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Constrained ordered weighted averaging (OWA) aggregation attempts to solve the OWA optimization problem subject to multiple constraints. The problem is nonlinear in nature due to the reordered variables of arguments in the objective function, and the solution approach via mixed integer linear programming is quite complex even in the problem with one restriction of which coefficients are all one. Recently, this has been relaxed to allow a constraint with variable coefficients but the solution approach is still abstruse. In this paper, we present a new intuitive method to constructing a problem with auxiliary symmetric constraints to convert it into linear programming problem. The side effect is that we encounter many small sub-problems to be solved. Interestingly, however, we discover that they share common symmetric features in the extreme points of the feasible region of each sub-problem. Consequently, we show that the structure of extreme points and the reordering process of input arguments peculiar to the OWA operator lead to a closed optimal solution to the constrained OWA optimization problem. Further, we extend our findings to the OWA optimization problem constrained by a range of order-preserving constraints and present the closed optimal solutions.