Research On Continuous-time Penalty-like Algorithms For Several Distributed Nonsmooth Optimization Problems

Recently,the distributed optimization over multi-agent network frequently arises in engineering and scientific fields such as smart grid,wireless sensor network,model predictive control and so on.Distributed optimization problems in reality usually have largescale and complex network structure.Therefore,it is a common issue for scholars to design fast and efficient distributed algorithms to investigate such kind of distributed optimization.Compared with the traditional centralized algorithm,distributed continuoustime algorithm overcomes many disadvantages of centralized algorithm,and then has outstanding advantages in solving large-scale and complex optimization problems.This paper designs several kinds of distributed continuous-time penalty-like algorithms for nonsmooth distributed convex or nonconvex optimization with inequality constraints and nonsmooth distributed resource allocation optimization under undirected graph by adopting penalty method and adaptive idea,and further investigates the convergence of these algorithms.The details are as follows:1.A continuous-time penality-like algorithm with time-varying penalty term is proposed for nonsmooth distributed convex optimization problems with inequality constraints.By using Lyapunov method,it is proved that the state solution of the algorithm finally achieves consensus and converges to an optimal solution.The existing penalty algorithms for solving this kind of optimization problems need to estimate the lower bound of accurate penalty parameters in advance,which often destroys the distributed structure of optimization problems.The continuous-time penalty-like algorithm designed in this part penalizes the inequality constraints by using a continuous function that tends infinite with time,so as to effectively avoid the disadvantage of estimating the lower bound of accurate penalty parameters.In addition,some numerical simulations verify the effectiveness of the proposed algorithm.2.An adaptive continuous-time penalty-like algorithm is proposed to investigate the nonsmooth distributed resource allocation convex optimization problem with coupling equality and convex inequality constraints.In order to avoid the disadvantages of introducing additional auxiliary variables of primal-dual theory,the proposed algorithm penalizes the inequality constraints combining with the adaptive ideas.By using nonsmooth analysis method,it is proved that the state solution of the algorithm converges to an optimal solution of the optimization problem.Compared with the existing algorithms for solving this optimization problems,the continuous-time algorithm designed in this part has a lower dimension of state solution space.Therefore,it is more flexible in selecting appropriate gain parameters.Meanwhile,some numerical simulation shows the effectiveness and feasibility of the designed algorithm.3.An adaptive continuous-time penalty-like algorithm is designed to solve the nonsmooth distributed convex optimization problem with inequality constraints by combining the penalty ideas with the consensus constraint.It is proved that the state solution of the algorithm can converge to an optimal solution of the optimization problem.By using the adaptive idea to penalty the consistency constraints,the algorithm designed in this part can ensures that each agent does not need to exchange private information of its own objectives and constraint sets with other agents,which effectively protects the privacy and reduces the communication burden.Meanwhile,some numerical simulation shows the feasibility of the proposed continuous-time optimization algorithm.4.A collective continuous-time penalty-like algorithm under the framework of particle swarm optimization algorithm is constructed to investigate the nonsmooth nonconvex distributed optimization problem with nonconvex objective function and convex inequality constraints.In order to overcome the challenge brought by the nonconvexity of the objective function,a continuous-time penalty-like algorithm is designed to solve the critical point of the optimization problem for the single particle.The overall idea of particle swarm optimization algorithm is further used to continuously evolve to find the global optimal solution of the optimization problem.Compared with other existing algorithms for solving this kind of optimization problem,the algorithm proposed can find the global optimum.Besides,numerical simulations verify the effectiveness of the proposed algorithm.