Convex Local Cost Functions Research Articles

Distributed finite‐time optimization algorithms for multi‐agent systems under directed graphs

AbstractExisting distributed finite‐time optimization algorithms for continuous‐time multi‐agent systems require either undirected graphs or weight balanced digraphs, and distributed finite‐time optimization problems for weight unbalanced digraphs are still great challenging. Thus, the distributed finite‐time optimization problems for continuous‐time multi‐agent systems with strongly convex local cost functions are investigated under directed graphs in this article. First, a distributed finite‐time gradient estimator is constructed by using non‐smooth analysis and algebraic graph theory, then distributed finite‐time optimization algorithms and piecewise distributed finite‐time optimization algorithms are proposed based on the designed gradient estimator. The new proposed distributed finite‐time optimization algorithms which only require strongly connected graphs relax the balanced requirement. Furthermore, the communication bandwidth of systems could be saved by deploying the proposed piecewise distributed finite‐time optimization algorithms since the information exchange in the optimization process is reduced. Finally, simulation examples are given to verify the effectiveness of proposed distributed optimization algorithms.

A Modular Optimal Formation Control Scheme of Multiagent Systems With Application to Multiple Mobile Robots

In this article, we studied an optimal formation control problem of general integrator chain multiagent systems with strongly convex local cost functions. The control goal is to design appropriate controllers for the agents such that they reach a desired formation shape and the global cost function is minimized. To achieve the goal, a modular control scheme with two modules is proposed. First, in Module I, an optimal formation signal generator is designed, whose outputs asymptotically converge to the minimizer of the global cost function. Second, in Module II, by employing the outputs into the feedback loop as the agents’ reference trajectories, some linear-tracking controllers are designed for the agents to track the references asymptotically. With both modules, the proposed control scheme realizes the optimal formation control goal asymptotically. Moreover, the proposed control scheme is also applied to solving an optimal formation control problem of multiple mobile robots, where the robots’ total “relative” weighted squared moving distance requires minimization. The simulations and experiments validate the effectiveness and practicality of the proposed control scheme.

Discrete-Time Algorithms for Distributed Constrained Convex Optimization With Linear Convergence Rates.

In this article, the constrained optimization problem with its global objective function being the sum of convex local cost functions and the constraint being a closed convex set is researched. The aim of this study is to solve the researched problem in a distributed manner, that is, using only local computations and local information exchanges. Toward this end, two gradient-tracking-based distributed optimization algorithms are designed for the considered problem over weight-balanced and weight-unbalanced graphs, respectively. Since the classical projection method is unsuitable to handle the closed convex set constraint under the gradient-tracking framework, a new indirect projection method is employed in this article to deal with the involved closed convex set constraint. Furthermore, two time scales are introduced to complete the convergence analyses. In addition, under the condition that all local cost functions are strongly convex and L -smooth, it is proved that the algorithms with well-selected fixed step sizes have linear convergence rates.

A distributed fixed-time optimization algorithm for multi-agent systems

In this paper, the distributed fixed-time optimization problem is investigated for first-order multi-agent systems with strongly convex local cost functions. To solve this problem, a two-piece distributed fixed-time optimization algorithm is proposed. In the first piece, some local optimization controllers are designed for the agents such that each agent converges to its local cost function’s minimizer in the unified fixed time under an arbitrary initial state. In the second piece, based on state information transmissions between neighboring agents and the local cost functions’ Hessian matrices, some distributed optimization controllers are developed for the agents such that they converge to the global cost function’s minimizer together in fixed time. Under the proposed algorithm, all the agents reach the global cost function’s minimizer in fixed time. Moreover, this fixed settling time is independent of the agents’ initial states and it can be predetermined according to the task demands. Numerical simulations demonstrate the effectiveness and advantages of the proposed distributed fixed-time optimization algorithm.

Privacy-Preserving Distributed Online Optimization Over Unbalanced Digraphs via Subgradient Rescaling

In this article, we investigate a distributed online constrained optimization problem with differential privacy where the network is modeled by an unbalanced digraph with a row-stochastic adjacency matrix. To address such a problem, a distributed differentially private algorithm without introducing a trusted third-party is proposed to preserve the privacy of the participating nodes. Under mild conditions, we show that the proposed algorithm attains an O(log T) expected regret bound for strongly convex local cost functions, where T is the time horizon. Moreover, we remove the need for knowing the time horizon T in advance by adopting doubling trick scheme, and derive an O(√T) expected regret bound for general convex local cost functions. Our results coincide with the best theoretical regrets that can be achieved in the state-of-the-art algorithms. Finally, simulation results are conducted to validate the efficiency of our proposed algorithm.

Distributed zero‐gradient‐sum algorithm for convex optimization with time‐varying communication delays and switching networks

SummaryThe distributed convex optimization problem subject to time‐varying communication delays and switching network topologies is addressed in this paper. Based on continuous‐time Zero‐Gradient‐Sum scheme, the novel distributed algorithms are proposed to minimize the global objective function which is composed of a sum of strictly convex local cost functions. In the fixed network topology case, by constructing a new Lyapunov‐Krasovskii function, two explicit sufficient conditions for the maximum admissible time delay are derived to guarantee that all agents' states converge to the optimal solution. In the switching network topology case, the stability condition is derived by the common Lyapunov function theory. In addition, two sufficient conditions about the maximum admissible time delays are also derived for the fixed and switching weight‐balanced network topologies, respectively. Several simulation tests are used to illustrate the effectiveness of our obtained theoretical results.

Semi-Decentralized Nash Equilibrium Seeking in Aggregative Games With Separable Coupling Constraints and Non-Differentiable Cost Functions

We study the Nash equilibrium seeking problem for noncooperative agents whose decision making process can be modeled as a generalized aggregative game. Specifically, we consider players with convex local cost functions, convex local constraints, and convex separable coupling constraints, and we extend the literature on generalized aggregative games by handling possibly non-differentiable cost functions. We recast the Nash equilibrium seeking problem as the problem to find a zero of a set-valued monotone operator and show that the variational Nash equilibria correspond to KKT solutions of the original game where the shared constraints have the same Lagrange multipliers for all the players. Finally, we design a semi-decentralized algorithm with global convergence guarantee for generalized Nash equilibrium seeking.

Finite‐time convergent distributed consensus optimisation over networks

In this study, a finite-time convergent distributed continuous-time algorithm is proposed to solve a network optimisation problem where the global cost function is the sum of strictly convex local cost functions under an undirected network with fixed topologies. The algorithm is inspired by finite-time consensus protocols and continuous-time zero-gradient-sum algorithms. Instead of the exponential convergence in existing works, the finite-time convergence is guaranteed based on the Lyapunov method. A numerical simulation example is provided to illustrate the effectiveness of the developed algorithm.