On the regularized method of barrier functions in the analysis of improper convex programs

The paper considers the problem of constructing approximations for a wide class of improper convex programs (ICPs). The original problem with an inconsistent system of constraints is immersed in a parametric family of feasible convex programming models, where the norm of the discrepancy of the constraint functions serves as a parameter. The minimum value of the parameter at which the feasible set of the problem becomes nonempty determines an optimal correction of the ICP. To solve the correction problem, one of the classical methods of regularization of ill-posed extremal problems is used — the method of stabilizing functions (Tikhonov's method). In this case, the original problem with constraints is initially reduced to the problem of unconstrained minimization of a certain penalty function. Instead of conventional external penalty functions, the paper uses the method of internal (barrier) functions. The design features of barrier functions can provide certain advantages in the numerical implementation of the correction method. Conditions for the solvability of problems arising at various stages of the proposed correction method are formulated, and the issues of matching the process parameters that ensure the required convergence are studied. The capabilities of the method when working with perturbed data are analyzed.