a non-negative convex functional vanishing at the origin. J. Math. Anal. Appl. 78 (1980), 367-377.
[224] I. Singer, Pseudo-conjugate functionals and pseudo-duality. In: Mathematical Methods in Operations Research (Invited lectures presented at the Internal.
References 341
Confer, held in Sofia, November 1980), Publ. House Bulg. Acad. Sci., Sofia (1981), 115-134.
[225] I. Singer, A characterization of constant-time linear control systems satisfying the Pontryagin maximum principle. J. Optim. Theory AppL 32 (1980), 379-384.
[226] I. Singer, Optimization and best approximation. In: Nonlinear Analysis, Theory and Applications (R. Kluge, ed.), Abhandl. Akad. Wiss. DDR, Abt. Math. Naturwiss.-Technik, 2N, Akademie-Verlag, Beriin (1981), 273-285.
[227] I. Singer, Duality theorems for perturbed convex optimization. J. Math. Anal. AppL 81 (1981), 437-452.
[228] I. Singer, On the perturbation and Lagrangian duality theories of Rockafellar and Kurcyusz. In: Vth Sympos. on Oper Res. Koln, 1980 (R. E. Burkard and T.H. Ellinger, eds.). Methods of Oper. Research 40, A. Hain Meisenheim GmbH, Konigstein/Ts. (1981), 153-156.
[229] I. Singer, Optimization by level set methods. Ill: Characterizations of solutions in the presence of duality. Numer Funct. Anal. Optim. 4(2) (1981-1982), 151-170.
[230] I. Singer, Optimization by level set methods. IV: Generalizations and complements. Numer Funct. Anal. Optim. 4(3) (1981-1982), 279-310.
[231] I. Singer, Optimization by level set methods. I. Duality formulae. In: Optimization: Theory and Algorithms (J.-B. Hiriart-Urruty, W. Oettli, and J. Stoer, eds.). Lecture Notes Pure Appl. Math. 86, Marcel Dekker, New York, 1983, 13-43.
[232] I. Singer, Abstract Pontryagin maximum principles for linear systems. Lin. Multilin. Alg. 13 (1983), 203-219.
[233] I. Singer, The lower semi-continuous quasi-convex hull as a normalized second conjugate. Nonlinear Anal. Theory, Methods, Appl. 1 (1983), 115-1121.
[234] I. Singer, Surrogate conjugate functionals and surrogate convexity. Applicable Anal. 16 (1983), 291-327.
[235] I. Singer, Optimization by level set methods. II: Further duality formulae in the case of essential constraints. In: Functional Analysis, Holomorphy and Approximation Theory. II (G.I. Zapata, ed.), Elsevier (North-Holland), Amsterdam-New York-Oxford, 1984, 383-411.
[236] I. Singer, Generalized convexity, functional hulls and applications to conjugate duality in optimization. In: Selected Topics in Operations Research and Mathematical Economics (G. Hammer and D. Pallaschke, eds.). Lecture Notes Econ. Math. Systems 226, Springer-Verlag, Berlin, Heidelberg, 1984, 49-79.
[237] I. Singer, Conjugation operators. In: Selected Topics in Operations Research and Mathematical Economics (G. Hammer and D. Pallaschke, eds.). Lecture Notes Econ. Math. Systems 226, Springer-Verlag, Beriin, Heidelberg, 1984, 80-97.
[238] I. Singer, Best approximation and optimization. J. Approx. Theory 40 (1984), 274-284.
342 References
[239] I. Singer, Optimization by level set methods. V: Duality theorems for perturbed optimization problems. Math. Operationsforsch. Stat. Ser. Optim. 15 (1984), 3-36.
[240] I. Singer, Surrogate dual problems and surrogate Lagrangians. J. Math. Anal. App/. 98 (1984), 31-71.
[241] I. Singer, A general theory of surrogate dual and perturbational extended surrogate dual optimization problems. J. Math. Anal. Appl. 104 (1984), 351-389.
[242] I. Singer, A general theory of dual optimization problems. / Math. Anal. Appl. 116 (1986), 75-130.
[243] I. Singer, Some relations between dualities, polarities, coupling functionals and conjugations. J. Math. Anal. Appl. 115 (1986), 1-22.
[244] I. Singer, Generalizations of convex supremization duality. In: Nonlinear and Convex Analysis (B.-L. Lin and S. Simons, eds.). Lecture Notes in Pure Appl. Math. 107, Marcel Dekker, New York, 1987, 253-270.
[245] L Singer, Optimization by level set methods. VL Generalizations of surrogate type reverse convex duality. Optimization 18 (1987), 485-499.
[246] L Singer, On duality and stability of parametrized optimization problems and related topics. In: Parametric Optimization and Related Topics (J. Guddat, H.Th. Jongen, B. Kummer, and F. Nozicka, eds.), Akademie-Verlag, Berlin (1987), 355-375.
[247] I. Singer, Abstract subdifferentials and some characterizations of optimal solutions. J. Optim. Theory Appl. 57 (1988), 361-368.
[248] I. Singer, A general theory of dual optimization problems. II: On the perturbational dual problem corresponding to an unperturbational dual problem. ZOR-Methods and Models of Open Res. 33 (1989), 241-258.
[249] I. Singer, Some relations between combinatorial min-max equalities and La-grangian duality equalities, via coupling functions. I: Cardinality results. Rev. Roum. Math. Pures Appl. 34 (1989), 455^91.
[250] I. Singer, Some relations between combinatorial min-max equalities and La-grangian duality equalities, via coupling functions. II: Further results. Rev. Roum. Math. Pures Appl 34 (1989), 661-692.
[251] I. Singer, Some general Lagrangian duality theorems. J. Math. Anal. Appl. 144 (1989), 26-51.
[252] I. Singer, Some further relations between unperturbational and perturbational dual optimization problems. Optimization 22 (1991), 317-339.
[253] I. Singer, Some further duality theorems for optimization problems with reverse convex constraint sets. J. Math. Anal. Appl. 171 (1992), 205-219.
[254] I. Singer, Abstract Convex Analysis. Wiley-Interscience, New York, 1997. [255] I. Singer, Duality for optimization and best approximation over finite inter
sections. Numer Funct. Anal. Optim. 19 (1998), 903-915. [256] I. Singer, Duality in quasi-convex supremization and reverse convex infimiza-
tion via abstract convex analysis, and applications to approximation. Optimization 45 (1999), 255-308.
[257] I. Singer, Dual representations of hulls for functions satisfying /(O) = inf /(Z\{0}). Optimization 45 (1999), 309-342.
References 343
[258] I. Singer, Lagrangian duality theorems for reverse convex infimization. Nu-mer. Funct. Anal. Optim. 21 (2000), 933-944.
[259] I. Singer, On duality for quasi-convex supremization and reverse convex infimization. In: Optimization Theory. Recent Developments from Mdtrahdza (F. Giannessi, P. Pardalos, and T. Rapcsak, eds.), Kluwer Acad. Publ., Dordrecht, 2001, 225-254.
[260] I. Singer, On suprema of abstract convex and quasi-convex hulls. In: Generalized Convexity and Generalized Monotonicity (N. Hadjisavvas, J.-E. Martinez-Legaz, and J.-P. Penot, eds.). Lecture Notes Econ. Math. Systems 502, Springer-Verlag, Berhn, Heidelberg, New York, Tokyo, 2001, 381-394.
[261] M. Sion, On general minimax theorems. Pacific J. Math. 8 (1958), 171-176. [262] J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimensions.
I. Springer-Verlag, Grundlehren Math. Wiss. 163, Berlin, Heidelberg, New York, 1970.
[263] A.S. Strekalovsky, On the global extremum problem. Soviet Math. Doklady 35 (1987), 194-198.
[264] A.S. Strekalovsky, On questions of global extremum in nonconvex extremal problems. Izv. Vysshih Uchebn. Zaved. 8 (1990), 74-80 [Russian].
[265] A.S. Strekalovsky, On search of global maximum of convex functions on a constraint set. Comput. Maths. Math. Phys. 33 (1993), 315-328.
[266] A.S. Strekalovsky, Extremal problems on complements of convex sets. Translated from Kibemetika i Sistemnyi Analiz, Nl, Plenum Publ. Corp., 1993, 88-100.
[267] A.S. Strekalovsky, On Global Optimality Conditions for D. C Programming Problems. Irkutsk State University, Irkutsk, 1997.
[268] A.S. Strekalovsky, Global optimality conditions for nonconvex optimization. J. Global Optim. 12 (1998), 415-434.
[269] J.-J. Strodiot, V.H. Nguyen, and N. Heukemes, e-optimal solutions in non-differentiable convex programming and some related questions. Math. Progn 25 (1983), 307-328.
[270] 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.
[271] P.T. Thach, Quasi-conjugates of functions, duality relationship between qua-siconvex minimization under a reverse convex constraint, and quasi-convex maximization under a convex constraint, and applications. J. Math. Anal. Appl. 159 (1991), 299-302.
[272] P.T. Thach, A nonconvex duality with zero gap and applications. SIAM J. Optim. 4 (1994), 44-64.
[273] P.T. Thach, A generalized duality and applications. J. Global Optim. 3 (1993), 311-324.
[274] P.T. Thach, Global optimality criterion and a duality with zero gap in non-convex optimization. SIAM J. Math. Anal. 24 (1993), 1537-1556.
[275] P.T. Thach, Diewert-Crouzeix conjugation for general quasiconvex duality and applications. J. Optim. Theory Appl. 86 (1995), 719-743.
344 References
[276] RT. Thach, R.E. Burkard, and W. Oettli, Mathematical programs with a two-dimensional reverse convex constraint. / Global Optim. 1 (1991), 145-154.
[277] V.M. Tikhomirov, Some Problems of Approximation Theory. Izdatelstvo Mosk. Gos. Univ., Moscow, 1976 [Russian].
[278] J. Tind, On duality in nonconvex and integer programming. Oper Res. Ver-fahren 32 (1979), 193-201.
[279] J. Tind and L.A. Wolsey, An elementary survey of general duality theory in mathematical programming. Math. Progr 21 (1981), 241-261.
[280] J.F. Toland, Duality in nonconvex optimization. J. Math. Anal. Appl 66 (1978), 399-415.
[281] J.F. Toland, A duality principle for nonconvex optimisation and the calculus of variations. Arch. Rat. Mech. Anal. 71 (1979), 41-61.
[282] J.F. Toland, On subdifferential calculus and duality in non-convex optimization. In: Analyse Non-convexe [1977, Pau], Bull. Soc. Math. France, Memoire 60(1979), 177-183.
[283] H. Tuy, D. c. optimization: theory, methods and algorithms. In [109], 149-216.
[284] H. Tuy, Convex Analysis and Global Optimization. Kluwer Acad. Publ., Dordrecht, 1998.
[285] H. Tuy and W. Oettli, On necessary and sufficient conditions for global opti-mality. Rev. Mat. Apl. 15 (1994), 39-41.
[286] 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.
[287] M. Voile, Conjugaison par tranches. Ann. Mat. Pura Appl. (4) 139 (1985), 279-311.
[288] M. Voile, Contributions a la dualite en optimisation et a I'epi-convergence. These. Univ. de Pau, 1986.
[289] M. Voile, Conjugaison par tranches et dualite de Toland. Optimization 18 (1987), 633-642.
[290] M. Voile, Quasiconvex duality for the max of two functions. In: Recent Advances in Optimization (P. Gritzmann, R. Horst, E. Sachs, and R. Tichatschke, eds.). Lecture Notes Econ. Math. Systems 452, Springer-Verlag, BerHn, Heidelberg, 1997, 365-379.
[291] M. Voile, A formula on the conjugate of the max of a convex function and a concave function. J. Math. Anal. Appl. 220 (1998), 313-321.
[292] M. Voile, Duality principles for optimization problems dealing with the difference of vector-valued convex mappings. J. Optim. Theory Appl. 114 (2002), 223-241.
[293] M. Walk, Theory of Duality in Mathematical Programming. Akademie-Verlag, Berlin, 1989.
[294] A. Wieczorek, An abstract symmetric framework for duality in mathematical programming. Optimization 28 (1994), 249-266.
[295] M. Wriedt, Konvexe Optimierungsoperatoren. Habilitationsschrift, Univ. of Kiel, 1976.
References 345
[296] Ya.I. Zabotin, A.I. Korablev, and R.F. Khabibullin, Conditions for an ex-tremum of a functional in the presence of constraints. Kibernetika 6 (1973), 65-79 [Russian].
[297] I. Zang and M. Avriel, On functions whose local minima are global. J. Optim. Theory AppL 16(1975), 183-190.
[298] I. Zang, E.U. Choo, and M. Avriel, A note on functions whose local minima are global. 7. Optim. Theory AppL 18 (1976), 555-559.
[299] K. Zimmermann, Conjugate optimization problems and algorithms in the extremal vector space. Ekon.-mat. obzor 10 (1974), 428^39.
[300] U. Zimmermann, On some extremal optimization problems. Ekon.-mat. oZ?-zor 15 (1979), 438-442.
[301] U. Zimmermann, Duality for algebraic linear programming. Linear Alg. AppL 32 (1980), 9-31.
[302] U. Zimmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures. Annals of Discrete Mathematics 10, North-Holland, Amsterdam, 1981.
Index
a^ 169 xc, 25 a M O l CC,6 (A, <, (8»), 280 CO C, 5 (A, <, (8), (g)), 281 coC, 10 Adif),6 co^iC,263 ^Cc(^o), 166 cowC, 260
core C, 18 ^A, 122 Conv(X), 79 y A, 122 ( o j ^
^A' 175 C°°, 20 -6;. 185 C,6 ^1'109 C(X),263
fie, 83 A, 27 )3„,288 A', 28 fi\nO A'A,28 y6M02,103 A^, 29, 265 /Stagr, 69 A^, 30, 265 /6Lgr'83 A3G,31 ,265
Aurr, 81 A4 , 33, 265 )S:u„, 83 A^;, 66 bd C, 6 A ^ , 69 Bx,2 A<",34,266
348 Index
A02 , 35, 266
A " , 34, 265
A'2, 34, 265
A '3 ,34 ,265
A'^, 127
^Ury 133 A^V, 133
Kj^ 135 A„'^, 135
K'T^ 135 9. /(2o) , 26
8(0, xo), 85
9/(zo), 23
9^<^)/(zo), 39
9 f <»'>/(xo), 321
d is t (C, ,C2) , 13
dist (xo, C), 13
dist (x,y),4
dom / , 22
DC(X) ,216
(£>.), 84
(£>,?), 83
(£»A), 122
( S A ) , 122
(/^k), 185
( 5 ^ ) , 185
(Z)J , 323
(£>2), 323
(Z)f), 323
(£>™), 288
( A U T ) , 80
eca C ,10
eco C, 10
ecowC, 261
epi / , 6
£ ' e A ( W ) , 2 6 2
£ 2 ( 1 ^ ) , 262
£CA(W), 260
£:/C(W), 260
Jco? ^ ^
/co(H^), 261
/^^<^\ 35
/ J '37 />^^37
/ ^ ^ 38
/ ; , 3 8
/ ^ ^ , 3 8
/eq, 26
fcqiW). 262
/eqa(Vy), 262 / ^ ^ ^ \ 38 jrMicp)^ 282, 321
/^(xo; X), 24
/ q ,26
/q ,26
/ qca 9 -^ '
/q(A'A),29,266
/q(A^)^ 263
fq(W), 262
/ ^ s ^ 282
/ ^ , 141
/ ^ « , 3 1 1
/ * , 2 1 , 3 6
/ * * , 2 1 , 3 6
^G(-^O) . 85
^ ( / ' < ) , 239
J^(P<), 232
^ ( P ^ , < ) , 241
G ^ 6 3
^o.i/» 7
'WG,.VO. 63
int C, 6
Index 349
Jix,u(x)),296 {Pe),2U
(PI, 169 (^i), 266 (P^), 101 iK2),266 (P,), 80, 130, 244
(^3)'2^6 (/><), 56, 232 '^(^)'260 (F^), 56, 232
iPK.<),24l Pc(xo),40
r, 170, 171
A^ 102, 103
'= ' 22 e(W),262 y^, 175 ^ i ,109 PC, 38 A' , 84 ^, 2 A", 83 p+, 56 surr, 80 /;++, 282
i (A) ,38 R\2 L(A)',38 L, ,84 a(X*,X),7 L",294 cf(X,X*),7 L^,294 SAf),6 L%„, 294 Supp / , 22 Lsurr, 81 supp / , 2
5 G ( / ) , 4 7
M A , 264 M G ( / ) , 101 0 '239
N{C\ Co), 25 u*, 76, 132 Ar,(C; Co), 25 U^j, 17 iv,(C;xo),25 f/|.d,8 N{C; xo), 25 f/|,,, 8
W,90 ^ ' 233 i ^^ 92 S2G,$, 102, 170 ^^(;,Q), 6
^G,w, 103, 171 fi„„109 i;(2), 80, 130,244 OG{XO), 100 V*, , 17
OG(/), 148 Vl„ 8
^Id. 8 (P),47,55 V,67,90 (/>.), 214 ^ ^ 9 2 {(P), (£>)}, 52 {(P), (£>„)}, 288 X*{f,k),\99
350 Index
+,20 +,20 X, 56 x,282 0 ^ 326 0,324 0,281 0,281
abstract convex analysis, 27, 259 addition, 1
lower, 20 upper, 20
adjoint of a mapping, 76, 132 associated pair, 301 axiomatic characterization of:
Fenchel-Moreau conjugations, 280
dual objective functions, 285, 291, 299
the Lagrangian functions associated with -, 285, 292, 299
barrier cone, 63 biconjugate, biconjugate function, 21
Fenchel-Moreau, 36 bipolar theorem, 19
canonical enlargement, 281 cavern, 6 conjugate, conjugate function:
concave, 21 convex, 21 Fenchel, 21 Fenchel-Moreau, 35 ofElQortobi, 282 of Rubinov and §im§ek, 282 of typeLau, 38
conjugation, 35, 281 surrogate, 298
constraint, constraint set, 47 abstract, 55 d.c, 214 essential, 61
general, 71 inessential, 64 primal, 47 structured, 54, 131, 190 surrogate,48, 60, 81, 102, 176
constraint qualification, 50 of Attouch-Brezis, 50 Slater, 57, 78, 236 (/,/)-, 227
constraints multifunction, see perturbation multifunction
coupling function, 35 additive, 307 multiplicative, 304 natural, 35
critical point, 247
d.c. optimization, 213 abstract, 275
deviation, 85 directional derivative, 24 distance
between two subsets, 13 between x and y, 4 from the empty set, 13 fromjco to C, 13
domain, effective domain, 22 dual constraint set, 48, 122, 184 dual objective function, 48, 72, 80,
103, 122, 130, 171, 175, 184, 192, 244
A-, 122, 185 dual of a polarity, 28 dual problem, duality, 46
unperturbational Lagrangian, 48, 50,56, 127, 135, 189, 198
perturbational Lagrangian, 72, 130 unperturbational surrogate, 60, 83,
102, 122, 170, 175, 185 perturbational surrogate, 80, 132 for d.c. infimization, 218, 226,
236, 244 for abstract quasi-convex
supremization, 267, 270
Index 351
for abstract reverse convex infimization 271, 273
for abstract d.c. infimization, 275 associated with a Lagrangian
function, 52 for infimization problems
involving maximum operators, 248, 252
TT-, 84
6>-, 83 A-, 122 PES-, perturbational extended
surrogate dual, 297 decomposed, 297
(Wz)-, 298 anomalous, 323 m-Lagrangian, 288 perturbational conjugate, 297 quasi-convex, 80
duality: for best approximation, 40 for worst approximation, 87 strong, 51, 69, 73, 83, 180,288 weak, 51, 69, 73, 114,178,181 V-, 280, 325 *-, associated with a binary
operation, 258, 296 (*, 5)-, 282, 326
duality equality: Lagrangian, 128 strong, 68 surrogate, 128 weak, 69, 178
duality gap, 51 duality inequality, 49
element: of best approximation, 40 of ^-approximation, 6:-best
approximation, 283 of worst approximation, 85
environment, 301 epigraph, 6 epigraphic methods, 48, 50 8-solution, £-optimal solution, 284
£-subdifferential, 25 excess, see deviation extremality relations, 247
farthest point, see element of worst approximation
Fenchel equality, 23 Fenchel inequality, 23 function:
Of-Holder continuous, 262 with constant N, 262
A'A-quasi-convex, 29 abstract convex, 260 abstract quasi-convex, 260 abstract d.c, 275 additive, 7 affine, 7 best, 207, 313 concave, 21 continuous, 6
at a point, 6 convex, 21 d.c, diff-convex, 213 elementary, 260 evenly quasi-coaffine, 27 evenly quasi-convex, 26
strongly, 139 homogeneous, 6 indicator, 24 Lagrange-type, 309 linear, 6 lower semicontinuous, 6
at a point, 5 A1-quasi-convex, 263 maximal, 45 min-type, 305
additive, 307 optimal, optimal dual solution, 99,
148, 165 optimal value, marginal, 74, 295 positively homogeneous, 21 pseudoconjugate, 38 proper, 21 quasi-concave, 26 quasi-convex, 26
352 Index
evenly, 26 strongly evenly, 139 strongly, 142
/?-evenly convex, 127 regular weak separation, RWS,
303 regular with respcect io(p, 116 regular with respect to if, 268 strictly increasing along segments
from 0, 142 subdifferentiable, 24 sublinear, 21 tangentially convex, 312 upper semicontinuous, 6
at a point, 6 W-convex, 261 VK-evenly quasi-coaffine, 262 V^-evenly quasi-convex, 262 W-quasi-convex, 262
group: lattice ordered, 281
conditionally complete, 281
half-space: closed, 8 open, 8 quasi-support, 16
half-space theorem: for infimization, 65 for quasi-convex supremization,
116, 119 for reverse convex infimization,
183 hull of a function:
A'A-quasi-convex, 29 evenly quasi-coaffine, 27 evenly quasi-convex, 26 lower semicontinuous convex, 22 A^-quasi-convex, 242 quasi-convex, 26
lower semi-continuous quasi-convex, 33
W-convex, 241 W-evenly quasi-coaffine, 242
W-evenly quasi-convex, 241 W-quasi-convex, 241
hull of a set: A A-convex, 28 closed convex, 10 convex, 5 evenly coaffine, 10 evenly convex, 10 A^-convex, 263 V^-convex, 260 W-evenly coaffine, 260 ly-evenly convex, 260
hull operator, 28 hyperplane, 8
best, 207 optimal, 99, 166 quasi-support, 11 support, 11
hyperplane theorem (of surrogate duahty):
for convex infimization, 63 for quasi-convex supremization,
163 for reverse convex infimization,
172
image set, 287, 303 incidence triple, 285 inequality constraint, 56, 198, 225,
232 inf-sup theorem:
of Moreau, 27 of Sion-Kneser-Fan, 27
infimal convolution, 236 inner product, 4 instance of a problem, 300 inversion, 314
Lagrangian, Lagrangian function: associated with a perturbation, 244 augmented, 51 for infimization, 52 for perturbed infimization, 73 for perturbed supremization, 130 for d.c. infimization, 244
Index 353
of Kurcyusz, 294 of type I, 322 of type II, 322 quasi-convex, 81 surrogate, 269
Lagrangian duality theorem, 48, 81, 127,135,189,228,244
general, 287 lattice:
conditionally complete, 281 level set, 6 linearization, 222
mapping: antitone, 28 convex, 54
vector-valued, 320 min-max equality:
combinatorial, 285 of all-cardinality
covering-packing type, 286 for A-cover packings, 286 for ^-colorings, 286 for weighted A-covers, 286 for weighted 5-packings, 286
minimax theorem, 27 minimum principle of Bauer, 311 Moore-Smith closure operator, 29 multiplication:
lower, 282 upper, 282 with a scalar, 1
nearest point, see element of best approximation
norm, 2 normal cone, 24
extended, 25
objective function, 47 open ball, 5 opposite element, 1 optimization problem, xi
convex, xi anticonvex, xi
convex-anticonvex, xi abstract d.c, 259 d.c.,213 extremal, 301 involving maximum operators,
247
penalize, 50 penalty term, 50 perturbation:
horizontal, 80, 293 normal, 292 vertical, 80, 293
perturbation function, parameterization, 72
objective-function separated, 298 perturbation multifunction, 294 PMP, Pontryagin maximum
principle, 295 abstract, 295
PMP„,296 polar set, 19 polarity, 27 primal parameters, 300 primal problem:
of infimization, 47 perturbed, parameterized, 72, 80,
129, 244, 293 structured, 54
constrained, 47 unconstrained, 47 of convex infimization, 47
structured, 55 of quasi-convex infimization, 47 of quasi-convex supremization,
101 of reverse convex infimization,
169 of d.c. infimization, diff-convex,
213 unconstrained, 213 constrained, d.c. constrained,
214 with a d.c. inequality constraint,
225
354 Index
with finitely many d.c. inequality constraints, 232
primal-dual pair, 51, 288 program, programming problem, 55
convex, 55 linear, 55
abstract, 301 algebraic, 301
mathematical, 132 minimization, in an environment,
301 quasi-convex, 82
quasi-conjugate, quasi-conjugate function:
of Greenberg-Pierskalla, 37 second, 37
normalized, 37 ofThach, 141,310
quasi-subdifferential, 300 quasi-support half-space, 16 quasi-support hyperplane, 11
reduction principle, 44, 64, 88, 106, 123, 125, 155, 173, 188
reverse convex best approximation, 153
saddle-value, 293 scheme of formal replacements, 299 semiconjugate, 38 separation:
min-type, 305 nonlinear, 287 by a function, 8
strict, 8 strong, 279
by a hyperplane, 8 strict, 8
by a set, 263 set:
abstract convex, 260 A^A-convex, 29 Chebyshev, 153 comprehensive, 307
conical, 266 conormal, reverse normal, 304 convex, 5 evenly coaffine, 10 evenly convex, 10 linearly open, 18 7V(-convex, 263 normal, 304 of ^-normal directions, 25 of extended £-normal directions,
25 of perturbations, of parameters,
72, 129, 244 polar, 19 proximinal, 149 semi-Chebyshev, 313 support, 22, 280
(X*, /?)-, 22 /^-evenly convex, 126 M -convex, 260 \y-evenly coaffine, 260 ly-evenly convex, 260
Slater condition, see Slater constraint qualification
solution, optimal solution: for quasi-convex minimization, 47 for quasi-convex maximization,
101, 137 for reverse convex minimization,
169, 203 for d.c. minimization, 221 global, 284,318 local, 284, 318 primal, 54, 75 dual, optimal dual, 54, 75 of (D^), 288
space: Banach, 2 conjugate, 7 Euclidean, 2 Hilbert, 4 linear, 2 locally convex, 5 normed linear, 2
complete, 2
Index 355
topological linear, 5 strong CHIP, strong conical hull
intersection property, 28 subdifferential, 22
of Balder, 300 ofThach, 141,283 surrogate, 300 with respect to a conjugation, 39,
283 with respect to a conjugation of
type Lau, 283 with respect to a primal-dual pair
of optimization problems, 300 (*,^K326
substitution method, 64, 128, 189, 198
subtraction, 2 system, 54
canonical, 323 constant time linear control, see
linear system convex, 54 equlibrium, 323, linear, 54
constitutively, 323 fully, 295 geometrically, 295
polar, 295
potential, 323 strictly reflexive, 323
target set, 54, 131 non-one point, 295
theorem of: separation, 8 strict separation, 9 Fenchel-Moreau, 22 Moreau and Pshenichnyi, 24 Moreau-Rockafellar, 32 Fenchel-Rockafellar, 76 Pshenichnyi-Rockafellar, 52
topology: general, 5 Hausdorff, 5 norm, 5 weak, 7 weak*, 7
unit ball, 2
value, optimal value, 51 surrogate dual, 80
vector operations, 1, 7
weak alternative theorem, 304