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The Penalty Functions Method and Multiplier Rules Based on the Mordukhovich Subdifferential

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Abstract

We show that the finite-dimensional Fritz John multiplier rule, which is based on the limiting/Mordukhovich subdifferential, can be proved by using differentiable penalty functions and the basic calculus tools in variational analysis. The corresponding Kuhn–Tucker multiplier rule is derived from the Fritz John multiplier rule by imposing a constraint qualification condition or the exactness of an ℓ1 penalty function. Complementing the existing proofs, our proofs provide another viewpoint on the fundamental multiplier rules employing the Mordukhovich subdifferential.

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References

  1. Borwein, J.M., Fitzpatrick, S.: Weak-star sequential compactness and bornological limit derivatives. J. Convex Anal. 2, 59–68 (1995)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley & Sons, Inc., New York (1983)

    MATH  Google Scholar 

  4. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  6. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  7. Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B, 116, 369–396 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mordukhovich, B.S., Wang, B.: Necessary suboptimality and optimality conditions via variational principles. SIAM J. Control Optim. 41, 623–640 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)

    Google Scholar 

  11. Ruszczynski, A.P.: Nonlinear Optimization. Princeton University Press, New Jersey (2006)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    Nguyen Thi Van Hang

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  1. Nguyen Thi Van Hang
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Correspondence to Nguyen Thi Van Hang.

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Van Hang, N.T. The Penalty Functions Method and Multiplier Rules Based on the Mordukhovich Subdifferential. Set-Valued Var. Anal 22, 299–312 (2014). https://doi.org/10.1007/s11228-013-0260-5

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  • DOI: https://doi.org/10.1007/s11228-013-0260-5

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