Dynamic distributed optimization and fixed-point seeking via over-the-air function computation

This thesis investigates the design and analysis of distributed algorithms for cooperative multiagent systems, where agents aim to solve time-varying optimization and fixed-point seeking problems under decentralized and noisy communication constraints. The problems of interest include minimizing the sum of local time-varying costs subject to local constraints, or finding points in the intersection of time-varying fixed-point sets, assuming that time-invariant solutions exists. Motivated by real-world applications in wireless sensor networks, the proposed framework addresses scenarios where agents communicate over time-varying networks with noisy and fading channels, and where tight coordination among the networked entities for synchronization or scheduling is infeasible. In this thesis, we study two-step iterative schemes that decouple local computations from network-driven information aggregation. Local update step incorporates current local information to the iterates, and it is modeled using set-theoretic methods, such as the adaptive projective subgradient method (APSM) or the general class of quasi-nonexpansive mappings. Whereas, the consensus step facilitates efficient aggregation of estimates across the agents over decentralized networks. A key contribution lies in the development of fully decentralized and scalable consensus implementation based on the over-the-air function computation (OTA-C) technology, which exploits the superposition property of wireless multiple-access channels (WMACs). Unlike prior OTA-C methods, the proposed protocol requires neither centralized coordination nor any channel-related information making it well-suited for time-sensitive, large-scale, and resource-constrained deployments. To solve the dynamic distributed optimization and fixed-point seeking problems considered in this thesis, we introduce a class of distributed algorithms that are both resilient to bounded random perturbations and compatible with the OTA-C based consensus steps. These algorithms extend existing frameworks by integrating advanced machine learning techniques such as superiorization with scalable physical-layer communication. The theoretical analysis establishes sufficient conditions for almost sure and mean-square convergence to feasible solutions under minimal assumptions on operator structure and communication dynamics. The convergence is established under two scenarios of practical relevance: (i) doubly-stochastic mixing, where the network graph is balanced in expectation at each iteration, and (ii) row-stochastic mixing, which is more general and allows agents to operate with minimum coordination. Numerical experiments on synthetic and real-world data demonstrate that the proposed algorithms are not only communication-efficient and scalable, but also enable significant energy savings during communication. The results provide new insights into distributed computation under realistic communication models and open promising directions for OTA-C based collaborative learning, decentralized control, and operator-theoretic algorithm design. The thesis concludes by highlighting several open problems and future research directions. These include extending the OTA-C consensus framework to achieve improved performance guarantees and explicit tail bounds, relaxing the i.i.d.~assumption on network graph realizations, analyzing convergence behavior in the absence of common time-invariant solutions, and exploring extensions related to inexact consensus and convergence rate analysis.