Two new closely related concepts are introduced that depend on a positive constant $\Gamma$. An iteration is proportional if the norm of violation of the Kuhn--Tucker conditions at active variables does not excessively exceed the norm of the part of the gradient that corresponds to free variables, while a progressive direction determines a descent direction that enables the released variables to move far enough from the boundary in a step called proportioning. An algorithm that uses the conjugate gradient method to explore the face of the region defined bythe current iterate until a disproportional iteration is generated is proposed. It then changes the face by means of the progressive direction. It is proved that for strictly convex problems, the proportioning is a spacer iteration so that the algorithm converges to the solution. If the solution is nondegenerate then the algorithm finds the solution in a finite number of steps. Moreover, a simple lower bound on $\Gamma$ is given to ensure finite termination even for problems with degenerate solutions. The theory covers a class of algorithms, allowing many constraints to be added or dropped at a time and accepting approximate solutions of auxiliary problems. Preliminary numerical results are promising.

Preconditioning of the conjugate gradient method by a conjugate projector has been suggested. We describe an algorithm and prove its correctness. An estimate of the preconditioning effect in terms of the gap between the invariant subspace of smooth eigenvectors of a matrix of original system and the complement of the range of the preconditioning projector is obtained. The paper is supplied with results of a numerical experiment which suggest that the method may be efficient in practical computations.