Browsing by Subject "Convergence rate"

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Open Access
Consensus in networks of anticipatory agents under transmission delays
(2024-07) Güven, Zeynep
This thesis examines the dynamics of a coupled system of linear delay differential equations, addressing the normalized consensus problem on undirected and connected graphs of anticipatory agents in the presence of a fixed information transmission delay. The anticipation rule, a first-order linear extrapolation, enables agents to predict the present states of their neighbours using past information, thereby introducing an additional delayed term into the formulation and resulting in a system of delay differential equations with two discrete delays. The main result of this study is the necessary and sufficient condition for the anticipatory consensus protocol under transmission delays to reach consensus. Simulations indicate that the convergence rate of the anticipatory protocol is superior to that of the protocol without anticipatory agents, both under transmission delays. As a natural extension, the findings are applied to the Kuramoto model of coupled phase oscillators to determine the local stability of synchronized states. It is demonstrated that the delay margin for achieving local stability is inversely proportional to the coupling strength between agents. Furthermore, it is shown that the synchronized frequency of the extended model remains the same as that of the original Kuramoto model, contrasting with other extended versions that involve single delays.
Open Access
Convergence analysis of a norm minimization-based convex vector optimization algorithm
(Society for Industrial and Applied Mathematics, 2024-07-25) Ararat, Çağın; Ulus, Firdevs; Umer, Muhammad
In this work, we propose an outer approximation algorithm for solving bounded convex vector optimization problems (CVOPs). The scalarization model solved iteratively within the algorithm is a modification of the norm-minimizing scalarization proposed in [\c C. Ararat, F. Ulus, and we prove that the algorithm terminates after finitely many iterations, and it returns a polyhedral outer approximation to the upper image of the CVOP such that the Hausdorff distance between the two is less than \epsilon . We show that for an arbitrary norm used in the scalarization models, the approximation error after k iterations decreases by the order of O(k1/(1-q)), where q is the dimension of the objective space. An improved convergence rate of O(k2/(1-q)) is proved for the special case of using the Euclidean norm.