This paper considers the continuation problem for the Helmholtz equation. The solution to the original problem is reduced to solving the inverse problem with respect to the direct ( well-posed) problem. The inverse problem is formulated to clarify the boundary condition with the help of additional information about the solution of the direct problem. The inverse problem is written in operator form. The solution of the operator equation is reduced to the problem of minimizing the objective functional. The paper also examines issues of convergence of gradient methods for solving the inverse problem. An algorithm has been developed for solving the inverse problem using the theory of conjugate optimization and the Landweber method. Detailed calculations for obtaining the associated problem are presented. The results obtained show that the use of the theory of conjugate optimization and the Landweber method makes it possible to effectively solve inverse problems.