A family F of reduced gradient algorithms for solving linearly constrained minimization problems is introduced. Its basic features are: (i) The variables with respect to which an unconstrained minimization is carried out are chosen according to a general rule. (ii) The unconstrained minimization is completed. (iii) Weak assumptions are considered on the unconstrained minimization algorithm. For any algorithm in F under suitable hypotheses convergence to a Kuhn-Tucker point is established. Further some convergence rate properties are found.
