Exact solution of the $A_{n-1}^{(1)}$ trigonometric vertex model with non-diagonal open boundaries

The $A_{n-1}^{(1)}$ trigonometric vertex model with {\it generic non-diagonal} boundaries is studied. The double-row transfer matrix of the model is diagonalized by algebraic Bethe ansatz method in terms of the intertwiner and the corresponding face-vertex relation. The eigenvalues and the corresponding Bethe ansatz equations are obtained.