A quasi-Riemannian approach is developed for constrained optimization in which the retraction and transport operators are only approximate. If n is the dimension of the admissible domain, and p the number of scalar equality constraints, the iteration is expressed in terms of a vector of reduced dimension n − p lying in the subspace tangent to the constraint manifold as optimization variable, whereas the minimized function is evaluated at a point, after retraction, that is approximately on the constraint manifold. Precisely, if h is the norm of the tangent vector, the distance between the point of evaluation of the function to be minimized, after retraction, is in general O(h4), while it would only be O(h2) if retraction were not applied. The construction only requires evaluation procedures for constraint functions and their gradients to be provided, and eludes the necessity of curvature information.