On the role of Lagrange multipliers and duality in ill-posed problems for constrained extremum. To the 60th anniversary of the Tikhonov regularization method

The important role of Lagrange multipliers and duality in the theory of ill-posed problems for a constrained extremum is discussed. The central attention is paid to the problem of stable approximate finding of a normal (minimum in norm) solution of the operator equation of the first kind $Az=u,$ $z\in {\mathcal D}\subseteq Z,$ where $A:\,Z\to U$ is a linear bounded operator, $u\in U$ is a given element, ${\mathcal D}\subseteq Z$ is a convex closed set, $Z,U$ are Hilbert spaces. As is known, this problem is classical for the theory of ill-posed problems. We consider two problems equivalent to it (from the point of view of the simultaneous existence of their unique solutions) for a constrained extremum, the first of which is the problem ($CE1$) with a functional inequality constraint $\|z\|^2\to\min,$ $\|Az-u\|^2\leq 0,$ $z\in {\mathcal D},$ and the second is the problem ($CE2$) with operator equality constraint $\|z\|^2\to\min,$ $Az=u,$ $z\in {\mathcal D}.$ First of all, we show that Tikhonov's regularization method can be naturally interpreted as a method of stable approximation of the exact solution by extremals of the Lagrange functional for problem ($CE1$) with simultaneous construction of a maximizing sequence of Lagrange multipliers in its dual problem. In this case, the Lagrange multiplier is the reciprocal of the regularization parameter in the Tikhonov method. In other words, the convergence theorem of the Tikhonov regularization method is given the form of a statement in the form of duality with respect to the problem ($CE1$). Next, we discuss the role of Tikhonov stabilization for general convex problems in solving problems for constrained extremum and a stable method based on Tikhonov stabilization of the problem dual to ($CE2$) for solving the original operator equation, which can be considered as a regularization method for the Lagrange multiplier rule for the problem ($CE2$). The paper discusses the features of each of the two above mentioned approaches to the regularization of solving the original operator equation.