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DOI: 10.23952/cot.2026.9
Received August 14, 2024; Accepted February 6, 2025; Published online January 26, 2026
Abstract. The regularization of the Lagrange multiplier rule (LMR) in nondifferential form in a convex constrained extremum problem (CEP) with an operator equality constraint in a Hilbert space and a finite number of functional inequality constraints is discussed. The set of admissible elements of the problem under consideration also belongs to a Hilbert space, and its constraints contain additively included parameters, which makes it possible to apply the so-called perturbation method to its study. The main purpose of the regularized LMR is the stable generation of generalized minimizing sequences (GMS) that approximate the exact solution of the problem by means of extremals of the regular Lagrange functional. The regularized LMR can be interpreted as an GMS-generating (regularizing) operator that assigns to each set of input data of the CEP the extremal of its regular Lagrange functional corresponding to this set. In this case, the dual variable in the Lagrange functional is generated in accordance with one or another procedure for stabilizing the dual problem. The main attention in the paper is paid to the discussion of: 1) problems associated with the ill-posedness properties of the classical LMR, as well as its applicability for solving CEPs; 2) the procedure of the dual regularization and its connection with the regularization of the LMR; 3) the procedure for obtaining the classical LMR as a limiting version of its regularized analogue; 4) the possibilities of using the regularized LMR to solve current ill-posed problems. The set of properties of the regularized LMR discussed in the paper allows us to speak about the universality of its classical analogue. On the one hand, the classical LMR is the generally recognized core of the entire theory of extremal problems, and on the other hand, it constitutes the fundamental basis for constructing stable algorithms for solving many ill-posed problems, including extremal ones.
How to Cite this Article:
M. Sumin, On regularization of the Lagrange multiplier rule in convex constrained extremum problems and on its universality, Commun. Optim. Theory 2026 (2026) 9.