Abstract

We introduce a new second order asymptotic function which gives information on the convexity (concavity) of the original function from its behavior at infinity. We establish several properties and calculus rules for this concept, which differs from previous notions of second order asymptotic function. Finally, we apply our new definition in order to obtain necessary and sufficient optimality conditions for quadratic programming and quadratic fractional programming.

Author information

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

Alfredo N. Iusem

Instituto de Matemática Pura e Aplicada, Estrada Doña Castorina 110, Jardim Botânico, Rio de Janeiro, CEP 22460-320, Brazil

iusp@impa.br

Keywords

  • Asymptotic cone
  • asymptotic function
  • second order asymptotic functions
  • generalized convexity
  • quadratic programming
  • quadratic fractional programming

Mathematics Subject Classification

90C20, 90C26, 90C32

References

  1. Ch. Amara: Directions de majoration d’une fonction quasiconvexe et applications, Serdica Math. J. 24 (1998) 289–306.Google Scholar
  2. A. Auslender, M. Teboulle: Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer-Verlag, Berlin (2003).Google Scholar
  3. A. Cambini, L. Carosi, L. Martein: On the supremum in quadratic fractional programming, in: Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas et al. (eds.), Lect. Notes Econ. Math. Systems 502, Springer-Verlag, Berlin (1999) 129–143.https://doi.org/10.1007/978-3-642-56645-5_8Google Scholar
  4. A. Cambini, L. Martein: Generalized Convexity and Optimization, Lect. Notes Econ. Math. Systems 616, Springer-Verlag, Berlin (2009).Google Scholar
  5. R. Cominetti, R. Correa: A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim. 28 (1990) 789–809.https://doi.org/10.1137/0328045Google Scholar
  6. J. Cotrina, F. Raupp, W. Sosa: Semicontinuous quadratic optimization: existence conditions and duality scheme, J. Global Optim. 63 (2015) 281–295.https://doi.org/10.1007/s10898-015-0306-3Google Scholar
  7. F. Flores-Bazán: Existence theory for finite-dimensional pseudomonotone equilibrium problems, Acta Appl. Math. 77 (2003) 249–297.https://doi.org/10.1023/A:1024971128483Google Scholar
  8. F. Flores-Bazán, F. Flores-Bazán, C. Vera: Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivates, J. Global Optim. 63 (2015) 99–123.https://doi.org/10.1007/s10898-015-0267-6Google Scholar
  9. F. Flores-Bazán, N. Hadjisavvas, F. Lara: Second order asymptotic analysis: basic theory, J. Convex Analysis 22 (2015) 1173–1196.https://doi.org/10.68381/jca22061Google Scholar
  10. F. Flores-Bazán, N. Hadjisavvas, F. Lara, I. Montenegro: First- and second-order asymptotic analysis with applications in quasiconvex optimization, J. Optim. Theory Appl. 170 (2016) 372–393.https://doi.org/10.1007/s10957-016-0938-6Google Scholar
  11. I. Ginchev, A. Guerraggio: Second order optimizality conditions in nonsmooth unconstrained optimization, Pliska Stud. Math. Bulgar. 12 (1998) 39–50.Google Scholar
  12. N. Hadjisavvas, D. T. Luc: Second order asymptotic directions of unbounded sets with application to optimization, J. Convex Analysis 18 (2011) 181–202.https://doi.org/10.68381/jca18010Google Scholar
  13. F. Lara: Generalized asymptotic functions in nonconvex multiobjective optimization problems, Optimization 66 (2017) 1259–1272.https://doi.org/10.1080/02331934.2016.1235162Google Scholar
  14. F. Lara, R. López: Formulas for asymptotic functions via conjugates, directional derivatives and subdifferentials, J. Optim. Theory Appl. 173 (2017) 793–811.https://doi.org/10.1007/s10957-017-1101-8Google Scholar
  15. G. M. Lee, N. N. Tam, N. D. Yen: Quadratic Programming and Affine Variational Inequalities. A Qualitative Study, Springer-Verlag, Berlin (2005).Google Scholar
  16. D. T. Luc: Theory of Vector Optimization, Lect. Notes Econ. Math. Systems 319, Springer-Verlag, Berlin (1989).https://doi.org/10.1007/978-3-642-50280-4Google Scholar
  17. D. T. Luc, J. P. Penot: Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001) 4095–4121.https://doi.org/10.1090/S0002-9947-01-02664-2Google Scholar
  18. P. Michel, J. P. Penot: Second order moderate derivatives, Nonlinear Anal. Theory, Methods Appl. 22 (1994), 809–821.https://doi.org/10.1016/0362-546X(94)90050-7Google Scholar
  19. J. P. Penot: What is quasiconvex analysis ?, Optimization 47 (2000) 35–110.https://doi.org/10.1080/02331930008844469Google Scholar
  20. J. P. Penot: Calculus Without Derivatives, Springer, New York (2013).https://doi.org/10.1007/978-1-4614-4538-8Google Scholar
  21. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  22. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer-Verlag, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar

Suggested citation

A. N. Iusem, F. Lara. “Second Order Asymptotic Functions and Applications to Quadratic Programming.” Journal of Convex Analysis 25 (2018), No. 1, 271–291. https://doi.org/10.68381/jca25016

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.