Optimality Functions and Lopsided Convergence

  • Johannes O. Royset
  • Roger J-B Wets


Optimality functions pioneered by E. Polak characterize stationary points, quantify the degree with which a point fails to be stationary, and play central roles in algorithm development. For optimization problems requiring approximations, optimality functions can be used to ensure consistency in approximations, with the consequence that optimal and stationary points of the approximate problems indeed are approximately optimal and stationary for an original problem. In this paper, we review the framework and illustrate its application to nonlinear programming and other areas. Moreover, we introduce lopsided convergence of bifunctions on metric spaces and show that this notion of convergence is instrumental in establishing consistency of approximations. Lopsided convergence also leads to further characterizations of stationary points under perturbations and approximations.


epi-Convergence Lopsided convergence Consistent approximations Optimality functions Optimality conditions 

Mathematics Subject Classification

90C46 49J53 



This material is based upon work supported in part by the US Army Research Laboratory and the US Army Research Office under Grant Numbers 00101-80683, W911NF-10-1-0246 and W911NF-12-1-0273.


  1. 1.
    Polak, E.: Optimization. Algorithms and Consistent Approximations, Applied Mathematical Sciences, vol. 124. Springer, New York (1997)MATHGoogle Scholar
  2. 2.
    Royset, J.O.: Optimality functions in stochastic programming. Math. Program. 135(1), 293–321 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Royset, J.O., Pee, E.Y.: Rate of convergence analysis of discretization and smoothing algorithms for semi-infinite minimax problems. J. Optim. Theory Appl. 155(3), 855–882 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Foraker, J.C., Royset, J.O., Kaminer, I.: Search-trajectory optimization: part 1, formulation and theory. J. Optim. Theory Appl. (2015)Google Scholar
  5. 5.
    Phelps, C., Royset, J.O., Gong, Q.: Optimal control of uncertain systems using sample average approximations. SIAM J. Control Optim. (2015)Google Scholar
  6. 6.
    Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)MathSciNetMATHGoogle Scholar
  7. 7.
    Zangwill, W.I.: Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs (1969)MATHGoogle Scholar
  8. 8.
    Royset, J.O., Szechtman, R.: Optimal budget allocation for sample average approximation. Oper. Res. 61, 762–776 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Royset, J.O.: On sample size control in sample average approximations for solving smooth stochastic programs. Comput. Optim. Appl. 55(2), 265–309 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pasupathy, R.: On choosing parameters in retrospective-approximation algorithms for stochastic root finding and simulation optimization. Oper. Res. 58(4), 889–901 (2010)CrossRefMATHGoogle Scholar
  11. 11.
    Polak, E.: On the use of models in the synthesis of optimization algorithms. In: Kuhn, H., Szego, G. (eds.) Differential Games and Related Topics, pp. 263–279. North Holland, Amsterdam (1971)Google Scholar
  12. 12.
    Polak, E.: Computational Methods in Optimization. A Unified Approach. Academic Press, New York (1971)Google Scholar
  13. 13.
    Attouch, H., Wets, R.: Convergence des points min/sup et de points fixes. C. R. Acad. Sci. Paris 296, 657–660 (1983)MathSciNetGoogle Scholar
  14. 14.
    Jofre, A., Wets, R.: Variational convergence of bivariate functions: lopsided convergence. Math. Program. B 116, 275–295 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jofre, A., Wets, R.: Variational convergence of bifunctions: motivating applications. SIAM J. Optim. 24(4), 1952–1979 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Sciences. Pitman, Boston (1984)MATHGoogle Scholar
  17. 17.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)MATHGoogle Scholar
  18. 18.
    Rockafellar, R.T., Wets, R.: Variational Analysis. Grundlehren der Mathematischen Wissenschaft, vol. 317, 3rd printing-2009 edn. Springer (1998)Google Scholar
  19. 19.
    Attouch, H., Wets, R.: A convergence theory for saddle functions. Trans. Am. Math. Soc. 280(1), 1–41 (1983)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Beer, G.: Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, vol. 268. Kluwer, Dordrecht (1992)Google Scholar
  21. 21.
    Bagh, A.: Epi/hypo-convergence: the slice topology and saddle points approximation. J. Appl. Anal. 4, 13–39 (1996)MathSciNetMATHGoogle Scholar
  22. 22.
    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities-III, pp. 103–113. Academic Press, New York (1972)Google Scholar

Copyright information

© Springer Science+Business Media New York (outside the US) 2015

Authors and Affiliations

  1. 1.Naval Postgraduate SchoolMontereyUSA
  2. 2.University of California, DavisDavisUSA

Personalised recommendations