Session: Decomposition Methods
Chair: Matthew Viens
Cluster: nan
Talk 1: A Dantzig-Wolfe Decomposition Method for Quasi-Variational Inequalities
Speaker: Manoel Jardim
Abstract: We propose an algorithm to solve quasi-variational inequality problems, based on the Dantzig-Wolfe decomposition paradigm. Our approach solves in the subproblems variational inequalities, which is a simpler problem, while restricting quasi-variational inequalities in the master subproblems, making them generally (much) smaller in size when the original problem is large-scale. We prove global convergence of our algorithm, assuming the the mapping of the quasi-variational inequality is either single-valued and continuous or it is set-vaued maximally monotone. Quasi-variational inequalities serve as a framework for several equilibrium problems, and we illustrate our algorithm on an important example in the field of economics, namely the Walrasian equilibrium problem formulated as a generalized Nash equilibrium problem. We show that the proposed method demonstrates good performance for the large-scale cases. Our numerical section tackles big problems in the theory of abstract economy, as well as some academic examples that have been previously employed in the literature.
Talk 2: Treating Uncertainty in Modeling with Multiple Solutions
Speaker: Matthew Viens
Abstract: Optimization problems can have multiple solutions that achieve an optimal or near-optimal objective value. We provide a theoretical foundation for characterizing multiple solutions combining sublevel set, epigraph, and KKT representations. We discuss how this theory enables generation of multiple solutions in two different problems: quadratic programming and Benders decomposition. We demonstrate how multiple solutions can provide additional insights into solution structure and tradeoffs for both problems. Further, we show how this additional insight is of especial value in models with data uncertainty and held-out objectives.
Talk 3: A general-purpose approach to multi-agent Bayesian optimization across decomposition methods
Speaker: Dinesh Krishnamoorthy
Abstract: Multi-agent decision-making problems, formulated as optimization problems, arise in a wide range of applications where multiple local decision-making agents collaborate to achieve a common goal. Examples include sensor and communication networks, where nodes coordinate to optimize performance and resource allocation. Distributed optimization techniques play a crucial role in these settings, enabling each agent to solve its local optimization problem while coordinating with others to achieve a system-wide optimum. Such coordination can be either decentralized (peer-to-peer) or facilitated by a central coordinator. However, traditional approaches typically require explicit analytical models linking local decision variables to objectives, which are often difficult or impractical to obtain in engineering applications. This necessitates black-box optimization methods, such as Bayesian optimization (BO), which can optimize unknown objective functions based on sampled observations. In multi-agent systems with interdependencies through shared variables or coupling constraints, standard BO methods fail to account for subsystem interactions effectively. Moreover, local objective function observations are often inaccessible to other agents, limiting the information available for updating probabilistic models and acquisition functions. Consequently, while BO has proven effective for single-agent optimization with unknown objectives, its extension to multi-agent settings remains underdeveloped. This talk will address this research gap by presenting a general-purpose multi-agent Bayesian optimization (MABO) framework that is compatible with a wide array of decomposition methods, with both centralized coordination and peer-to-peer coordination. whereby we augment traditional BO acquisition functions with suitably derived coordinating terms to facilitate coordination among subsystems without sharing local data. Regret analysis reveals that the cumulative regret of MABO is the sum of individual regrets and remains unaffected by the coordinating terms, thus bridging advancements in distributed optimization and Bayesian optimization methodologies. Numerical experiments on vehicle platooning and offshore oil production optimization examples validate the effectiveness of the proposed MABO framework for different classes of decomposition methods. This talk is based on the paper published in Optimizationa and Engineering. Krishnamoorthy, D. A general-purpose approach to multi-agent Bayesian optimization across decomposition methods. Optim Eng (2025). https://doi.org/10.1007/s11081-024-09953-w
