Abstract
Another formulation and method of solving weak constraint problem in variational objective analysis is shown. Weak constraint problem is solved by introducing new variables and imposing strong constraints which are the relations between old variables and new variables. The Euler-Lagrange equation obtained by the proposed formulation of weak constraint differ a little in an additional perturbation term from those obtained by strong constraint, while those obtained by the conventional formulation of weak constraint differ much from those obtained by strong constraint. Then this method of solving weak constraint problem serves a practical purpose.
Weak constraints are viewed as transformation of old variables into new variables, the weights of which are given in the weighted least square error expression. The formulation of weak constraint shows the relationship between weak constaint and strong constraint more clearly than the conventional formulation of weak constraint. Especially when the weight of weak constraint becomes larger, in other words, weak constraint becomes stronger, the alternative solving method shows more naturally that the solution of weak constraint tends to the solution of strong constraint than the conventional one does.
It also gives a unified expression of weak and strong constraints. The unified expression of weak and strong constraints gives us flexibility in solving some problems, and we can change weak constraint into strong constraint and vice versa in the expression.
