Abstract:
In this paper the author continues his research on the application of penalty functions of
composite type for solving linear programming problems. The term “composite” is
explained by the fact that the graphs of such functions are obtained by the operation of
smooth gluing of different-type graphs of a number of usual functions of internal and
external penalties.
Such an operation allows one to preserve the positive qualities of the glued components
and eliminate their specific shortcomings. In particular, these constructions preserve the
smoothness properties, allowing the use of second-order methods for their minimization,
and at the same time are applicable not only to problems whose admissible regions have
a non-empty interior, but also to ill-posed (improper, contradictory, poorly posed) problems
that do not have admissible plans at all; for the latter, composite functions are capable of
finding their so-called approximation solutions. The author proposes a rigorous axiomatization
of such functions, thus extending their list, and also proves convergence theorems corresponding
to the new axiomatization of the method.
Keywords:linear programming, combinations of the methods, interior penalties, exterior penalties, improper programs, generalizes solutions.
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