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Relational mathematics, as it is studied in fields like mathematical economics and social choice theory for some time, provides a rich and general framework and appears to be a natural and direct way to paraphrase optimization goals, to represent user preferences, to justify fairness criterions, to cope with QoS or to valuate utility. Here, we will focus on the specific application aspects of formal relations in network design and control problems and provide the general concept of relational optimization. In relational optimization, we represent the optimization problem by a formal relation, and the solution by the set of maximal (or non-dominated) elements of this relation. This appears to be a natural extension of standard optimization, and covers other notions of optimality as well. Along with this, we will provide a set of fairness relations that can serve as maximizing relations in relational optimization according to various application needs, and we specify a meta-heuristic approach derived from evolutionary multi-objective optimization algorithms to approximate their maximum sets.

Relational Optimization and Its Application: From Bottleneck Flow Control to Wireless Channel Allocation