This paper presents a method for finding an $$L_q$$ L q -closest-point to a set of affine subspaces, that is a point for which the sum of the q-th power of orthogonal distances to all the subspaces is minimized, where $$1 \le q < 2$$ 1 ≤ q < 2 . We give a theoretical proof for the convergence of the proposed algorithm to a unique $$L_q$$ L q minimum. The proposed method is motivated by the $$L_q$$ L q Weiszfeld algorithm, an extremely simple and rapid averaging algorithm, that finds the $$L_q$$ L q mean of a set of given points in a Euclidean space. The proposed algorithm is applied to the triangulation problem in computer vision by finding the $$L_q$$ L q -closest-point to a set of lines in 3D. Our experimental results for the triangulation problem confirm that the $$L_q$$ L q -closest-point method, for $$1 \le q < 2$$ 1 ≤ q < 2 , is more robust to outliers than the $$L_2$$ L 2 -closest-point method.