Boundary value inverse problems deal with the estimation of unknown boundary values on incompletely prescribed boundaries using over-prescribed boundary values on other boundaries or in the domain. Like many other inverse problems, this kind of inverse problem is usually ill-conditioned. In the present paper, a regularization scheme was proposed for solving the ill-conditioned boundary value inverse problems. The scheme was based on a multivariable constrained optimization algorithm for determining the most plausible solution satisfying inequality constraints deduced from information on unknown variables available in advance. To demonstrate the applicability of the proposed scheme, it was applied to a boundary value inverse problem with trusslike structures. The nonpositiveness or nonnegativeness of unknown variables was used as the constraint. It was found that the scheme using the constraint was effective in obtaining reasonable estimates of the unknown boundary values and was rather insensitive to error in input data, while the unconstrained scheme was not.