Research On Continuous-Time Optimization Algorithm And Convergence Rate Of Distributed Resource Allocation Problem

Distributed optimization has broad application prospects in large-scale complex networked systems ranging from smart grid,sensor networks,machine learning,and others,which is a research hotspot in recent years.Distributed resource allocation is a kind of important distributed optimization problem,which considers not only the coupled constraints of decision variables,but also the constraints of communication network and control performance.Thus,the design and analysis of optimization algorithm for distributed resource allocation problem are more challenging.At present,the distributed resource allocation algorithm mainly solves the distributed processing problem of coupled constraints.However,the distributed resource allocation problem under the constraints of communication network and control performance has not been fully investigated.Based on the existing research results,this paper studies continuous-time optimization algorithms and convergence rates for several types of distributed resource allocation problems with communication networks or control performance constraints.The main works of this paper are summarized as follows:(1)For the distributed resource allocation problem with coupled inequality constraints under unbalanced directed graphs,a distributed optimization algorithm with asymptotic convergence is studied.In order to counteract the asymmetry of unbalanced directed graphs and ensure the nonnegativity and consensus of Lagrange multipliers,a surplus feedback based distributed continuous-time primal-dual algorithm is proposed.The proposed algorithm uses fixed-time projection to deal with the nonnegativity of Lagrange multipliers and the local constraints of decision variables,and the dual problem under directed graphs is solved by combing with surplus feedback,consensus method and gradient descent.The convergence of the algorithm is analyzed by using saddle point theory and eigenvalue perturbation theory.The proposed distributed continuous-time optimization algorithm asymptotically converges to an optimal solution under the condition that the local objective function and constraint functions are convex functions and unbalanced directed graph is strongly connected.The effectiveness of the theoretical results is further verified by numerical simulation.(2)For the distributed resource allocation problem under communication bandwidth constraint,distributed optimization algorithms with exponential convergence are studied.Based on the continuous-time distributed weighted gradient algorithm,two eventtriggered communication mechanisms are synthesized through a new Lyapunov function,one is a continuous evaluation based event trigger,and the other is a periodic evaluation based event trigger.The proposed event-triggered mechanisms do not require agents to continuously or periodically access the state information of their communication neighbors,and thus it can reduce the communication number among agents and the update frequency of control.The stability analysis shows that the eventtriggered distributed optimization algorithm converges to the optimal solution at an exponential rate.At the same time,the new analysis method is also suitable for resource allocation under time-varying demand.Numerical simulations verify the effectiveness of the event-triggered distributed weighted gradient algorithm.(3)For the distributed resource allocation problem under bounded disturbances,a fixed-time convergent distributed optimization algorithm is studied.Firstly,an improved distributed weighted gradient optimization algorithm is proposed for undisturbed resource allocation problem.The nonlinear polynomial feedback is used to achieve the fixed-time convergence,and an auxiliary variable is used to realize the initialization-free of the algorithm.In addition,the algorithm is also suitable for the case where the control gain is heterogeneous.Furthermore,a distributed robust optimization algorithm based on fixed-time sliding mode control is proposed for the resource allocation problem with bounded disturbances.The stability of the undisturbed and disturbed optimization algorithm is analyzed by applying the fixed-time Lyapunov theory,and the upper bound of the convergence time of the algorithms is estimated.The convergence time is only related to the control parameters and does not depend on the initial values of the system states,which makes it possible to design the convergence time in advance according to the task requirements.The effectiveness and superiority of the algorithms are verified by several numerical simulations.(4)For the distributed resource allocation problem with time-varying coupled equality constraint and bounded disturbances,a robust distributed fixed-time convergent optimization algorithm is studied.According to the characteristics of the optimal solution of the time-varying resource allocation problem,a class of distributed optimization algorithms based on fixed-time dynamic weighted average consensus is proposed.Firstly,the variable structure property of signum function is used to realize the complete suppression of bounded disturbances and the accurate tracking of timevarying optimal solution.Then,the linear function is used to replace the signum function to obtain the algorithm of continuous control input,which realizes the partial suppression of the disturbances and the uniformly bounded tracking of the time-varying optimal solution.The stability of the proposed algorithms is analyzed based on the fixed-time Lyapunov theory,which not only estimated the upper bound of the convergence time of the proposed algorithms,but also gave the relationships between controller parameters and tracking performance.The tracking performance of the proposed optimization algorithms is verified by numerical simulations.