Summary: The REQP algorithm solves constrained minimization problems using a sequential quadratic programming technique based on the properties of penalty functions. The convergence of REQP has been studied elsewhere [see the first author, IMA J. Numer. Anal. 8, No. 2, 253-271 (1988; Zbl 0643.65030); J. Inst. Math. Appl. 21, 67-71 (1978; Zbl 0373.90056)]. This paper uses a novel approach to the analysis of the method near to the solution, based on the use of conjugate subspaces. The step \(p\) taken by a constrained minimization algorithm can be thought of as having two components, \(h\) in the subspace tangential to the constraints and \(v\) in the subspace spanned by the constraint normals. It is usual for \(h\) and \(v\) to be orthogonal components. Recently, L. C. W. Dixon [‘Conjugate bases for constrained optimization’, Tech. Rep. 181, Numer. Optimization Centre, Hatfield Polytechnic (1987)] has suggested constructing \(p\) from components which are not orthogonal. That is, we write \(p= h'+ v'\), where \(h'\) is in the subspace tangential to the constraints and where \(v'\) and \(h'\) are conjugate with respect to the Hessian of the Lagrangian function. By looking at the conjugate components of the REQP search directions, it is possible to simplify the analysis of the behavior near the solution and to obtain new results about the local rate of convergence of the method.