Numerical Algorithms

, Volume 80, Issue 2, pp 397–427 | Cite as

An infeasible bundle method for nonconvex constrained optimization with application to semi-infinite programming problems

  • Jian Lv
  • Li-Ping PangEmail author
  • Na Xu
  • Ze-Hao Xiao
Original Paper


The main difficulty for solving semi-infinite programming (SIP) problem is precisely that it has infinitely many constraints. By using a maximum function, the SIP problem can be rewritten as a nonconvex nonsmooth constrained optimization (NNCO) problem. Global convergence in most of constrained optimization algorithms has traditionally been enforced by the use of a penalty function or filter strategy. In this paper, we propose an infeasible bundle method for NNCO problem based on the so-called improvement functions, without a penalty function and filter strategy. The method appears to be more direct and easier to implement, in the sense that it is closer in spirit and structure to the well-developed unconstrained bundle methods. Under a special constraint qualification, the sequence generated by this algorithm converges to the KKT point of the NNCO problem as well as the SIP problems. Preliminary numerical results show that this algorithm is robust and efficient for NNCO problems and SIP problems.


Semi-infinite programming Constrained optimization Nonconvex optimization Nonsmooth optimization Improvement function Bundle method 

Mathematics Subject Classifications (2010)

90C26 49J52 93B40 


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  1. 1.
    Bhattacharjee, B., Lemonidis, P., Green, W.H. Jr, Barton, P.I.: Global solution of semi-infinite programs. Math. Program. 103, 283–307 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonnans, J.F., Gilbert, J.C., Lemarchal, C., Sagastizbal, C.: Numerical Optimization: Theoretical and Pratical Aspects, 2nd edn. Springer, Berlin (2000)Google Scholar
  3. 3.
    Cánovas, M.J., Hantoute, A., Láopez, M.A., Parra, J.: Stability of indices in the KKT conditions and metric regularity in convex semi-infinite optimization. J. Optim. Theory Appl. 139, 485–500 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Clarke, F.H., Ledyaev, Y.u.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)zbMATHGoogle Scholar
  5. 5.
    Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–275 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ferrier, C.: Computation of the distance to semi-algebraic sets. ESAIM Control Optim. Calc. Var. 5, 139–156 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fletcher, R., Leyffer, S.: A Bundle Filter Method for Nonsmooth Nonlinear Optimization, Numerical Analysis Report NA/195, Department of Mathematics. The University of Dundee, Scotland (1999)Google Scholar
  8. 8.
    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program. 91, 239–269 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Floudas, C.A., Stein, O.: The adaptive convexification algorithm: a feasible point method for semi-infinite programming. SIAM J. Optim. 18, 1187–1208 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fuduli, A., Gaudioso, M., Giallombardo, G.: A DC piecewise and model and a bundling technique in nonconvex nonsmooth minimization. Optim. Methods Softw. 19, 89–102 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, New-York (1998)zbMATHGoogle Scholar
  12. 12.
    Haarala, N., Miettinen, K., Mäkelä, M.M.: Globally convergent limited memory bundle method for large-scale nonsmooth optimization. Math. Program. 109, 181–205 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hare, W., Poliquin, R.A.: Prox-regularity and stability of the proximal mapping. J. Convex Anal. 14, 589–606 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hare, W., Sagastizbal, C.: Computing proximal points of nonconvex functions. Math Program. 116, 221–258 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hare, W., Sagastizábal, C.: A redistributed proximal bundlemethod for nonconvex optimization. SIAM J. Optim. 20, 2442–2473 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hare, W., Sagastizábal, C., Solodov, M.: A proximal bundle method for nonsmooth nonconvex functions with inexact information. Comput. Optim. Appl. 63, 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming: Theory, methods and applications. SIAM Rev. 35, 380–429 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Karmitsa, N., Mäkelä, M.M.: Adaptive limited memory bundle method for bound constrained largescale nonsmooth optimization. Optimization 59, 945–962 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Karmitsa, N., Mäkelä, M.M.: Limited memory bundle method for large bound constrained nonsmooth optimization: convergence analysis. Optim. Methods Softw. 25, 895–916 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Karmitsa, N., Mäkelä, M.M., Ali, M.M.: Limited memory interior point bundle method for large inequality constrained nonsmooth minimization. Appl. Math. Comput. 198, 382–400 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics. Springer-Verlag, Berlin (1985)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kiwiel, K.C.: An exact penalty function algorithm for nonsmooth convex constrained minimization problems. IMA J. Numer. Anal. 5, 111–119 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kiwiel, K.C.: Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization. Math. Program. 52, 285–302 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kortanek, K.O., No, H.: A central cutting plane algorithm for convex semi-infinite programming problems. SIAM J. Optim. 3, 901–918 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kuntsevich, A., Kappel, F.: SolvOpt-The solver for local nonlinear optimization problems: Matlab, C and fortran source codes, institute for mathematics. Karl-Franzens University of Graz (1997)Google Scholar
  26. 26.
    Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69, 111–147 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    López, M.A., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lv, J., Pang, L.P., Wang, J.H.: Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization. Appl. Math. Comput. 265, 635–651 (2015)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific Publishing Co., Singapore (1992)CrossRefzbMATHGoogle Scholar
  30. 30.
    Mäkelä, M.M., Karmitsa, N., Wilppu, O.: Proximal bundle method for nonsmooth and nonconvex multiobjective optimization. In: Tuovinen, T., Repin, S., Neittaanmäki, P. (eds.) Mathematical Modeling and Optimization of Complex Structures. Computational Methods in Applied Sciences, vol. 40, pp 191–204. Springer (2016)Google Scholar
  31. 31.
    Mehrotra, S., Papp, D.: A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization. SIAM J. Optim. 24, 1670–1697 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mifflin, R.: An algorithm for constrained optimization with semismooth functions. Math. Oper. Res. 2, 191–207 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ni, Q., Ling, C., Qi, L., Teo, K.L.: A truncated projected Newton-type algorithm for large-scale semi-infinite programming. SIAM J. Optim. 16, 1137–1154 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Pang, L.P., Lv, J., Wang, J.H.: Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems. Comput. Optim. Appl. 64, 433–465 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Qi, L., Ling, C., Tong, X.J., Zhou, G.: A smoothing projected Newton-type algorithm for semi-infinite programming. Comput. Optim. Appl. 42, 1–30 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)Google Scholar
  37. 37.
    Rockafellar, R.T., Wets, J.J.-B.: Variational Analysis, Volume 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1998)Google Scholar
  38. 38.
    Rückmann, J.-J., Stein, O.: On convex lower level problems in generalized semi-infinite optimization. In: Goberna, M. A., López, M. A. (eds.) Semi-infinite Programming Recent Advances, pp 121–134. Kluwer, Dordrecht (2001)Google Scholar
  39. 39.
    Sagastizábal, C., Solodov, M.: An infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter. SIAM J. Optim. 16, 146–169 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2, 121–152 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Stein, O., Still, G.: On generalized semi-infinite optimization and bilevel optimization. European J. Oper. Res. 142, 444–462 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Stein, O.: Bi-level Strategies in Semi-infinite Programming. Kluwer, Boston (2003)CrossRefzbMATHGoogle Scholar
  43. 43.
    Stein, O.: On constraint qualifications in nonsmooth optimization. J. Optim. Theory Appl. 121, 647–671 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Still, G.: Discretization in semi-infinite programming: the rate of convergence. Math. Program. 91, 53–69 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Teo, K.L., Yang, X.Q., Jennings, L.S.: Computational discretization algorithms for functional inequality constrained optimization. Ann. Oper. Res. 28, 215–234 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Tichatschke, R., Nebeling, V.: A cutting plane method for quadratic semi-infinite programming. Probl. Optim. 19, 803–817 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Wu, S.-Y., Li, D.H., Qi, L., Zhou, G.: An iterative method for solving KKT system of the semi-infinite programming. Optim, Methods Soft. 20, 629–643 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Yang, Y., Pang, L.P., Ma, X.F., Shen, J.: Constrained nonconvex nonsmooth optimization via proximal bundle method. J. Optim. Theory Appl. 163, 900–925 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Zhang, L.P., Wu, S.-Y., López, M.A.: A new exchange method for convex semi-infinite programming. SIAM J. Optim. 20, 2959–2977 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of FinanceZhejiang University of Finance & EconomicsHangzhouChina
  2. 2.School of Mathematical SciencesDalian University of TechnologyDalianChina

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