This paper studies analysis and synthesis problems for diffusively coupled multi-agent systems. We focus on networks comprised of multi-input multi-output nonlinear systems that possess a property we term maximal equilibrium-independent cyclically monotone passivity (MEICMP), which is an extension of recent passivity results. We demonstrate that networks comprised of MEICMP systems are related to a pair of dual network optimization problems. In particular, we show that the steady-state behavior of the multi-agent system correspond to the minimizers of appropriately defined network optimization problems. This optimization perspective leads to a synthesis procedure for designing the network controllers to achieve a desired output. We provide detailed examples of networked systems satisfying these properties and demonstrate the results for a network of damped planar oscillators.