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A new smoothing Newton-type algorithm for semi-infinite programming

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Abstract

We consider a semismooth reformulation of the KKT system arising from the semi-infinite programming (SIP) problem. Based upon this reformulation, we present a new smoothing Newton-type method for the solution of SIP problem. The main properties of this method are: (a) it is globally convergent at least to a stationary point of the SIP problem, (b) it is locally superlinearly convergent under a certain regularity condition, (c) the feasibility is ensured via the aggregated constraint, and (d) it has to solve just one linear system of equations at each iteration. Preliminary numerical results are reported.

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References

  1. Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    Google Scholar 

  2. Coope I.D., Watson G.A.: A projected Lagrangian algorithm for semi-infinite programming. Math. Prog. 32, 337–356 (1985)

    Article  Google Scholar 

  3. Facchinei F., Soares J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225–247 (1997)

    Article  Google Scholar 

  4. Fang S.C., Wu S.Y.: An inexact approach to solving linear semi-infinite programming problems. Optimization 28, 291–299 (1994)

    Article  Google Scholar 

  5. Goberna M.A., López M.A.: Optimal value function in semi-infinite programming. J. Optim. Theory Appl. 59, 261–279 (1988)

    Google Scholar 

  6. Goberna M.A., López M.A.: Semi-Infinite Programming: Recent Advances. Kluwer, Dordrecht (2001)

    Google Scholar 

  7. Gustafson S.A.: On numerical analysis in semi-infinite programming. In: Hettich, R. (eds) Semi-Infinite Programming, Lecture Notes in Control and Information Science, vol 15, pp. 51–65. Springer, New York (1979)

    Google Scholar 

  8. Gustafson S.A.: A three-phase algorithm for semi-infinite programming. In: Fiacco, A.V., Kortanek, K.O. (eds) Semi-Infinite Programming and Applications, pp. 138–157. Springer, Berlin (1983)

    Google Scholar 

  9. Hettich R., Kortanek K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Review 35, 380–429 (1993)

    Article  Google Scholar 

  10. Hettich R., Zencke P.: Numerische Methoden der Approximation und Semi-infinite Optimierung. Teubner, Stuttgart (1982)

    Google Scholar 

  11. Ito S., Lin Y., Teo K.L.: A dual parametrization method for convex semi-infinite programming. Ann. Oper. Res. 98, 189–214 (2000)

    Article  Google Scholar 

  12. Kanzow C., Qi H.D.: A QP-free constrained Newton-type method for variational inequality problems. Math. Prog. 85, 81–106 (1999)

    Article  Google Scholar 

  13. Lai H.C., Wu S.Y.: On linear semi-infinite programming problems: an algorithm. Numer. Funct. Anal. Optim. 13, 287–304 (1992)

    Article  Google Scholar 

  14. Li D.H., Qi L., Tam J., Wu S.Y.: A smoothing Newton method for semi-infinite programming. J. Global Optim. 30, 169–194 (2004)

    Article  Google Scholar 

  15. Lin C.J., Fang S.C., Wu S.Y.: A dual affine scaling based algorithm for solving linear semi-infinite programming problems. In: Du, D.Z., Sun, J. (eds) Advances in Optimization and Approximation, pp. 217–233. Kluwer, London (1994)

    Google Scholar 

  16. Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977)

    Article  Google Scholar 

  17. Moré J.J., Sorensen D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4, 553–572 (1983)

    Article  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Pang J.S., Qi L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)

    Article  Google Scholar 

  20. Polak E., Tits A.L.: A recursive quadratic programming algorithm for semi-infinite programming problems. Appl. Math. Optim. 8, 325–349 (1982)

    Article  Google Scholar 

  21. Qi L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  Google Scholar 

  22. 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)

    Article  Google Scholar 

  23. Qi L., A. Shapiro, Ling C.: Differentiability and semismoothness properties of integral functions and their applications. Math. Prog. 102, 223–248 (2005)

    Article  Google Scholar 

  24. Qi L., Sun D., Zhou G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Prog. 87, 1–35 (2000)

    Google Scholar 

  25. Qi L., Sun J.: A nonsmooth version of Newton’s method. Math. Prog. 58, 353–367 (1993)

    Article  Google Scholar 

  26. Qi L., Wu S.Y., Zhou G.: Semismooth Newton methods for solving semi-infinite programming problems. J. Global Optim. 27, 215–232 (2003)

    Article  Google Scholar 

  27. Roleff K.: A stable multiple exchange algorithm for linear SIP. In: Hettich, R. (eds) Semi-Infinite Programming, Lecture Notes in Control and Information Science, vol 15, pp. 83–96. Springer, New York (1979)

    Google Scholar 

  28. Rückmann J.J., Shapiro A.: Second order optimality conditions in generalized semi-infinite programming. Set-Valued Anal. 9, 169–186 (2001)

    Article  Google Scholar 

  29. Shapiro A.: First and second order optimality conditions and perturbation analysis of semi-infinite programming problems. In: Reemtsen, R., Rükmann, J. (eds) Semi-Infinite Programming, pp. 103–133. Kluwer, Boston (1998)

    Google Scholar 

  30. Stein O., Tezel A.: The semismooth approach for semi-infinite programming under the reduction Ansatz. J. Global Optim. 41, 245–266 (2008)

    Article  Google Scholar 

  31. Still G.: Discretization in semi-infinite programming: the rate of convergence. Math. Prog. 91, 53–69 (2001)

    Google Scholar 

  32. Sheu R.L., Wu S.Y., Fang S.C.: A primal-dual infeasible-interior-point algorithm for linear semi-infinite programming. Comput. Math. Applic. 29, 7–18 (1995)

    Article  Google Scholar 

  33. Tanaka Y., Fukushima M., Ibaraki T.: A globally convergent SQP method for semi-infinite nonlinear optimization. J. Comp. Appl. Math. 23, 141–153 (1988)

    Article  Google Scholar 

  34. Teo K.L., Rehbock V., Jennings L.S.: A new computational algorithm for functional inequality constrained optimization problems. Automatica 29, 789–792 (1993)

    Article  Google Scholar 

  35. Teo K.L., Yang X.Q., Jennings L.S.: Computational discretization algorithms for functional inequality constrained optimization. Ann. Oper. Res. 98, 215–234 (2000)

    Article  Google Scholar 

  36. Todd M.J.: Interior-point algorithms for semi-infinite programming. Math. Prog. 65, 217–245 (1994)

    Article  Google Scholar 

  37. Watson G.A.: A multiple exchange algorithm for multivariate Chebyshev approximation. SIAM J. Numer. Anal. 12, 46–52 (1975)

    Article  Google Scholar 

  38. Watson G.A.: Numerical experiments with globally convergent methods for semi-infinite programming problems. In: Fiacco, A.V., Kortanek, K.O. (eds) Semi-Infinite Programming and Applications, pp. 193–205. Springer, Berlin (1983)

    Google Scholar 

  39. Wu S.Y., Li D.H., Qi L., Zhou G.: An iterative method for solving KKT system of the semi-infinite programming. Optim. Methods Softw. 20, 629–643 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhejiang University of Finance and Economics, 310018, Hangzhou, China

    Chen Ling

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, China

    Qin Ni

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Chen Ling & Liqun Qi

  4. Department of Mathematics, National Cheng Kung University/National Center for Theoretical Sciences, Tainan, Taiwan

    Soon-Yi Wu

Authors
  1. Chen Ling
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  2. Qin Ni
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  3. Liqun Qi
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  4. Soon-Yi Wu
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Corresponding author

Correspondence to Liqun Qi.

Additional information

Chen Ling work was supported by the National Natural Science Foundation of China (10871168), the Zhejiang Provincial National Science Foundation of China (Y606168) and a Hong Kong Polytechnic University Postdoctoral Fellowship. Qin Ni work was supported by the National Natural Science Foundation of China (grant 10471062). Liqun Qi work was supported by the Hong Kong Research Grant Council (Grant PolyU 102307) and a Chair Professor Fund of the Hong Kong Polytechnic University.

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Ling, C., Ni, Q., Qi, L. et al. A new smoothing Newton-type algorithm for semi-infinite programming. J Glob Optim 47, 133–159 (2010). https://doi.org/10.1007/s10898-009-9462-7

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  • DOI: https://doi.org/10.1007/s10898-009-9462-7

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