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A Variational Approach to Lagrange Multipliers

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Abstract

We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied.

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Notes

  1. Explicitly named only in the last century by Feynman and others, the principle states that the path taken in a mechanical system will be the one which is stationary with respect to the action (which of course must be specified) [3, 5].

  2. We are assuming that f is everywhere finite, if not we must also require that \(f({\hat{x}}) < +\infty \).

  3. The use of the term “primal” is much more recent than the term “dual” and was suggested by George Dantzig’s father Tobias when linear programming was being developed in the 1940s.

  4. In infinite dimensions, we also assume that f, g are lsc and that A is continuous.

  5. Actually this term arises because in the Boltzmann–Shannon case one is minimizing the negative of the entropy.

  6. Since \(\gamma _{C_1}(x)-1<0\) for \(x\in \mathrm{int}C_1\), \(\lambda \ne 0\).

References

  1. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)

    MATH  Google Scholar 

  2. Gale, D.: A geometric duality theorem with economic applications. Rev. Econ. Stud. 34, 19–24 (1967)

    Article  Google Scholar 

  3. Goldstine, H.H.: A History of the Calculus of Variations from the 17th through 19th Century. Springer, New York (1980)

    Book  MATH  Google Scholar 

  4. Gregory, J., Lin, C.: Constrained Optimization in the Calculus of Variations and Optimal Control Theory. Chapman and Hall, London (1992)

    Book  MATH  Google Scholar 

  5. Hanca, J., Taylor, E.F.: From conservation of energy to the principle of least action: a story line. Am. J. Phys. 72, 514–521 (2004)

    Article  Google Scholar 

  6. Weinstock, R.: Calculus of Variations with Applications to Physics and Engineering. Dover, Mineola (1974)

    MATH  Google Scholar 

  7. Benoist, J., Borwein, J.M., Popovici, N.: A characterization of quasiconvex vector-valued functions. Proc. AMS 131, 1109–1113 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borwein, J.M.: Generalized linear complementary problems treated without fixed point theorem. JOTA 43, 343–356 (1984)

    Article  MATH  Google Scholar 

  9. Borwein, J.M., Zhu, Q.J.: Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity. SIAM J. Control Optim. 34, 1568–1591 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borwein, J.M., Ioffe, A.D.: Proximal analysis in smooth Banach spaces. Set-valued Anal. 4, 1–24 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Spivak, M.: Calculus on Manifolds: A Modern Approach to Classical Theorem of Advanced Calculus, 4th edn. Publish or Perish, Berkeley (2008)

    Google Scholar 

  12. Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, Salt Lake City, UT (1973)

    MATH  Google Scholar 

  13. Giannessi, F.: Constrained Optimization and Image Space Analysis I. Springer, New York (2005)

    MATH  Google Scholar 

  14. Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Avakov, E.R., Magaril-Il’yaev, G.G., Tikhomirov, V.M.: Lagrange’s principle in extremum problems with constraints. Russian Math. Surv. 68, 401–433 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Co., Amsterdam (1979)

    Google Scholar 

  17. Magaril-Il’yaev, G.G., Tikhomirov, V.M.: Newton’s method, differential equations, and the Lagrangian principle for necessary extremum conditions. Proc. Steklov Inst. Math. 262, 149–169 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I, II. Springer, New York (2006)

    Google Scholar 

  19. Penot, J.-P.: Calculus Without Derivatives. Springer, New York (2014)

    MATH  Google Scholar 

  20. Borwein, J.M., Zhuang, D.: On Fan’s minimax theorem. Math. Program. 34, 232–234 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shapiro, A., Nemirovski, A.: Duality of linear conic problems. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA (2003). http://www.optimization-online.org/DB_HTML/2003/12/793.html

  22. Zǎlinescu, C.: On duality gaps in linear conic problems. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA (2010). http://www.optimization-online.org/DB_HTML/2010/09/2737.html

  23. Clarke, F.H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1983)

    Google Scholar 

  24. Demyanov, V.F., Vasilev, and L.V.: Nondifferentiable Optimization. (Translations Series in Mathematics and Engineering). Springer, New York (1986)

  25. Borwein, J.M., Treiman, J.S., Zhu, Q.J.: Necessary conditionsfor constrained optimization problem with semicontinuous andcontinuous data. Trans. AMS 350, 2409–2429 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Borwein, J.M., Zhu, Q.J.: A survey of subdifferential calculus with applications. Nonlinear Anal.: TMA 38, 687–773 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nghia, T.T.A.: A nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions. Nonlinear Anal. 75, 6379–6390 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhu, Q.J.: Necessary conditions for constrained optimization problem in smooth Banach spaces with applications. SIAM J. Optim. 12, 1032–1047 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mangasarian, O.L., Fromovitz, S.: The Fritz John necessary conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kruger, A.Y., Minchenko, L., Outrata, J.V.: On relaxing the Mangasarian–Fromovitz constraint qualification. Positivity 18, 171–189 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Borwein, J.M.: Convex relations in optimization and analysis. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Convexity inOptimization and Economics, pp. 335–377. Academic Press, London (1981)

    Google Scholar 

  32. Borwein, J.M.: Maximum entropy and feasibility methods for convex and non-convex inverse problems. Optimization 61, 1–33 (2012). doi:10.1080/02331934.2011.632502

    Article  MathSciNet  MATH  Google Scholar 

  33. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2 (2005) edn. Springer, New york (2000)

    Book  MATH  Google Scholar 

  34. Ioffe, A.D., Levin, V.L.: Subdifferentials of convex functions. Trudy Mosk. Mat. Obšč 26, 3–73 (1972)

    MathSciNet  MATH  Google Scholar 

  35. Borwein, J.M.: A very complicated proof of the minimax theorem. Minimax Theory and its Applications, vol. 1 (2015)

  36. Jameson, G.J.O.: Topology and Normed Spaces. Chapman and Hall, London (1974)

    MATH  Google Scholar 

  37. Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, Hoboken (1972)

    MATH  Google Scholar 

  38. Rockafellar, R.T., Wolenski, P.R.: Convexity in Hamilton–Jacobi theory 1: dynamics and duality. SIAM J. Control Optim. 40, 1323–1350 (2001)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The research is partially supported by an Australian Research Council Discovery Projects grant. We thank several of our colleagues for helpful input during the preparation of this work. We especially thank the referee whose incisive and detailed comments have substantially improved the final version of our manuscript.

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Authors and Affiliations

  1. Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, 2308, Australia

    Jonathan M. Borwein

  2. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Qiji J. Zhu

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  1. Jonathan M. Borwein
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  2. Qiji J. Zhu
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Correspondence to Qiji J. Zhu.

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Borwein, J.M., Zhu, Q.J. A Variational Approach to Lagrange Multipliers. J Optim Theory Appl 171, 727–756 (2016). https://doi.org/10.1007/s10957-015-0756-2

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