This paper discusses the crucial role of the Lagrange multiplier rule (LMR) in its non-differential form in the theory of constrained optimization. It provides a natural link between the familiar approach to classical optimality conditions within the theory of extremal problems, in which the initial data of problems are treated as “rigidly” specified mathematical objects, and an approach according to which we regard the same extremal problems as ill-posed, implying that their initial data may be specified with errors. The paper substantiates the validity of saying that the LMR in its non-differential form organically unites both directions of optimization theory, corresponding to the two approaches mentioned above. In the first case, under the classical approach, it serves as the familiar foundation of the entire theory of extremal problems. If we consider the ill-posedness of extremal problems, then the classical rule retains this role, but in a regularized form. Thus, we can speak about the universality of the classical LMR and obtain additional evidence in favor of its fundamental nature: the “internal potential” of the classical LMR turns out to be such that, with the appropriate constructive transformation-regularization, it is transformed into a universal means of stable solving ill-posed problems for constrained extremum.
Keywords: convex problem for constrained extremum, Lagrange multiplier rule, ill-posedness, regularization, perturbations method, value function, subdifferential, dual problem, generalized minimizing sequence, regularizing algorithm
Received January 12, 2026
Revised February 20, 2026
Accepted February 23, 2026
Mikhail Iosifovich Sumin, Dr. Phys.-Math. Sci., Prof., Leading Researcher, Derzhavin Tambov State University, Tambov, 392000 Russia, e-mail: m.sumin@mail.ru
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Cite this article as: M.I. Sumin On the universality of the Lagrange multiplier rule in convex problems for constrained extremum. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 242–258.
