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We consider the problem of elastic-demand network equilibrium in communication networks which support both multirate multicast sessions and unicast sessions. Extending existing work for unicast sessions, we first present the elastic-demand network equilibrium models in different multicast sessions, which can be formulated as a convex programming problem. To solve the convex programming problem we use the augmented Lagrangian multiplier algorithm in which the attractive features of the exterior penalty with primal-dual methods and Lagrangian multipliers concepts are combined while curtailing the disadvantage of both. Some numerical results about unicast sessions and multirate multicast sessions are demonstrated through efficient implementations of the augmented Lagrangian multiplier algorithm.