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This technical paper presents a distributed continuous-time algorithm to solve multi-agent optimization problem with the team objective being the sum of all local convex objective functions while subject to an equality constraint. The optimal solutions are achieved within fixed time which is independent of the initial conditions of agents. This advantage makes it possible to off-line preassign the settling time according to task requirements. The fixed-time convergence for the proposed algorithm is rigorously proved with the aid of convex optimization and fixed-time Lyapunov theory. Finally, the algorithm is valuated via an example. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical framework for the design and analysis of gradient descent-based algorithms for coverage control tasks involving robot swarms. We adopt a multiscale approach to analysis and design to ensure consistency of the algorithms in the large-scale limit. First, we represent the macroscopic configuration of the swarm as a probability measure and formulate the macroscopic coverage task as the minimization of a convex objective function over probability measures. We then construct a macroscopic dynamics for swarm coverage, which takes the form of a proximal descent scheme in the L 2 -Wasserstein space. Our analysis exploits the generalized geodesic convexity of the coverage objective function, proving convergence in the L 2 -Wasserstein sense to the target probability measure. We then obtain a consistent gradient descent algorithm in the Euclidean space that is implementable by a finite collection of agents, via a "variational" discretization of the macroscopic coverage objective function. We establish the convergence properties of the gradient descent and its behavior in the continuous-time and large-scale limits. Furthermore, we establish a connection with well-known Lloyd-based algorithms, seen as a particular class of algorithms within our framework, and demonstrate our results via numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a class of optimization problems is investigated, where the objective function is the sum of N convex functions viewed as local functions and the constraints are N nonidentical closed convex sets. Additionally, it is aimed to solve the considered optimization problem in a distributed manner and thus a sequence of time-varying unbalanced directed graphs is introduced first to depict the information connection topologies. Then, the novel push-sum based constrained optimization algorithm (PSCOA) is developed, where the new gradient descent-like method is applied to settle the involved closed convex set constraints. Furthermore, the rigorous convergence analysis is shown under some standard and common assumptions and it is proved that the developed distributed discrete-time algorithm owns a convergence rate of O ( ln t t ) in general case. Specially, the convergence rate of O ( 1 t ) can be further obtained under the assumption that at least one objective function is strongly convex. Finally, simulation results are given to demonstrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents two distributed event-triggered algorithms under directed communication networks to solve a class of convex optimization problems such as the economic dispatch problem (EDP) with equality constraint, while the objective of optimization is the sum of all locally convex functions. One is distributed continuous-time event-triggered optimization algorithm, and the other is discrete-time algorithm based on iteration scheme. Continuous communication is not required by adopting event-triggered communication strategy. It means that the communication cost can be reduced and unnecessary waste of network resources can be avoided. Moreover, the convergence for the proposed algorithms are rigorously proved with the aid of Lyapunov stability theory under the strongly connected and weight-balanced network topology. Finally, four numerical simulations show the effectiveness and advantages of the two novel distributed event-triggered optimization algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we exploit the possibility of combining nonsmooth control jointly with projected gradient approaches to solve convex optimization problems with linear equality constraints. Our development gives rise to a sign projected gradient method employing only "coarse" information, which has finite-time convergence capability and preserves the simplicity and elegance of gradient descent. Our primary objective is to understand the design principle of the projection matrix in the nonsmooth setting and to investigate some key features of the method, such as convergence time. Specifically, we offer a set of convergence results via nonsmooth analysis, revealing that the algorithm guarantees finite-time convergence for a wide range of cost functions, including strongly convex, strictly convex, and convex functions, provided that the proposed design criteria and conditions are satisfied. The results suggest that the choice of the projection matrix might not be unique, which motivates the optimal design issue of the projection matrix to minimize convergence time. We show that the problem can be formulated as a convex optimization problem, which can be solved readily by existing optimization tools. We discuss the possibility of employing the proposed method to solve unconstrained optimization problems, which in turn gives an initial feasible solution to the sign projected gradient flow. We apply the algorithm to solve a formation control problem, and the numerical results show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]