Abstract
We present an elementary proof of the Karush–Kuhn–Tucker Theorem for the problem with nonlinear inequality constraints and linear equality constraints. Most proofs in the literature rely on advanced optimization concepts such as linear programming duality, the convex separation theorem, or a theorem of the alternative for systems of linear inequalities. By contrast, the proof given here uses only basic facts from linear algebra and the definition of differentiability.
Similar content being viewed by others
References
Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming: Theory and Algorithms, 2nd edn. Wiley, New York (1993)
Beltrami E.J.: A constructive proof of the Kuhn–Tucker multiplier rule. J. Math. Anal. Appl. 26, 297–306 (1969)
Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Boston (1995)
Birbil S.I., Frenk J.B.G., Still G.J.: An elementary proof of the Fritz–John and Karush–Kuhn–Tucker conditions in nonlinear programming. Eur. J. Oper. Res. 180, 479–484 (2007)
Karush, W.: Minima of Functions of Several Variables with Inequalities as Side Conditions. Master’s thesis. Department of Mathematics, University of Chicago (1939)
Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1950)
Luenberger D.G.: Introduction to Linear and Nonlinear Programming. Addison-Wesley, Reading (1973)
McShane E.J.: The Lagrange multiplier rule. Am. Math. Monthly 80, 922–925 (1973)
Pourciau B.H.: Modern multiplier rules. Am. Math. Monthly 87, 433–452 (1980)
Prékopa A.: On the development of optimization theory. Am. Math. Monthly 87, 527–542 (1980)
Rockafellar R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brezhneva, O.A., Tret’yakov, A.A. & Wright, S.E. A simple and elementary proof of the Karush–Kuhn–Tucker theorem for inequality-constrained optimization. Optim Lett 3, 7–10 (2009). https://doi.org/10.1007/s11590-008-0096-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-008-0096-3