Skip to main content

Part of the book series: Springer Monographs in Mathematics ((SMM))

  • 3163 Accesses

Abstract

This chapter is concerned with basic principles of convex programming in Banach spaces, that is, with the minimization of lower-semicontinuous convex functions on closed convex sets.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abadie M (1965) Problèmes d’Optimisation. Institut Blaise Pascal, Paris

    Google Scholar 

  2. Altman M (1970) A general separation theorem for mappings, saddle-points, duality and conjugate functions. Stud Math 36:131–167

    MathSciNet  MATH  Google Scholar 

  3. Arrow KJ, Hurvicz L, Uzawa H (1958) Studies in linear and non-linear programming. Stanford University Press, Stanford

    MATH  Google Scholar 

  4. Asimow L (1978) Best approximation by gauges on a Banach space. J Math Anal Appl 62:571–580

    Article  MathSciNet  Google Scholar 

  5. Asimow L, Simoson A (1979) Decomposability and dual optimization in Banach spaces. Preprint, Univ of Wyoming

    Google Scholar 

  6. Asplund E (1966) Farthest points in reflexive locally uniformly rotund Banach spaces. Isr J Math 4:213–216

    Article  MathSciNet  MATH  Google Scholar 

  7. Aubin JP, Clarke FH (1977) Multiplicateurs de Lagrange en optimisation non convexe et applications. C R Acad Sci Paris 285:451–453

    MathSciNet  MATH  Google Scholar 

  8. Balaganskii VS (1995) On approximation properties of sets with convex complements. Math Notes 57:26–29

    Article  MathSciNet  Google Scholar 

  9. Balaganskii VS (1998) On nearest and farthest points. Math Notes 63:250–252

    Article  MathSciNet  Google Scholar 

  10. Balakrishnan AV (1971) Introduction to optimization theory in a Hilbert space. Springer, Berlin

    Book  MATH  Google Scholar 

  11. Balder EJ (1977) An extension of duality–stability relations. SIAM J Control Optim 15:329–343

    Article  MathSciNet  MATH  Google Scholar 

  12. Baronti M, Papini PL (2001) Remotal sets revisited. Taiwan J Math 5:367–373

    MathSciNet  MATH  Google Scholar 

  13. Bazaraa MS, Goode J (1972) Necessary optimality criteria in mathematical programming in the presence of differentiability. J Math Anal Appl 40:609–621

    Article  MathSciNet  MATH  Google Scholar 

  14. Bazaraa MS, Shetty CM (1976) Foundations of optimization. Lecture notes in economics and mathematical systems, vol 122. Springer, Berlin

    Book  MATH  Google Scholar 

  15. Bazaraa MS, Goode J, Nashed MZ (1972) A nonlinear complementary problem in mathematical programming in Banach spaces. Proc Am Math Soc 35:165–170

    Article  MathSciNet  MATH  Google Scholar 

  16. Beattie R (1980) Continuous convergence and the closed–graph theorem. Math Nachr 99:87–94

    Article  MathSciNet  MATH  Google Scholar 

  17. Blater J (1969) Werteste punkte and Nachste punkte. Rev Roum Math Pures Appl 4:615–621

    Google Scholar 

  18. Borwein J (1977) Proper efficient points for maximizations with respect to cones. SIAM J Control Optim 15:57–63

    Article  MATH  Google Scholar 

  19. Borwein J (1978) Weak tangent cones and optimization in a Banach space. SIAM J Control Optim 16:512–522

    Article  MATH  Google Scholar 

  20. Boţ I, Wanka G (2006) A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal, Theory Methods Appl 64(12):2787–2804

    Article  MATH  Google Scholar 

  21. Boţ I, Grad S, Wanka G (2009) Generalized Moreau–Rockafellar results for composed convex functions. Optimization 58:917–933

    Article  MathSciNet  MATH  Google Scholar 

  22. Brans JP, Claesen G (1970) Minimax and duality for convex–concave functions. Cah Cent étud Rech Opér 12:149–163

    MathSciNet  MATH  Google Scholar 

  23. Burachik R, Jeyakumar V (2005) A dual condition for the convex subdifferential sum formula with applications. J Convex Anal 12:279–290

    MathSciNet  MATH  Google Scholar 

  24. Claesen G (1974) A characterization of the saddle points of convex–concave functions. Cah Cent étud Rech Opér 14:127–152

    MathSciNet  Google Scholar 

  25. Clarke FH (1973) Necessary conditions for nonsmooth problems in optimal control and the calculus of variations. Thesis, Univ Washington

    Google Scholar 

  26. Clarke FH (1975) Generalized gradients and applications. Trans Am Math Soc 205:247–262

    Article  MATH  Google Scholar 

  27. Cobzaş St (2005) Geometric properties of Banach spaces and the existence of nearest and farthest points. Abstr Appl Anal 3:259–285

    Article  Google Scholar 

  28. Dedieu J-P (1977) Cône asymptote d’un ensemble non convexe. Applications à l’optimisation. C R Acad Sci Paris 185:501–503

    MathSciNet  Google Scholar 

  29. Dedieu J-P (1978) Critères de femeture pour l’image d’un fermé non convexe par une multiplication. C R Acad Sci Paris 287:941–943

    MathSciNet  MATH  Google Scholar 

  30. Deumlich R, Elster KH (1980) Duality theorems and optimality conditions for nonconvex problems. Math Operforsch Stat, Ser Optim 11:181–219

    MathSciNet  MATH  Google Scholar 

  31. Dieter U (1966) Optimierungsaufgaben in topologische Vectorräumen. I. Dualitatstheorie. Z Wahrscheinlichkeitstheor Verw Geb 5:89–117

    Article  MathSciNet  MATH  Google Scholar 

  32. Dieudonné J (1966) Sur la séparation des ensembles convexes. Math Ann 163:1–3

    Article  MathSciNet  Google Scholar 

  33. Dolecki S, Kurcyusz S (1978) On Φ-convexity in extremal problems. SIAM J Control Optim 16:277–300

    Article  MathSciNet  MATH  Google Scholar 

  34. Duffin J (1973) Convex analysis treated by linear programming. Math Program 4:125–143

    Article  MathSciNet  MATH  Google Scholar 

  35. Duvaut G, Lions JL (1972) Sur les inéqualitions en mécanique et en physique. Dunod, Paris

    Google Scholar 

  36. Edelstein M (1966) Farthest points of sets in uniformly convex Banach space. Isr J Math 4:171–176

    Article  MathSciNet  MATH  Google Scholar 

  37. Ekeland I (1974) On the variational principle. J Math Anal Appl 47:324–353

    Article  MathSciNet  MATH  Google Scholar 

  38. Ekeland I (1979) Nonconvex minimization problems. Bull Am Math Soc 1:443–474

    Article  MathSciNet  MATH  Google Scholar 

  39. Ekeland I, Temam R (1974) Analyse convexe et problèmes variationnels. Dunod, Gauthier-Villars, Paris

    MATH  Google Scholar 

  40. El-Hodiri MA (1971) Constrained extrema. Introduction to the differentiable case with economic applications. Lecture notes in oper res and math systems. Springer, Berlin

    Book  MATH  Google Scholar 

  41. Floret K (1978) On the sum of two closed convex sets. Math Methods Oper Res 39:73–85

    Google Scholar 

  42. Fortmann TE, Athans M (1974) Filter design subject to output sidelobe constraints: theoretical considerations. J Optim Theory Appl 14:179–198

    Article  MathSciNet  MATH  Google Scholar 

  43. Franchetti C, Papini PL (1981) Approximation properties of sets with bounded complements. Proc R Soc Edinb A 89:75–86

    Article  MathSciNet  MATH  Google Scholar 

  44. Garkavi L (1961) Duality theorems for approximation by elements of convex sets. Usp Mat Nauk 16:141–145 (Russian)

    MathSciNet  MATH  Google Scholar 

  45. Godini G (1973) Characterizations of proximinal subspaces in normed linear spaces. Rev Roum Math Pures Appl 18:900–906

    Google Scholar 

  46. Guinard M (1969) Generalized Kuhn–Tucker conditions for mathematical programming problems in a Banach space. SIAM J Control 7:232–241

    Article  MathSciNet  Google Scholar 

  47. Gwinner J (1977) Closed images of convex multivalued mappings in linear topological spaces with applications. J Math Anal Appl 60:75–86

    Article  MathSciNet  MATH  Google Scholar 

  48. Halkin H, Neustadt LW (1966) General necessary conditions for optimizations problems. Proc Natl Acad Sci USA 56:1066–1071

    Article  MathSciNet  MATH  Google Scholar 

  49. Hestenes MR (1975) Optimization theory: the finite dimensional case. Wiley, New York

    MATH  Google Scholar 

  50. Hiriart-Urruty JB (1977) Contributions à la programmation mathématique. Thèse, Université de Clermont–Ferrand

    Google Scholar 

  51. Hiriart-Urruty JB (1989) From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality. In: Clarke FH, Demyanov VF, Giannesi F (eds) Nonsmooth optimization and related topics. Plenum, New York, pp 219–239

    Google Scholar 

  52. Hiriart-Urruty JB (2005) La conjecture des points les plus éloingnés revisitée. Ann Sci Math Qué 29:197–214

    MathSciNet  MATH  Google Scholar 

  53. Holmes RB (1972) A course on optimization and best approximation. Lecture notes in oper res and math systems. Springer, Berlin

    MATH  Google Scholar 

  54. Holmes RB (1975) Geometric functional analysis and its applications. Springer, Berlin

    Book  MATH  Google Scholar 

  55. Ioffe AD, Levin VL (1972) Subdifferential of convex functions. Trudi Mosc Mat Obsc 26:3–73 (Russian)

    MathSciNet  MATH  Google Scholar 

  56. James RC (1957) Reflexivity and the supremum of linear functionals. Ann Math 66:159–169

    Article  MATH  Google Scholar 

  57. James RC (1964) Characterization of reflexivity. Stud Math 23:205–216

    MATH  Google Scholar 

  58. Kallina C, Williams AC (1971) Linear programming in reflexive spaces. SIAM Rev 13:350–376

    Article  MathSciNet  MATH  Google Scholar 

  59. Karamardian S (1967) Strictly quasi-convex (concave) functions and duality in mathematical programming. J Math Anal Appl 20:344–358

    Article  MathSciNet  MATH  Google Scholar 

  60. Kortanek KO, Soyster AL (1972) On refinements of some duality theorems in linear programming over cones. Oper Res 20:137–142

    Article  MathSciNet  MATH  Google Scholar 

  61. Köthe G (1969) Topological vector spaces. I. Springer, Berlin

    Book  MATH  Google Scholar 

  62. Krabs W (1969) Duality in nonlinear approximation. J Approx Theory 2:136–151

    Article  MathSciNet  MATH  Google Scholar 

  63. Krabs W (1979) Optimization and approximation. Wiley, Chichester

    MATH  Google Scholar 

  64. Kretschmer KS (1961) Programmes in paired spaces. Can J Math 13:221–238

    Article  MathSciNet  MATH  Google Scholar 

  65. Kurcyush S (1976) On the existence and nonexistence of Lagrange multiplier in Banach space. J Optim Theory Appl 20:81–110

    Article  Google Scholar 

  66. Kutateladze SS (1977) Formulas for computing subdifferentials. Dokl Akad Nauk SSSR 232:770–772 (Russian)

    MathSciNet  Google Scholar 

  67. Ky F (1953) Minimax theorems. Proc Natl Acad Sci USA 39:42–47

    Article  MATH  Google Scholar 

  68. Ky F (1969) Asymptotic cones and duality. J Approx Theory 2:152–169

    Article  Google Scholar 

  69. Lau K-S (1975) Farthest points in weakly compact sets. Isr J Math 2:165–174

    Google Scholar 

  70. Laurent PJ (1972) Approximation and optimization. Herman, Paris

    Google Scholar 

  71. Levine L, Pomerol JCh (1974) Infinite programming and duality in topological vector spaces. J Math Anal Appl 46:75–81

    Article  MathSciNet  MATH  Google Scholar 

  72. Levine L, Pomerol JCh (1976) C-closed mappings and Kuhn–Tucker vectors in convex programming. CORE Disc Papers 7620, Univ Louvain

    Google Scholar 

  73. Levine L, Pomerol JCh (1979) Sufficient conditions for Kuhn–Tucker vectors in convex programming. SIAM J Control Optim 17:689–699

    Article  MathSciNet  MATH  Google Scholar 

  74. Linberg PO (1980) Duality from LP duality. Math Operforsch Stat, Ser Optim 11:171–180

    Google Scholar 

  75. Mangasarian OL, Fromovitz S (1967) The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J Math Anal Appl 17:37–47

    Article  MathSciNet  MATH  Google Scholar 

  76. Mangasarian OL, Ponstein J (1965) Minimax and duality in nonlinear programming. J Math Anal Appl 11:504–518

    Article  MathSciNet  MATH  Google Scholar 

  77. Mennicken R, Sergaloff B (1979/1980) On Banach’s closed range theorem. Arch Math 33:461–465

    Article  Google Scholar 

  78. Moreau JJ (1966–1967) Fonctionelles convexes. Séminaire sur les équations aux dérivées partielles. College de France

    Google Scholar 

  79. Nagahisa Y, Sakawa Y (1969) Nonlinear programming in Banach spaces. J Optim Theory Appl 4:182–190

    Article  MathSciNet  MATH  Google Scholar 

  80. Nakamura T, Yamasaki M (1979) Sufficient conditions for duality theorem in infinite linear programming problems. Hiroshima Math J 9:323–334

    MathSciNet  MATH  Google Scholar 

  81. Norris DO (1971) A generalized Lagrange multiplier rule for equality constraints in normed linear spaces. SIAM J Control 9:561–567

    Article  MathSciNet  MATH  Google Scholar 

  82. Panda BB, Kapoor OP (1978) On farthest points of sets. J Math Anal Appl 62:345–353

    Article  MathSciNet  MATH  Google Scholar 

  83. Precupanu T (1980) Duality in best approximation problem. An St Univ Iaşi 26:23–30

    MathSciNet  Google Scholar 

  84. Precupanu T (1981) Some duality results in convex optimization. Rev Roum Math Pures Appl 26:769–780

    MathSciNet  MATH  Google Scholar 

  85. Precupanu T (1982) On the stability in Fenchel–Rockafellar duality. An St Univ Iaşi 28:19–24

    MathSciNet  Google Scholar 

  86. Precupanu T (1984) Closedness conditions for the optimality of a family of nonconvex optimization problems. Math Operforsch Stat, Ser Optim 15:339–346

    MathSciNet  MATH  Google Scholar 

  87. Precupanu T (1984) Global sufficient optimality conditions for a family of non-convex optimization problems. An St Univ Iaşi 30:51–58

    MathSciNet  MATH  Google Scholar 

  88. Precupanu T (1994) Sur l’existence des solutions optimales pour une famille des problèmes d’optimisation. An St Univ Iaşi 40:359–366

    MathSciNet  Google Scholar 

  89. Precupanu T (2007) Some mappings associated to the farthest point problem and optimality properties. An St Univ Timişoara 45:125–133

    MathSciNet  MATH  Google Scholar 

  90. Precupanu T (2011) Relationships between the farthest point problem and the best approximation problem. An St Univ Iaşi 57:1–12

    MathSciNet  MATH  Google Scholar 

  91. Precupanu T (2012) Characterizations of pointwise additivity of subdifferential (to appear)

    Google Scholar 

  92. Precupanu A, Precupanu T (2000) Proximinality and antiproximinality for a family of optimization problems. In: Proc Natl Conf Math Anal Appl, pp. 295–308, Timisoara

    Google Scholar 

  93. Pshenichny BN (1965) Convex programming in linear normed spaces. Kibernetika 1:46–54 (Russian)

    Google Scholar 

  94. Raffin VL (1969) Sur les programmes convexes définis dans des espaces vectoriels topologiques. C R Acad Sci Paris 268:738–741; Ann Inst Fourier 20:457–491 (1970)

    MathSciNet  MATH  Google Scholar 

  95. Revalski J, Théra M (1999) Generalized sums of monotone operators. C R Acad Sci, Ser 1 Math 329:979–984

    MATH  Google Scholar 

  96. Ritter K (1967) Duality for nonlinear programming in a Banach space. SIAM J Appl Math 15:294–302

    Article  MathSciNet  MATH  Google Scholar 

  97. Robinson SM (1976) Regularity and stability of convex multivalued functions. Math Oper Res 1:130–143

    Article  MathSciNet  MATH  Google Scholar 

  98. Rockafellar RT (1966) Extension of Fenchel’s duality theorems for convex functions. Duke Math J 33:81–90

    Article  MathSciNet  MATH  Google Scholar 

  99. Rockafellar RT (1967) Duality and stability in extremum problems involving convex functions. Pac J Math 21:167–187

    Article  MathSciNet  MATH  Google Scholar 

  100. Rockafellar RT (1969) Convex analysis. Princeton University Press, Princeton

    Google Scholar 

  101. Rockafellar RT (1971) Saddle-points and convex analysis. In: Kuhn HW, Szegö GP (eds) Differential games and related topics. North-Holland, Amsterdam, pp 109–128

    Google Scholar 

  102. Rockafellar RT (1974) Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J Control 12:268–285

    Article  MathSciNet  MATH  Google Scholar 

  103. Rosinger R (1978) Multiobjective duality without convexity. J Math Anal Appl 66:442–450

    Article  MathSciNet  MATH  Google Scholar 

  104. Schechter M (1972) Linear programs in topological linear spaces. J Math Anal Appl 37:492–500

    Article  MathSciNet  MATH  Google Scholar 

  105. Singer I (1971) Best approximation in normed linear spaces by elements of linear subspaces. Springer, Berlin

    Google Scholar 

  106. Singer I (1980) Maximization of lower semi-continuous convex functionals on bounded subsets on locally convex spaces. II. Quasi-Lagrangian duality theorems. Results Math 3:235–248

    Article  MATH  Google Scholar 

  107. Singer I (2006) Duality for nonconvex approximation and optimization. Springer, Berlin

    MATH  Google Scholar 

  108. Stegall C (1978) Optimization of functions on certain subsets of Banach spaces. Math Annalen 236:171–176

    Article  MathSciNet  MATH  Google Scholar 

  109. Stoer J (1963) Duality in nonlinear programming and the minimax theorems. Numer Math 5:371–379

    Article  MathSciNet  MATH  Google Scholar 

  110. Stoer J, Witzgall C (1970) Convexity and optimization in finite dimension. Springer, Berlin

    Book  Google Scholar 

  111. Toland JF (1978) Duality in nonconvex optimization. J Math Anal Appl 66:339–354

    Article  MathSciNet  Google Scholar 

  112. Tuy H (1964) Sur une classe des programmes nonlinéaires. Bull Acad Pol 12:213–215

    Google Scholar 

  113. Tuy H (1977) Stability property of a system of inequalities. Math Operforsch Stat, Ser Optim 8:27–39

    Google Scholar 

  114. Ursescu C (1973) Sur une généralisation de la notion de différentiabilité. Atti Accad Naz Lincei, Rend Cl Sci Fis Mat Nat 54:199–204

    MathSciNet  MATH  Google Scholar 

  115. Ursescu C (1975) A differentiable dependence on the right-hand side of solutions of ordinary differential equations. Ann Pol Math 31:191–195

    MathSciNet  MATH  Google Scholar 

  116. Ursescu C (1975) Multifunctions with closed convex graph. Czechoslov Math J 25:438–441

    MathSciNet  Google Scholar 

  117. Valadier M (1972) Sous-différentiabilité de fonctions convexes à valeurs dans un espace vectoriel ordonné. Math Scand 30:65–74

    MathSciNet  MATH  Google Scholar 

  118. van Slyke R, Wets R (1968) A duality theory for abstract mathematical programs with applications to optimal control theory. J Math Anal Appl 22:679–706

    Article  MathSciNet  MATH  Google Scholar 

  119. Varaiya (1967) Nonlinear programming in Banach space. SIAM J Appl Math 15:284–293

    Article  MathSciNet  MATH  Google Scholar 

  120. Zălinescu C (1978) A generalization of Farkas lemma and applications to convex programming. J Math Anal Appl 66:651–678

    Article  MathSciNet  MATH  Google Scholar 

  121. Zălinescu C (1983) Duality for vectorial nonconvex optimization by convexification and applications. An ştiinţ Univ Al I Cuz Iaşi 29:15–34

    Google Scholar 

  122. Zlobek (1970) Asymptotic Kuhn–Tucker conditions for mathematical programming in a Banach space. SIAM J Control 8:505–512

    Article  MathSciNet  Google Scholar 

  123. Zowe J (1974) Subdifferentiability of convex functions with values in an ordered vector space. Math Scand 34:63–83

    MathSciNet  Google Scholar 

  124. Zowe J, Kurcyusz S (1979) Regularity and stability for the mathematical programming problems in Banach spaces. Appl Math Optim 5:49–62

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Viorel Barbu .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Barbu, V., Precupanu, T. (2012). Convex Programming. In: Convexity and Optimization in Banach Spaces. Springer Monographs in Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2247-7_3

Download citation

Publish with us

Policies and ethics