Journal of Optimization Theory and Applications

, Volume 171, Issue 2, pp 617–642 | Cite as

On the Problem of Minimizing a Difference of Polyhedral Convex Functions Under Linear Constraints

  • Nguyen Thi Van Hang
  • Nguyen Dong YenEmail author


This paper is concerned with two d.p. (difference of polyhedral convex functions) programming models, unconstrained and linearly constrained, in a finite-dimensional setting. We obtain exact formulae for the Fréchet and Mordukhovich subdifferentials of a d.p. function. We establish optimality conditions via subdifferentials in the sense of convex analysis, of Fréchet and of Mordukhovich, and describe their relationships. Existence and computation of descent and steepest descent directions for both the models are also studied.


d.p. programming Subdifferential Optimality conditions  Stationary point Density Active index set  Extreme point 

Mathematics Subject Classification

49J52 90C26 90C46 



The authors would like to thank the reviewer for comments and suggestions improving the present paper. They would particularly like to thank Professor Boris S. Mordukhovich for supplying them valuable references and for a great encouragement. Financial support from National Foundation for Science and Technology Development (NAFOSTED, Vietnam) under Grant 101.01-2014.37 is gratefully acknowledged.


  1. 1.
    Kiwiel, K.C.: An aggregate subgradient method for nonsmooth and nonconvex minimization. J. Comput. Appl. Math. 14, 391–400 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Vlček, J., Lukšan, L.: Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization. J. Optim. Theory Appl. 111, 407–430 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fuduli, A., Gaudioso, M., Giallombardo, G.: Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14(3), 743–756 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to d.c. programming theory, algorithms and applications. Acta Math. Vietnam. 22(1), 289–355 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Polyakova, P.L.: On global unconstrained minimization of the difference of polyhedral functions. J. Glob. Optim. 50(2), 179–195 (2011)Google Scholar
  6. 6.
    Demy’anov, V. F., Vasil’ev, L. V.: Nondifferentiable Optimization. Translated from the Russian by Tetsushi Sasagawa. Translations series in mathematics and engineering. Optimization Software Inc., New York (1985)Google Scholar
  7. 7.
    Demy’anov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Approximation & Optimization, vol. 7. Peter Lang, Frankfurt am Main (1995)Google Scholar
  8. 8.
    Demy’anov, V.F., Rubinov, A.M.: An introduction to quasidifferential calculus. Quasidifferentiability and Related Topics, 1–31, Nonconvex Optimization and Its Applications, vol. 43. Kluwer, Dordrecht (2000)Google Scholar
  9. 9.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)Google Scholar
  10. 10.
    Roshchina, V.A.: Mordukhovich subdifferential of pointwise minimum of approximate convex functions. Optim. Methods Softw. 25(1), 129–141 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, New Jersey (1970)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. I. Fundamentals. Springer, Berlin (1993)zbMATHGoogle Scholar
  13. 13.
    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55(5–6), 685–708 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  15. 15.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

Personalised recommendations