An Improvement of the Lagrange Multiplier Rule for Smooth Optimization Problems
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The famous Lagrange multiplier rule was introduced in “Lagrange, J. L., Mécanique analytique I-II, Paris, 1788” for minimizing a function subject to equality constraints, and thus, for finding the stable equilibrium of a mechanical system. Since then, a great many applications of wide variety have been published in the fields of theory (e.g., physics, mathematics, operations research, management science etc.), methodology and practice without changing its scope from the point of view of mathematics. (Prékopa 1980) stated that “This ingenious method was unable to attract mathematics student and teachers for a long time. The method of presentation used nowadays by many instructors is to prove the necessary condition first in the case of inequality constraints, give a geometric meaning to this, and refer to the case of equality constraints. This can make the students more enthusiastic about this theory.” Here, an improvement of the sufficiency part of the Lagrange multiplier rule is presented. That is to say that a result stronger than that of Lagrange, with respect to the sufficiency part, can be proved under the same type of conditions he used, based on a different, perhaps a deeper mathematical investigation. Moreover, the geometric meaning of the Lagrange theorem is brought into the limelight.
KeywordsRiemannian Manifold Inequality Constraint Lagrangian Function Global Minimum Point Strict Local Minimum
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