Convex Constrained Programmes with Unattained Infima

We consider a problem of minimization of convex function f x. over the convex region R where the objective function and the feasible region have a common direction of recession. In cases when one of these directions is not in the constancy space of the objective function, then the minimal solution is not achieved even if the function f x. is bounded below over the region R. Many algorithms, if applied to this class of programmes, do not guarantee convergence to the global infimum. Our approach to this problem leads to derivation of the equation of the feasible parametrized curve C t., such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region R. We show that if all functions defining the program are analytic, then C t. is also an analytic function. The equation of the curve can be successfully used to determine the global infimum in particular, unboundedness.of the convex constrained programmes in cases when the application of classical methods, such as the steepest descent method, fails to converge to the global infimum