Optimization underpins many of the challenges that science and technology face on a daily basis. Recent years have witnessed a major shift from traditional optimization paradigms grounded on batch algorithms for medium-scale problems to challenging dynamic, time-varying, and even huge-size settings. This is driven by technological transformations that converted infrastructural and social platforms into complex and dynamic networked systems with even pervasive sensing and computing capabilities. This article reviews a broad class of state-of-the-art algorithms for time-varying optimization, with an eye to performing both algorithmic development and performance analysis. It offers a comprehensive overview of available tools and methods and unveils open challenges in application domains of broad range of interest. The real-world examples presented include smart power systems, robotics, machine learning, and data analytics, highlighting domain-specific issues and solutions. The ultimate goal is to exemplify wide engineering relevance of analytical tools and pertinent theoretical foundations.

Network-science related applications frequently deal with inference of spatio-temporal processes. Such inference tasks can be aided by a graph whose topology contributes to the underlying spatio-temporal dependencies. Contemporary approaches extrapolate dynamic processes relying on a fixed dynamical model, that is not adaptive to changes in the dynamics. Alleviating this limitation, the present work adopts a candidate set of graph-adaptive dynamical models with one active at any given time. Given partially observed nodal samples, a scalable Bayesian tracker is leveraged to infer the graph processes and learn the active dynamical model simultaneously in a data-driven fashion. The resulting algorithm is termed graph-adaptive interacting multiple dynamical models (Grad-IMDM). Numerical tests with synthetic and real data corroborate that the proposed Grad-IMDM is capable of tracking the graph processes and adapting to the dynamical model that best fits the data.

Graphs are widely adopted for modeling complex systems, including financial, biological, and social networks. Nodes in networks usually entail attributes, such as the age or gender of users in a social network. However, real-world networks can have very large size, and nodal attributes can be unavailable to a number of nodes, e.g., due to privacy concerns. Moreover, new nodes can emerge over time, which can necessitate real-time evaluation of their nodal attributes. In this con, this paper deals with scalable learning of nodal attributes by estimating a nodal function based on noisy observations at a subset of nodes. A multikernel-based approach is developed, which is scalable to large-size networks. Unlike most existing methods that re-solve the function estimation problem over all existing nodes whenever a new node joins the network, the novel method is capable of providing real-time evaluation of the function values on newly joining nodes without resorting to a batch solver. Interestingly, the novel scheme only relies on an encrypted version of each node's connectivity in order to learn the nodal attributes, which promotes privacy. Experiments on both synthetic and real datasets corroborate the effectiveness of the proposed methods.