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The authors aim to enhance the end-to-end rate of a general dual-hop relay network with multiple channels and finite modulation formats, by jointly optimizing channel pairing, power allocation, and integer bit loading. Such an optimization problem has both a discrete feasible region, due to the combinatoric nature of channel pairing, and a discrete objective, due to the bit loading requirement. For this type of mixed-integer programming problems, the Lagrange dual method generally is inapplicable, due to the non-zero duality gap. However, by exploring the structure of their problem, they are able to bound the gap to within one bit, allowing the extraction of an exact optimal integer solution.
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