Set-Valued and Variational Analysis

, Volume 22, Issue 2, pp 299–312 | Cite as

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

  • Nguyen Thi Van HangEmail author


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.


Finite-dimensional nondifferentiable mathematical programming problem Multiplier rule Penalty function method Mordukhovich subdifferential Basic calculus tools 

AMS Subject Classifications (2000)

90C47 49J52 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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