Abstract
Mathematical programs with vanishing constraints constitute a new class of difficult optimization problems with important applications in optimal topology design of mechanical structures. Vanishing constraints usually violate standard constraint qualifications, which gives rise to serious difficulties in theoretical and numerical treatment of these problems. In this work, we suggest several globalization strategies for the active-set Newton-type methods developed earlier by the authors for this problem class, preserving superlinear convergence rate of these methods under weak assumptions. Preliminary numerical results demonstrate that our approach is rather promising and competitive with respect to the existing alternatives.
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References
Achtziger, W., Kanzow, C.: Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math. Program. 114, 69–99 (2007)
Achtziger, W., Hoheisel, T., Kanzow, C.: A smoothing-regularization approach to mathematical programs with vanishing constraints. Preprint 284, Institute of Mathematics, University of Würzburg (2008)
Bonnans, J.F.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994)
Chen, Y.D., Gao, Y., Liu, Y.-J.: An inexact SQP Newton method for convex SC1 minimization problems. J. Optim. Theory Appl. 146, 33–49 (2010)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
De Luca, T., Facchinei, F., Kanzow, C.: A theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Optim. Appl. 16, 173–205 (2000)
Dolan, E., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Facchinei, F., Fischer, A., Kanzow, C.: Regularity properties of a semismooth reformulation of variational inequalities. SIAM J. Optim. 8, 850–869 (1998)
Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM J. Optim. 9, 14–32 (1999)
Fischer, A.: Local behaviour of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94, 91–124 (2002)
Hager, W.W., Gowda, M.S.: Stability in the presence of degeneracy and error estimation. Math. Program. 85, 181–192 (1999)
Hoheisel, T., Kanzow, C.: First- and second-order optimality conditions for mathematical programs with vanishing constraints. Appl. Math. 52, 495–514 (2007)
Hoheisel, T., Kanzow, C.: Stationarity conditions for mathematical programs with vanishing constraints using weak constraint qualifications. J. Math. Anal. Appl. 337, 292–310 (2008)
Hoheisel, T., Kanzow, C.: On the Abadie and Guignard constraint qualifications for mathematical programs with vanishing constraints. Optimization 58, 431–448 (2009)
Hoheisel, T., Kanzow, C., Outrata, J.: Exact penalty results for mathematical programs with vanishing constraints. Nonlinear Anal. 72, 2514–2526 (2010)
Hoheisel, T., Kanzow, C., Schwartz, A.: Convergence of a local regularization approach for mathematical programs with complementarity or vanishing constraints. Optim. Methods Softw. doi:10.1080/10556788.2010.535170
Izmailov, A.F., Pogosyan, A.L.: Optimality conditions and Newton-type methods for mathematical programs with vanishing constraints. Comput. Math. Math. Phys. 49, 1128–1140 (2009)
Izmailov, A.F., Pogosyan, A.L.: On active-set methods for mathematical programs with vanishing constraints. In: Bereznyov, V.A. (ed.) Theoretical and Applied Problems of Nonlinear Analysis, pp. 18–49. CCAS, Moscow (2009) (In Russian)
Izmailov, A.F., Solodov, M.V.: An active-set Newton method for mathematical programs with complementarity constraints. SIAM J. Optim. 19, 1003–1027 (2008)
Izmailov, A.F., Solodov, M.V.: Mathematical programs with vanishing constraints: optimality conditions, sensitivity, and a relaxation method. J. Optim. Theory Appl. 142, 501–532 (2009)
Izmailov, A.F., Pogosyan, A.L., Solodov, M.V.: Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints. Comput. Optim. Appl. 51, 199–221 (2012)
Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge Univ. Press, Cambridge (1996)
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)
Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic, Boston (1998)
Tseng, P.: Growth behavior of a class of merit functions for the nonlinear complementarity problem. J. Optim. Theory Appl. 89, 17–37 (1996)
Acknowledgements
The authors thank the anonymous referees for useful comments. In particular, Remark 4.1 is in response to the question raised by one of the referees.
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This research is supported by the Russian Foundation for Basic Research Grant 10-01-00251.
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Izmailov, A.F., Pogosyan, A.L. Active-set Newton methods for mathematical programs with vanishing constraints. Comput Optim Appl 53, 425–452 (2012). https://doi.org/10.1007/s10589-012-9467-x
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DOI: https://doi.org/10.1007/s10589-012-9467-x