• Mohammad Fathi
  • Hassan Bevrani


One common approach towards solving an optimization problem, in general, is to transfer it from the primal domain into a dual domain. This is sometimes of great advantage as the problem in the dual domain is simple enough to solve. In this chapter, this transformation is introduced and, based on the achievements from the dual domain, Karush–Kuhn–Tucker (KKT) conditions are derived to find optimal solutions in general optimization problems. Moreover, based on KKT conditions, Lagrangian algorithm is introduced to be implemented as an iterative search method to solve convex problems. Finally, some application examples in electrical engineering are given, accordingly.


Duality Dual decomposition KKT conditions Lagrangian algorithm Sensitivity analysis 


  1. 1.
    S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, U.K, 2004)Google Scholar
  2. 2.
    H.W. Kuhn, A.W. Tucker, Nonlinear programming, in Proceedings of 2nd Berkeley Symposium (University of California Press, Berkeley, 1951), pp. 481–492Google Scholar
  3. 3.
    Y. Nesterov, A. Nemirovsky, Interior Point Polynomial Algorithms in Convex Programming. (SIAM, Philadelphia, 1994)Google Scholar
  4. 4.
    H. Hindi, A tutorial on convex optimization ii: duality and interior point methods, in Proc. American Control Conference (2006)Google Scholar
  5. 5.
    A. Goldsmith, Wireless communications (Cambridge University Press, New York, 2005)CrossRefGoogle Scholar
  6. 6.
    J.F. Kurose, K.W. Ross, Computer Networking: A Top-Down Approach (Pearson, New Jersey, 2012)Google Scholar
  7. 7.
    A.S. Tanenbaum, D.J. Wetherall, Computer Networks (Prentice Hall, Upper Saddle River, 2010)Google Scholar
  8. 8.
    R. Srikant, L. Ying, Communication Networks: An Optimization, Control, and Stochastic Networks Perspective (Cambridge University Press, Cambridge, 2014)zbMATHGoogle Scholar
  9. 9.
    M. Chiang, Balancing transport and physical layers in wireless multihop networks: jointly optimal congestion control and power control. Proc. IEEE 23(1), 104–116 (2005)Google Scholar
  10. 10.
    W. Stanczak, M. Wiczanowski, H. Boche, Fundamentals of Resource Allocation in Wireless Networks: Theory and Algorithms (Springer, Berlin, 2008)CrossRefGoogle Scholar
  11. 11.
    S. Shakkottai, T.S. Rappaport, P.C. Karlsson, Cross-layer design for wireless networks. IEEE Commun. Mag. 41(10), 74–80 (2003)CrossRefGoogle Scholar
  12. 12.
    M. Chiang, S.H. Low, A.R. Calderbank, J.C. Doyle, Layering as optimization decomposition: a mathematical theory of network architectures. Proc. IEEE 95(1), 255–312 (2007)CrossRefGoogle Scholar
  13. 13.
    F.P. Kelly, A. Maulloo, D. Tan, Rate control for communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49(3), 237–252 (1998)CrossRefGoogle Scholar
  14. 14.
    R. Srikant, The Mathematics of Internet Congestion Control (Birkhauser, Basel, 2004)CrossRefGoogle Scholar
  15. 15.
    S.H. Low, A duality model of TCP and queue management algorithms. IEEE Trans. Net. 11(4), 525–536 (2003)CrossRefGoogle Scholar
  16. 16.
    W. Yu, R. Lui, Dual methods for nonconvex spectrum optimization of multicarrier systems. IEEE Trans. Commun. 54(7), 1310–1322 (2006)CrossRefGoogle Scholar
  17. 17.
    M. Fathi, H. Taheri, M. Mehrjoo, Cross-layer joint rate control and scheduling for OFDMA wireless mesh networks. IEEE Trans. Veh. Technol. 59(8), 3933–3941 (2010)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mohammad Fathi
    • 1
  • Hassan Bevrani
    • 1
  1. 1.University of KurdistanKurdistanIran

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