This dissertation presents a theory of network optimization based on the lexicographic criterion. The objective is to identify the transmission rates of each user that optimally utilize the network resources. Optimality is defined in two senses: on one hand, transmission rates are to be maximized; on the other hand, fairness among all the users must be guaranteed. The lexicographically largest point is the preferred one since, by definition, such solution recursively maximizes the rate of those users that are most poorly treated. The theorems of bottleneck optimality condition and projection optimality condition are derived, which allows the design of practical distributed protocols in three different scenarios: unicast, multicast and discrete rate communications. While the complexity of these protocols is minimal (logarithmic), their convergence time is shown to be two times faster than previous works. The theorem of precedent link is presented. This results into a theory of bottleneck order, which we use in two ways: to unveil the bottleneck structure of a network and to measure the complexity of a protocol.
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