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A semismooth sequential quadratic programming method for lifted mathematical programs with vanishing constraints

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Abstract

Mathematical programs with vanishing constraints are a difficult class of optimization problems with important applications to optimal topology design problems of mechanical structures. Recently, they have attracted increasingly more attention of experts. The basic difficulty in the analysis and numerical solution of such problems is that their constraints are usually nonregular at the solution. In this paper, a new approach to the numerical solution of these problems is proposed. It is based on their reduction to the so-called lifted mathematical programs with conventional equality and inequality constraints. Special versions of the sequential quadratic programming method are proposed for solving lifted problems. Preliminary numerical results indicate the competitiveness of this approach.

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References

  1. W. Achtziger and C. Kanzow, “Mathematical Programs with Vanishing Constraints: Optimality Conditions and Constraint Qualifications,” Math. Program. 114(1), 69–99 (2007).

    Article  MathSciNet  Google Scholar 

  2. W. Achtziger, T. Hoheisel, and C. Kanzow, “A Smoothing-Regularization Approach to Mathematical Programs with Vanishing Constraints,” 10.1080/10556788.2010.535170. Preprint No. 284 (Inst. Math. Univ. Würzburg, Würzburg, 2008).

    Google Scholar 

  3. T. Hoheisel and C. Kanzow, “First- and Second-Order Optimality Conditions for Mathematical Programs with Vanishing Constraints,” Appl. Math. 52, 495–514 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Hoheisel and C. Kanzow, “Stationarity Conditions for Mathematical Programs with Vanishing Constraints Using Weak Constraint Qualifications,” J. Math. Anal. Appl. 337, 292–310 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Hoheisel and C. Kanzow, “On the Abadie and Guignard Constraint Qualifications for Mathematical Programs with Vanishing Constraints,” Optimization 58, 431–448 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. F. Izmailov and M. V. Solodov, “Mathematical Programs with Vanishing Constraints: Optimality Conditions, Sensitivity, and a Relaxation Method,” J. Optim. Theory Appl. 142, 501–532 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. F. Izmailov and A. L. Pogosyan, “Optimality Conditions and Newton-Type Methods for Mathematical Programs with Vanishing Constraints,” Zh. Vychisl. Mat. Mat. Fiz. 49, 1184–1196 (2009) [Comput. Math. Math. Phys. 49, 1128–1140 (2009)].

    MathSciNet  MATH  Google Scholar 

  8. A. F. Izmailov and A. L. Pogosyan, “On Active-Set Methods for Mathematical Programs with Vanishing Constraints,” in Theoretical and Applied Problems in Nonlinear Analysis (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2009), pp. 18–49 [in Russian].

    Google Scholar 

  9. T. Hoheisel, C. Kanzow, and A. Schwartz, “Convergence of a Local Regularization Approach for Mathematical Programs with Complementarity or Vanishing Constraints,” Optim. Methods Software. DOI: 10.1080/10556788.2010.535170.

  10. Z.-Q. Luo, J.-S. Pang, and D. Ralph, Mathematical Programs with Equilibrium Constraints (Cambridge Univ. Press, Cambridge, 1996).

    Google Scholar 

  11. J. V. Outrata, M. Kocvara, and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications, and Numerical Results (Kluwer, Boston, 1998).

    MATH  Google Scholar 

  12. A. F. Izmailov, “Mathematical Programs with Complementarity Constraints: Regularity, Optimality Conditions, and Sensitivity,” Zh. Vychisl. Mat. Mat. Fiz. 44, 1209–1228 (2004) [Comput. Math. Math. Phys. 44, 1145–1164 (2004)].

    MathSciNet  MATH  Google Scholar 

  13. A. F. Izmailov, Sensitivity in Optimization (Fizmatlit, Moscow, 2006) [in Russian].

    Google Scholar 

  14. O. Stein, “Lifting Mathematical Programs with Complementarity Constraints,” Math. Program. (2010). DOI 10/1007/s10107-010-0345-y..

  15. A. F. Izmailov, A. L. Pogosyan, and M. V. Solodov, “Semismooth Newton Method for the Lifted Reformulation of Mathematical Programs with Complementarity Constraints,” Comput. Optim. Appl. DOI 10.1007/s10589-010-9341-7.

  16. A. F. Izmailov and M. V. Solodov, Numerical Optimization Methods, 2nd. ed. (Fizmatlit, Moscow, 2008) [in Russian].

    MATH  Google Scholar 

  17. L. Qi, “Superlinearly Convergent Approximate Newton Methods for LC1 Optimization Problems,” Math. Program. 64, 277–294 (1994).

    Article  MATH  Google Scholar 

  18. J. Han and D. Sun, “Superlinear Convergence of Approximate Newton Methods for LC1 Optimization Problems without Strict Complementarity,” Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1993), pp. 353–367.

    Google Scholar 

  19. A. F. Izmailov, A. L. Pogosyan, and M. V. Solodov, Preprint A 675/2010/IMPA (Rio de Janeiro, 2010) (available at http://www.preprint.impa.br:80/Shadows/SERIE A/2010/675.html).

  20. J. F. Bonnans, “Local Analysis of Newton-Type Methods for Variational Inequalities and Nonlinear Programming,” Appl. Math. Optim. 29, 161–186 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, New York, 2006).

    MATH  Google Scholar 

  22. D. Fernández, A. F. Izmailov, and M. V. Solodov, “Sharp Primal Superlinear Convergence Results for Some Newtonian Methods for Constrained Optimization,” SIAM J. Optim. 20, 3312–3334 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Tseng, “Growth Behavior of a Class of Merit Functions for the Nonlinear Complementarity Problem,” J. Optim. Theory Appl. 89, 17–37 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  24. J. F. Bonnans, J. Ch. Gilbert, C. Lemaréchal, and C. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects (Springer-Verlag, Berlin, 2006).

    MATH  Google Scholar 

  25. E. Dolan and J. Moré, “Benchmarking Optimization Software with Performance Profiles,” Math. Program. 91, 201–213 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. N. Dar’ina and A. F. Izmailov, “Semismooth Newton Method for Quadratic Programs with Bound Constraints,” Zh. Vychisl. Mat. Mat. Fiz. 49, 1785–1795 (2009) [Comput. Math. Math. Phys. 49, 1706–1716 (2009)].

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia

    A. F. Izmailov & A. L. Pogosyan

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  1. A. F. Izmailov
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  2. A. L. Pogosyan
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Correspondence to A. F. Izmailov.

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Original Russian Text © A.F. Izmailov, A.L. Pogosyan, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 6, pp. 983–1006.

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Izmailov, A.F., Pogosyan, A.L. A semismooth sequential quadratic programming method for lifted mathematical programs with vanishing constraints. Comput. Math. and Math. Phys. 51, 919–941 (2011). https://doi.org/10.1134/S0965542511060108

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