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This paper considers a time-varying optimization problem associated with anetwork of systems, with each of the systems shared by (and affecting) a numberof individuals. The objective is to minimize cost functions associated with theindividuals' preferences, which are unknown, subject to time-varyingconstraints that capture physical or operational limits of the network. To thisend, the paper develops a distributed online optimization algorithm withconcurrent learning of the cost functions. The cost functions are learnedon-the-fly based on the users' feedback (provided at irregular intervals) byleveraging tools from shape-constrained Gaussian Process. The online algorithmis based on a primal-dual method, and acts effectively in a closed-loop fashionwhere: i) users' feedback is utilized to estimate the cost, and ii)measurements from the network are utilized in the algorithmic steps to bypassthe need for sensing of (unknown) exogenous inputs of the network. Theperformance of the algorithm is analyzed in terms of dynamic network regret andconstraint violation. Numerical examples are presented in the context ofreal-time optimization of distributed energy resources.

Time-Varying Optimization of Networked Systems with Human Preferences