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   Su, L; Vaidya, N.H. (January 2021, IEEE transactions on automatic control)

   null (Ed.)

   We consider the problem of multiagent optimization wherein an unknown subset of agents suffer Byzantine faults and thus behave adversarially. We assume that each agent i has a local cost function fi , and the overarching goal of the good agents is to collaboratively minimize a global objective that properly aggregates these local cost functions. To the best of our knowledge, we are among the first to study Byzantine-resilient optimization where no central coordinating agent exists, and we are the first to characterize the structures of the convex coefficients of the achievable global objectives. Dealing with Byzantine faults is very challenging. For example, in contrast to fault-free networks, reaching Byzantine-resilient agreement even in the simplest setting is far from trivial. We take a step toward solving the proposed Byzantine-resilient multiagent optimization problem by focusing on scalar local cost functions. Our results might provide useful insights for the general local cost functions. 

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   The multi-objective optimization is to optimize several objective functions over a common feasible set. Because the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the geometry for (weakly) Pareto values and represent Pareto front as the boundary of a convex set. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them and how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check whether a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points. Funding: J. Nie is partially supported by the National Science Foundation [Grant DMS-2110780]. 

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