Abstract

We consider an operation on subsets of a topological vector space which is closely related to what has been called the inverse addition by R. T. Rockafellar. Applied to closed convex sets, it appears as the operation corresponding to the addition under polarity. However, our study is not limited to the convex case. Crucial tools for it are the gauges one can associate with a subset. We stress the role played by asymptotic cones in such a context. We present an application to the calculus of conjugate functions for one of the most fruitful dualities for quasiconvex problems. We also present an extension of the well-known rule for the computation of the normal cone to a convex set defined by a convex inequality.

Author information

Contact details are reproduced from the original publication and may be historical.

Jean-Paul Penot

Dép. de Mathématiques, Faculté des Sciences, Université de Pau, 64000 Pau, France

Constantin Zălinescu

Faculté de Mathématiques, Université "Al. I. Cuza", 6600 Iaşi, Romania

Keywords

  • Conjugate function
  • convex set
  • duality
  • gauge
  • harmonic sum
  • inverse sum
  • normal cone
  • shady set
  • star-shaped set

Mathematics Subject Classification

52A05; 52A30, 26B25, 90C25

References

  1. M. Atteia: Analyse convexe projective, C. R. Acad. Sci. Paris Sér. I Math. 276 (1973) 795-798, ibidem 855-858.Google Scholar
  2. D. Azé, M. Volle: A stability result in quasi-convex programming, J. Optim. Theory Appl. 67 (1990) 175-184.https://doi.org/10.1007/BF00939742Google Scholar
  3. A. Barbara, J.-P. Crouzeix: Concave gauge functions and applications, Math. Methods Oper. Res. 40 (1994) 43-74.https://doi.org/10.1007/BF01414029Google Scholar
  4. J. L. Kelley, I. Namioka: Linear Topological Spaces, Van Nostrand, New York, 1963.https://doi.org/10.1007/978-3-662-41914-4Google Scholar
  5. J.-P. Penot: Duality for star-shaped and shady programs, Acta Math. Vietnam. 22 (2) (1997) 541-566.Google Scholar
  6. J.-P. Penot: Duality for anticonvex programs, to appear in J. Global Optim.Google Scholar
  7. J.-P. Penot: What is quasiconvex analysis?, Optimization 47 (2000) 35-110.https://doi.org/10.1080/02331930008844469Google Scholar
  8. J.-P. Penot: Transposition of relations, Canadian Math. Soc. Conferences Proc., to appear.Google Scholar
  9. J.-P. Penot, M. Volle: Dualité de Fenchel et quasi-convexité, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987) 269-272.Google Scholar
  10. J.-P. Penot, M. Volle: Inversion of real-valued functions and applications, Math. Methods Oper. Res. 34 (1990) 117-141.https://doi.org/10.1007/BF01415975Google Scholar
  11. J.-P. Penot, M. Volle: On quasi-convex duality, Math. Oper. Res. 15 (1990) 597-625.https://doi.org/10.1287/moor.15.4.597Google Scholar
  12. J.-P. Penot, M. Volle: Another duality scheme for quasiconvex problems, in "Trends in Mathematical Optimization," K. H. Hoffmann et al. (eds.), Int. Series Numer. Math. 84, Birkhäuser, Basel (1988) 259-275.https://doi.org/10.1007/978-3-0348-9297-1_17Google Scholar
  13. J.-P. Penot, C. Zălinescu: Elements of quasiconvex subdifferential calculus, J. Convex Analysis 7 (2000), Number 2.https://doi.org/10.68381/jca07013Google Scholar
  14. J.-P. Penot, C. Zălinescu: Regularization of quasiconvex functions, in preparation.Google Scholar
  15. R. T. Rockafellar: Monotone Processes of Convex and Concave Type, Memoirs Amer. Math. Soc. 77, Providence (1967).https://doi.org/10.1090/memo/0077Google Scholar
  16. R. T. Rockafellar: Convex Analysis, Princeton Univ. Press, Princeton, N. J., 1970.https://doi.org/10.1515/9781400873173Google Scholar
  17. R. T. Rockafellar, R. J-B Wets: Variational Analysis, Springer, Berlin, 1998.https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  18. A. Rubinov: Antihomogeneous conjugacy operators in convex analysis, J. Convex Anal. 2 (1995) 291-307.https://doi.org/10.68381/jca02020Google Scholar
  19. A. M. Rubinov, B. M. Glover: On generalized quasiconvex conjugation, Contemp. Math. 204 (1997) 199-216.https://doi.org/10.1090/conm/204/02631Google Scholar
  20. A. M. Rubinov, A. A. Yagubov: The space of star-shaped sets and its applications in nonsmooth optimization, Math. Programming Study 29 (1986) 176-202.https://doi.org/10.1007/BFb0121146Google Scholar
  21. A. Seeger: Direct and inverse addition in convex analysis and applications, J. Math. Anal. Appl. 148 (1990) 317-349.https://doi.org/10.1016/0022-247X(90)90004-YGoogle Scholar
  22. A. Seeger, M. Volle: On a convolution operation obtained by adding level sets: classical and new results, RAIRO Rech. Opér. 29 (1995) 131-154.https://doi.org/10.1051/ro/1995290201311Google Scholar
  23. P. T. Thach: Global optimality criterion and a duality with a zero gap in nonconvex optimization, SIAM J. Math. Anal. 24 (6) (1993) 1537-1556.https://doi.org/10.1137/0524087Google Scholar
  24. P. T. Thach: A nonconvex duality with zero gap and applications, SIAM J. Optim. 4 (1994) 44-64.https://doi.org/10.1137/0804002Google Scholar
  25. M. Volle: Duality for the level sum of quasiconvex functions and applications, ESAIM Control Optim. Calc. Var. 3 (1998) 329-343 (electronic). URL: http://www.emath.fr/cocv/https://doi.org/10.1051/cocv:1998114Google Scholar

Suggested citation

J.-P. Penot, C. Zălinescu. “Harmonic Sum and Duality.” Journal of Convex Analysis 7 (2000), No. 1, 95–113. https://doi.org/10.68381/jca07005

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Published by Heldermann Verlag, 2000. Rights now held by Banach Press.