On the application of the residual method for the correction of inconsistent problems of convex programming

For the correction of a convex programming problem with potentially inconsistent constraint system (an improper problem), we apply the residual method, which is a standard regularization procedure for ill-posed optimization models. A problem statement typical for the residual method is reduced to a minimization problem for an appropriate penalty function. We apply two classical penalty functions: the quadratic penalty function and the exact Eremin-Zangwill penalty function. For each of the approaches, we establish convergence conditions and bounds for the approximation error.