Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Conditional Subgradient Method for Solving Nonsmooth Multi-Objective Optimization Problems

Discontinuity, Nonlinearity, and Complexity 11(1) (2022) 125--132 | DOI:10.5890/DNC.2022.03.010

A. Lahmdani$^1$ , M. Tifroute$^2$

$^1$ Faculty of Applied Sciences - Ait Melloul, Ibn Zohr University. IMI Laboratory - FSA, Morocco

$^2$ Laboratory of Engineering Systems and Information Technologies, ENSA - Ibn Zohr University, PO Box 1136, Agadir, Morocco

Download Full Text PDF



In this paper, we propose a method for solving quasiconvex nonsmooth multiobjective optimization problems. The proposed method extends the conditional subgradient method, from mono-objective to multiobjective optimization problems. We show that the sequence generated using this method converge to Pareto optimal points of the problem.


  1. [1]  Miettinen K. (1999), Nonlinear multiobjective optimization. Boston: Kluwer.
  2. [2]  Polyak, B.T. (1967), A general method for solving extremum problems, Soviet Mathematics Doklady, 8, 593-597.
  3. [3]  Ermoliev, Y.M. (1996), Methods of solution of nonlinear extremal problems, Cybern. Syst. Anal., 2, 1-14.
  4. [4]  Shor, N.Z. (1991), The development of numerical methods for nonsmooth optimization in the USSR, in: J.K. Lenstra, A.H.G. Rinnoy Kan and A. Schrijver (eds.), History of Mathematical Programming: A Collection of Personal Reminiscences, North-Holland, Amsterdam, 135-139.
  5. [5]  Hu, Y., Yu, C.K.W., Li, C., and Yang, X. (2016), Conditional subgradient methods for constrained quasi-convex optimization problems, J. Nonlinear Convex Anal., 17, 2143-2158.
  6. [6]  Larsson, T. Patriksson, M., and Stromberg, A.B. (1996), Conditional subgradient optimization---theory and applications, Euro. J. Oper. Res., 88, 382-403.
  7. [7]  Kiwiel, K.C. (2004), Convergence of approximate and incremental subgradient methods for convex optimization, SIAM J. Optim., 14, 807-840.
  8. [8]  Hu, Y.H., Sim, C.K., and Yang, X.Q. (2015), A subgradient method based on gradient sampling for solving convex optimization problems, Numer. Func. Anal. Opt., 36, 1559-1584.
  9. [9]  Bertsekas, D.P. (1999), Nonlinear Programming, Athena Scientific, Cambridge.
  10. [10]  Avriel, M. (2003), Nonlinear Programming: Analysis and Methods, Dover, New York.
  11. [11]  Bazaraa, M.S., Sherali, H.D., and Shetty, C.M. (2013), Nonlinear programming: theory and algorithms, John Wiley and Sons.
  12. [12]  Gutierez, J.M. (1984), D{\i}ez, Infragradients and directions of decrease, Rev. Real Acad. Cienc. Exact. F{\iys. Natur. Madrid}, 78, 523-532.
  13. [13]  Li, C. and Wang, J.H. (2006), Newton's method on Riemannian manifolds: Smale's point estimate theory under the condition, IMA J. Numer. Anal., 26(2), 228-251.
  14. [14]  Dem'janov, V.E. and Somesova, V.K. (1978),Conditional subdifferentials of convex functions, Soviet Mathematics Doklady 19, 1181-1185.
  15. [15]  Dem'yanov, V.E. and Somesova, V.K. (1980), Subdifferentials of functions on sets, Cybernetics, 16/I, 24-31.
  16. [16]  Xu, H., Rubinov, A.M., and Glover, B.M. (1999), lower subdifferentiability and applications, J. Aust. Math. Soc. Ser. B, Appl. Math., 40, 379-391.
  17. [17]  Burachik, R.S., Gran\~{a} Drummond, L.M., Iusem, A.N., and Svaiter, B.F. (1995), Full convergence of the steepest descent method with inexact line search, Optimization, 32, 137-146.
  18. [18]  Gran\~{a} Drummond, L,M. and Svaiter, B.F. (2005), A steepest descent method for vector optimization, Journal of Computational and Applied Mathematics, 175, 395-414.
  19. [19]  da Cruz Neto, J.X., da Silva, G.J.P., Ferreira, O.P., and Lopes, J.O. (2013), A subgradient method for multiobjective optimization, Comput. Optim. Appl., 54(3), 461-472.
  20. [20]  Loridan, P. (1984), $\epsilon$-solutions in vector minimization problems, J. Optim. Theory Appl., 43(2), 265-276.
  21. [21]  White, D. J. (1986), Epsilon effciency, J. Optim. Theory Appl., 49(2), 319-337.
  22. [22]  Helbig, S. and Pateva, D. (1994), On several concepts for $\epsilon$-effciency, Operations-Research- Spektrum, 16(3), 179-186.
  23. [23]  Dhara, A. and Dutta, J. (2011), Optimality conditions in convex optimization: a finite- dimensional view. CRC Press.
  24. [24]  Azagra, D., Ferrera, J., and L{o}pez-Mesas, M. (2005), Nonsmooth analysis and Hamilton---Jacobi equations on Riemannian manifolds, J. Funct. Anal., 220, 304-361.