Abstract
In this chapter we investigate optimization problems with constraints in the form of inequalities and equalities. For such constrained problems we formulate a multiplier rule as a necessary optimality condition and we give assumptions under which this multiplier rule is also a sufficient optimality condition. The optimality condition presented generalizes the known multiplier rule published by Lagrange in 1797. With the aid of this optimality condition we deduce then the Pontryagin maximum principle known from control theory.
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Notes
- 1.
S.M. Robinson, “Stability theory for systems of inequalities in nonlinear programming, part II: differentiable nonlinear systems”, SIAM J. Numer. Anal. 13 (1976) 497–513.
J. Zowe and S. Kurcyusz, “Regularity and stability for the mathematical programming problem in Banach spaces”, Appl. Math. Optim. 5 (1979) 49–62.
- 2.
F. John, “Extremum problems with inequalities as side conditions”, in: K.O. Friedrichs, O.E. Neugebauer and J.J. Stoker (eds.), Studies and Essays, Courant Anniversary Volume (Interscience, New York, 1948).
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W.E. Karush, Minima of functions of several variables with inequalities as side conditions (Master’s Dissertation, University of Chicago, 1939).
H.W. Kuhn and A.W. Tucker, “Nonlinear programming”, in: J. Neyman (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, 1951), p. 481–492.
References
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Jahn, J. (2020). Generalized Lagrange Multiplier Rule. In: Introduction to the Theory of Nonlinear Optimization. Springer, Cham. https://doi.org/10.1007/978-3-030-42760-3_5
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DOI: https://doi.org/10.1007/978-3-030-42760-3_5
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