Penalty Theories And Continuous-time Algorithms For Several Types Of Nonsmooth Convex Optimizations

In recent years,nonsmooth convex optimization problems have frequently appeared in engineering technology fields such as artificial intelligence,automatic control,and image processing.According to different communication methods,nonsmooth convex optimization problems can be divided into centralized and distributed types,and their difficulty in solving is influenced by various factors such as communication cost,network size,and the complexity of the problem itself.Therefore,how to accurately and efficiently solve nonsmooth convex optimization problems has received widespread attention from scholars.Compared with traditional numerical calculation methods,continuoustime algorithms have the performance of independent iteration step selection,parallel computation,and fast convergence,thus having significant advantages in solving largescale complex optimization problems in the engineering field.This dissertation proposes corresponding penalty theories,designs continuous-time algorithms,and explores their convergence for several types of nonsmooth convex optimization problems with different constraints.The specific research content is as follows:1.For a class of nonsmooth convex optimization problems constrained by inequalities and general closed convex sets,a projection friendly continuous time algorithm with lower computational complexity is designed to accurately calculate the projection on a projection friendly closed convex set,penalize the remaining inequality constraints in the feasible domain.By utilizing the properties of projection operators,it is proven that the algorithm state solution converges to the optimal solution of the studied problem.Unlike currently known projection based algorithms,this algorithm avoids the drawbacks of high error and complexity caused by numerical approximation of projections on general complex sets,thanks to the division of feasible regions.In addition,the algorithm is applied to deal with image fusion problems,and the efficiency and accuracy of the algorithm are verified through typical numerical examples and model comparisons.2.For a class of nonsmooth interval optimization problems constrained by interval partial order relationships and general closed convex sets,in order to overcome the drawback of high computational load caused by the projection of set-valued maps for computing subdifferentials,this section proposes an adaptive continuous-time algorithm based on adaptive techniques to punish interval partial order relationships and general closed convex set constraints.By utilizing nonsmooth analysis and Lyapunov method,it is proved that the algorithm can obtain the optimal interval of interval optimization problems under specific weight parameters.The adaptive algorithm proposed in this section avoids estimating the lower bound of precise penalty parameters and having a lower solution space dimension by selecting appropriate gain parameters for punishment.At the same time,the feasibility of the algorithm is demonstrated through numerical simulation and an example in investment decision-making problems.3.For a class of nonsmooth distributed convex optimization problems that are only constrained by consistency,utilizing the superior properties of the consistency constraint penalty function,this section proposes a nonautonomous continuous-time algorithm with low communication cost.Under the undirected connected topology,the algorithm’s state solution will reach consensus and eventually converge,and achieve exponential convergence rate when the the objective function is strongly convex.This section applies the nonautonomous penalty-like method to consistency constraints,making the algorithm independent of global information,ensuring that each agent does not need to exchange private information such as local targets with other agents,thus achieving protection of individual privacy and saving communication costs.Finally,several typical numerical examples and quantile regression problems are used to illustrate the effectiveness of the algorithm.4.For a class of nonsmooth distributed convex optimization problems subject to inequality,general closed convex sets,and consistency constraints,in order to overcome the high model complexity caused by penalty for inequality constraints,this section precisely punishes closed convex sets and consistency constraints.Based on the Lagrange multiplier method,a continuous-time exact penalty algorithm is proposed and its convergence is proven.Compared with the traditional algorithm,this algorithm has the advantages of low model complexity and easy discretization.Design a projection based precise penalty algorithm under specific constraint conditions,where the state solution enters the constraint set in finite time to avoid potential risks of convergence points being infeasible.At the same time,this section verifies the effectiveness of the algorithm in several typical numerical examples.