Journal of Optimization Theory and Applications

, Volume 144, Issue 2, pp 291–318 | Cite as

An Interior-Point Algorithm for Nonlinear Minimax Problems

  • E. Obasanjo
  • G. Tzallas-Regas
  • B. Rustem


We present a primal-dual interior-point method for constrained nonlinear, discrete minimax problems where the objective functions and constraints are not necessarily convex. The algorithm uses two merit functions to ensure progress toward the points satisfying the first-order optimality conditions of the original problem. Convergence properties are described and numerical results provided.


Discrete min-max Constrained nonlinear programming Primal-dual interior-point methods Stepsize strategies 


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  1. 1.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    El-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the Newton interior-point method for nonlinear programming. J. Optim. Theory Appl. 89(3), 507–541 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point method for linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming, pp. 29–47. Springer, New York (1989) Google Scholar
  4. 4.
    Benson, H.Y., Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: jamming and numerical testing. Math. Program. Ser. A 99(1), 35–48 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Du, D.-Z., Pardalos, P.M.: Minimax and Applications. Nonconvex Optimization and Its Applications. Springer, Dordrecht (1995) Google Scholar
  6. 6.
    Rustem, B., Howe, M.: Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, Princeton (2002) zbMATHGoogle Scholar
  7. 7.
    Akrotirianakis, I., Rustem, B.: A globally convergent interior point algorithm for non-linear problems. J. Optim. Theory Appl. 125(3), 497–521 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Rustem, B.: A constrained min-max algorithm for rival models of the same economic system. Math. Program. Ser. A 53(3), 279–295 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Medanić, J., Andjelić, M.: On a class of differential games without saddle-point solutions. J. Optim. Theory Appl. 8, 413–430 (1971) zbMATHCrossRefGoogle Scholar
  10. 10.
    Medanić, J., Andjelić, M.: Minimax solution of the multiple-target problem. IEEE Trans. Automat. Control 17(5), 597–604 (1972) zbMATHCrossRefGoogle Scholar
  11. 11.
    Vanderbei, R.J., Shanno, D.F.: An interior point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chamberlain, R.M., Powell, M.J.D., Lemarechal, C., Pedersen, H.C.: The watchdog technique for forcing convergence in algorithms for constrained optimization. Math. Program. Stud. 16, 1–17 (1982) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Gay, D.M., Overton, M.L., Wright, M.H.: A primal-dual interior method for nonconvex nonlinear programming. In: Yuan, Y. (ed.) Advances in Nonlinear Programming, Beijing, 1996. Appl. Optim., vol. 14, pp. 31–56. Kluwer Academic, Dordrecht (1998) Google Scholar
  14. 14.
    Rustem, B.: Convergent stepsizes for constrained optimization algorithms. J. Optim. Theory Appl. 49, 135–160 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rustem, B.: Equality and inequality constrained optimization algorithms with convergent stepsizes. J. Optim. Theory Appl. 76(3), 429–453 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rustem, B., Nguyen, Q.: An algorithm for the inequality-constrained discrete min-max problem. SIAM J. Optim. 8(1), 265–283 (1998) (electronic) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Yamashita, H.: A globally convergent primal–dual interior point method for constrained optimization. Optim. Methods Softw. 10, 443–469 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Yamashita, H., Yabe, H.: Superlinear and quadratic convergence of some primal–dual interior point methods for constrained optimization. Math. Program. 75, 377–397 (1996) MathSciNetGoogle Scholar
  19. 19.
    Tzallas-Regas, G.: Switching stepsize strategies for nonlinear programming. Ph.D. Thesis. University of London, Imperial College London, London (2007) Google Scholar
  20. 20.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL—A Modeling Language for Mathematical Programming. The Scientific Press, London (1993) Google Scholar
  21. 21.
    Benson, H.: The Cute AMPL models, (1996)

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Equity Derivatives Group–Systematic TradingBarclays CapitalLondonUK
  2. 2.Department of ComputingImperial College LondonLondonUK

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