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Smooth exact penalty functions: a general approach

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Abstract

In the article, we present a new perspective on the method of smooth exact penalty functions that is becoming more and more popular tool for solving constrained optimization problems. In particular, our approach to smooth exact penalty functions allows one to apply previously unused tools (namely, parametric optimization) to the study of these functions. We give a new simple proof of local exactness of smooth penalty functions that significantly generalizes all similar results existing in the literature. We also provide new necessary and sufficient conditions for a smooth penalty function to be globally exact.

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References

  1. Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM. J. Control Optim. 27, 1333–1360 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burke, J.V.: An exact penalization viewpoint on constrained optimization. SIAM. J. Control Optim. 29, 968–998 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Demyanov, V.F.: Nonsmooth optimization. In: Di Pillo, G., Schoen, F. (eds.) Nonlinear Optimization. Lecture Notes in Mathematics, vol. 1989, pp. 55–164. Springer, Heidelberg (2010)

  4. Huyer, W., Neumaier, A.: A new exact penalty function. SIAM. J. Optim. 13, 1141–1158 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bingzhuang, L., Wenling, Z.: A modified exact smooth penalty function for nonlinear constrained optimization. J. Inequal. Appl. 1, 173 (2012)

    Article  MathSciNet  Google Scholar 

  6. Wang, C., Ma, C., Zhou, J.: A new class of exact penalty functions and penalty algorithms. J. Glob. Optim. 58, 51–73 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ma, C., Li, X., Cedric Yiu, K.-F., Zhang, L.-S.: New exact penalty function for solving constrained finite min-max problems. Appl. Math. Mech. Engl. Ed. 33, 253–270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lin, Q., Loxton, R., Teo, K.L., Wu, Y.H., Yu, C.: A new exact penalty method for semi-infinite programming problems. J. Comput. Appl. Math. 261, 271–286 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, B., Yu, C.J., Teo, K.L., Duan, G.R.: An exact penalty function method for continuous inequality constrained optimal control problem. J. Optim. Theory. Appl. 151, 260–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jiang, C., Lin, Q., Yu, C., Teo, K.L., Duan, G.-R.: An exact penalty method for free terminal time optimal control problem with continuous inequality constraints. J. Optim. Theory. Appl. 154, 30–53 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lin, Q., Loxton, R., Teo, K.L., Wu, Y.H.: Optimal feedback control for dynamic systems with state constraints: an exact penalty approach. Optim. Lett. 8, 1535–1551 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Clarke, F.H.: A new approach to lagrange multipliers. Math. Oper. Res. 1, 165–174 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Clarke, F.H.: Optimization nonsmooth analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  14. Burke, J.V.: Calmness and exact penalization. SIAM. J. Control Optim. 29, 493–497 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Uderzo, A.: Exact penalty functions and calmness for mathematical programming under nonlinear perturbations. Nonlinear. Anal. 73, 1596–1609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Penot, J.-P.: Calmness and stability properties of marginal and performance functions. Numer. Funct. Anal. Optim. 25, 287–308 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Azé, D.: A: unified theory for metric regularity of multifunctions. J. Convex. Anal. 13, 225–252 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Kruger, A.Y.: Error bounds and metric subregularity. Optim. (2014). doi:10.1080/02331934.2014.938074

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Acknowledgments

The author was supported by the Russian Foundation for Basic Research (Project No. 14-01-31521 mol_a) and Saint Petersburg State University (Project No. 9.38.205.2014).

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Correspondence to Maksim V. Dolgopolik.

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Dolgopolik, M.V. Smooth exact penalty functions: a general approach. Optim Lett 10, 635–648 (2016). https://doi.org/10.1007/s11590-015-0886-3

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  • DOI: https://doi.org/10.1007/s11590-015-0886-3

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