Distributed Optimization Algorithms And Its Applications Based On Network Dynamics Theory

With the development of artificial intelligence and big data technologies,the scale and complexity of many optimization problems are increasing.The development of network science provides new inspiration for solving large-scale optimization problems Different from traditional centralized algorithms,distributed network optimization can achieve global objective through multiple agents’ cooperation.Related results have been successfully applied in smart grid,resource scheduling,machine learning,distributed sensor estimation and other fields.Based on communication network and local compu-tation units,the distributed optimization algorithm has the excellent characteristics of scalability,security,and low complexity.In addition,most of the current distributed op-timization algorithms are based on continuous or discrete-time dynamics over undirected communication topologies,and suffering from slow convergence speed,which limits the scope of application.In view of the problems that exist in the field of resent network dynamics optimization,this dissertation studies the finite-time stability of discontinuous and continuous network dynamics under a more general communication topology,and further applies them to the design of distributed optimization algorithms.Besides,a more generalized distributed network optimization algorithm is designed and applied to the distributed dynamic traffic assignment problems.The main contents of this disser-tation are summarized as follows(1)For a class of continuous network systems with nonlinear dynamics and hetero-geneous coefficients,the finite/fixed-time stability is studied.Several criteria are given for guaranteeing the finite/fixed-time stability and the upper bound estimation of the finite/fixed settling time is obtained.With the obtained results,several finite/fixed-time distributed protocols are designed for solving linear equations Ax=b in presence of dif-ferent initialization conditions.Next,a push-sum based continuous algorithm is designed for distributed optimization problems over directed graphs(2)For a class of discontinuous network systems with switching(sign)dynamics,the finite-time stability is studied and further applied to finite-time network consensus problems and finite-time distributed optimization problems.Using non-smooth analysis,the sufficiency criteria of several classes of discontinuous network systems with switching dynamics for achieving finite-time stability are given,and the explicit expression of the upper bound of finite settling time is derived.The system can be considered as a generalization of discontinuous consensus protocols in the existing literature.Under a unified framework,the finite-time convergence of such discontinuous systems under general signed networks is investigated and further applied to the network modulus consensus problem over signed digraphs.In addition,for some special systems,the uniqueness of the non-sliding dynamics is proved in the sense of Filippov solution,and the range of sliding dynamics on the possible continuum of equilibria is given.Then,for a special class of distributed optimization problems,a discontinuous distributed algorithm with finite-time convergence is presented.(3)For a special class of distributed convex optimization problems—distributed parametric consensus optimization problem(DPCOP),a two-stage optimization method including "primal decomposition" and "distributed consensus" is provided.Different from traditional distributed optimization problems which aim to drive all the local states to a common value,DPCOP aims to solve a system-wide problem with partial common parameters shared amongst local agents in a distributed way.To relax the restriction on the topology,a distributed projected subgradient method(DPSM)is applied in dis-tributed consensus stage to achieve the consensus of local estimated parameters,while the subgradients can be obtained by solving a multi-parametric problem locally.For a special class of DPCOP,a discrete-time distributed algorithm with exponential rate of convergence is provided.Furthermore,the proposed two-stage optimization method is applied to a distributed model predictive consensus(D-MPCS)problem in order to reach an optimal output consensus at equilibrium points for all agents.The stability analysis for the proposed algorithm is further given.(4)For a class of generalized distributed convex optimization problems with lo-cally coupled decision variables,a continuous-time distributed optimization algorithm based on multi-agent population dynamics is presented.Different from the traditional distributed optimization algorithm designed to achieve global consensus of individual states,the generalized distributed optimization algorithm considered the partial con-sensus of locally coupled states,which is more suitable for the practical system and meanwhile saves communication costs and protects customers’ privacy.Inspired by the neurodynamic method,a distributed continuous protocol for solving network op-timization problems with general constraints and objective functions is presented.As an application,from the perspective of multi-agents,the above collective neurodynamic algorithm is applied to solve distributed system optimum dynamic traffic assignment problems in intelligent transportation systems.