On the constructive solution of convex programming problems in separable form

Abstract

We present a constructive approach to solving convex programming problems in separable form and new constructive methods for simultaneous solving pairs of primal and dual geometric programming problems. These methods are based on the new principle of accumulation of approximate functions, which involves an approximation of the objective functions by a special function (which is minorant for the primal problem and majorant for the dual problem) constructed in a form that uses certain information about some preceding steps of the method. The choice of the optimal direction for current iteration for solution of primal or dual problem is performed basing on extreme properties of this function. The construction of the approximate functions is based on the convexity, concavity, and separability properties of objective functions and on duality results. The support of the problem which is the index set of independent variables is modified from one iteration to another. The method involves iterations of primal and dual type. During the iterations of primal, type we resolve the primal problem using information about the best dual approximation obtained up to this moment. During the iterations of dual type the dual problem is solved and the best primal approximation is used. The method interlaces primal and dual iterations providing the possibility of their interaction.