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A quadratic minimization problem is considered in Hilbert spaces under constraints given by a linear operator equation and a convex quadratic inequality. The main feature of the problem statement is that the practically available approximations to the exact linear operators specifying the criterion and the constraints converge to them only strongly pointwise rather than in the uniform operator norm, which makes it impossible to justify the use of the classical regularization methods. We propose a regularization method that is applicable in the presence of error estimates for approximate operators in pairs of other operator norms, which are weaker than the original ones. For each of the operators, the pair of corresponding weakened operator norms is obtained by strengthening the norm in the domain of the operator and weakening the norm in its range. The weakening of operator norms usually makes it possible to estimate errors in operators where this was fundamentally impossible in the original norms, for example, in the finite-dimensional approximation of a noncompact operator. From the original optimization formulation, a transition is made to the problem of finding a saddle point of the Lagrange function. The proposed numerical method for finding a saddle point is an iterative regularized extragradient two-stage procedure. At the first stage of each iteration, an approximation to the optimal value of the criterion is refined; at the second stage, the approximate solution with respect to the main variable is refined. Compared to the methods previously developed by the authors and working under similar information conditions, this method is preferable for practical implementation, since it does not require the gradient step size to converge to zero. The main result of the work is the proof of the strong convergence of the approximations generated by the method to one of the exact solutions to the original problem in the norm of the original space. [ABSTRACT FROM AUTHOR]

A quadratic minimization problem is considered in Hilbert spaces in the presence of a linear operator constraint of the equality type and a quadratic constraint of the inequality type. The problem is reformulated as the problem of finding the saddle point of the normal Lagrange function. For the numerical solution of this problem, a regularized gradient-type method is proposed that iterates both in primal and in dual variables. Approximate solutions are constructed under nonclassical information conditions, when the approximations to the exact operators from the problem statement available to the computer approximate these operators only strongly pointwise and there are no corresponding error estimates in the operator norms of the original spaces. Instead, a priori information is used about the error levels that become accessible when the norms are changed in the domains or ranges of the operators. Estimates of the first type appear after strengthening norms in the domain spaces of the operators, and estimates of the second type appear after weakening norms in their range spaces. At each iteration of the proposed method, two main actions are performed. First, the next approximation to the minimum value of the functional is computed using error estimates of the first type. Then, the next approximation to the optimal solution is found using error estimates of the second type. It is proved that the approximations generated by the proposed method converge to the solution of the original optimization problem in the norm of the original space. [ABSTRACT FROM AUTHOR]

Abstract: For a pair of dual inconsistent linear programming problems, the existence and uniqueness of a correction vector that is optimal in the norm is proved. It is shown that the correction problem is reduced to the problem of finding a saddle point of a regularized Lagrange function. A modified extragradient method is proposed for solving the latter problem, and its convergence is proved. [ABSTRACT FROM AUTHOR]