So, we first investigate the primal objective function and approximate it with a series of convex ones.
Via the Lagrangian relaxation, we remove the coupling QoS constraints and incorporate them into the primal objective function.
The difference is called the duality gap of iteration, and it upper bounds the distance from the so far best found primal objective function value to the supremum of the primal objective function.
The power constraint is assumed to be large enough, so that solving the primal objective is equivalent to solving an unconstrained problem.
Note: As mentioned before, formula (9) only represents the optimal solution for the primal objective when the power constraint ∑ j = 1 N x i, j ≤ 1 for i = 1,2 holds.
By comparing the primal objective value obtained from each subgradient iteration with the optimal solution obtained under UPA assumption, we see that our proposed algorithm taking power allocation into account improves the system throughput.
It is indicated that Zmin in (26) is the optimal solution of the primal objective only when the power constraint ∑ k i = 1 K i x i, k i ≤ 1 for i ∈ {1,...,K} holds.
Since the multi-path routing usually results in the non-strict concavity of the primal objective function, we first introduce the Proximal Optimization Algorithm to get around such difficulty.
In mathematic terms, the constraint in the primal objective should be changes to ∑ j = 1 N x 1, j = 1 and ∑ j = 1 N x 2, j ≤ 1, or to ∑ j = 1 N x 2, j = 1 and ∑ j = 1 N x 1, j ≤ 1.
In order to obtain a closed-form expression for the required power allocation, we divide the constrained problem into two different cases: a) The power constraint is assumed to be large enough, so that solving the primal objective is equivalent to solving an unconstrained problem.
