Fixed-Time Gradient Flows for Solving Constrained Optimization:A Unified ApproachOACSTPCDEI

The accelerated method in solving optimization problems has always been an absorbing topic.Based on the fixed-time(FxT)stability of nonlinear dynamical systems,we provide a unified approach for designing FxT gradient flows(FxTGFs).First,a general class of nonlinear functions in designing FxTGFs is provided.A unified method for designing first-order FxTGFs is shown under Polyak-Łjasiewicz inequality assumption,a weaker condition than strong convexity.When there exist both bounded and vanishing disturbances in the gradient flow,a specific class of nonsmooth robust FxTGFs with disturbance rejection is pre-sented.Under the strict convexity assumption,Newton-based FxTGFs is given and further extended to solve time-varying opti-mization.Besides,the proposed FxTGFs are further used for solving equation-constrained optimization.Moreover,an FxT proximal gradient flow with a wide range of parameters is pro-vided for solving nonsmooth composite optimization.To show the effectiveness of various FxTGFs,the static regret analyses for sev-eral typical FxTGFs are also provided in detail.Finally,the pro-posed FxTGFs are applied to solve two network problems,i.e.,the network consensus problem and solving a system linear equa-tions,respectively,from the perspective of optimization.Particu-larly,by choosing component-wisely sign-preserving functions,these problems can be solved in a distributed way,which extends the existing results.The accelerated convergence and robustness of the proposed FxTGFs are validated in several numerical examples stemming from practical applications.