Summary: This paper deals with methods for choosing search directions in the iterative solution of constrained minimization problems. The popular technique of calculating orthogonal components of the search direction (which are respectively tangential and normal to the constraints) is discussed and contrasted with the idea of constructing the search direction from two moves which are conjugate with respect to the Hessian of the Lagrangian function. Minimization algorithms which use search directions obtained by these two approaches are described and numerical results suggest that there are clear advantages in building steps from conjugate components.