Asynchronous Bundle Level Method For Minimax Problem

Nonsmooth optimization problem is a kind of problem with wide application background and profound theoretical research.Among them,theory and numerical algorithms have been widely used in big data,engineering applications,social services and many other fields.For a long time,they have been widely concerned by scholars at home and abroad.Nonsmooth optimization theory and algorithm is widely used in the field of machine learning,engineering design and optimization,image denoising,image restoration,generating unit optimization and the financial domain,etc.Minimax problem considering the objective function for the maximization of component functions,and it has a special distributed structure and the characteristics of nonsmooth.Minimax problem is produced in optimal control,environmental protection and energy,and many other fields.Some classic nonsmooth algorithms are naturally suitable for solving this problem.Since the Minimax problem has a special distributed structure,it is also an important direction to design a reasonable and efficient algorithm using its special structure.In this thesis,aiming at a nonsmooth optimization problem,namely Minimax problem,two asynchronous bundle level methods are proposed.Firstly,a completely asynchronous bundle level method for solving Minimax is proposed.On the basis of the classic bundle level method,due to the distributed structure of the Minimax problem,in each iteration,one only needs the first-order information of a certain component function to establish the lower approximation of the objective function.Therefore,the introduction of an asynchronous strategy enables the algorithm to effectively avoid the delay and waiting of different component functions when calculating the first-order information,and effectively improves the calculation efficiency of the algorithm;the related properties of the algorithm and the global convergence of the algorithm are analyzed.Secondly,a localizer-based asynchronous bundel level method with coordination is proposed.In the above method,at each iteration point,only a linearization of a certain component function is used to approximate the objective function,and the objective function cannot be approximated well which leads to the instability of the algorithm.By introducing the coordination step,the algorithm can generate a coordination step under certain conditions.At the coordination step,one can obtain the complete first-order information about the objective function and balance the instability of the algorithm.In addition,we consider using non-Euclidean distance instead of Euclidean distance.On the one hand,the structure of the feasible set can be reasonably used.On the other hand,the introduction of the localizer sequence in the algorithm can effectively adjust and control the number constraints of subproblem.And this technology can effectively avoid the storage pressure of the computer;the related properties of the algorithm and the global convergence of the algorithm are analyzed.Finally,some preliminary numerical experiments are done on the two asynchronous bundle level methods proposed in this thesis,and they are compared with the bundle level methods.The numerical results show the effectiveness of the asynchronous strategy and the rationality of the coordination step and localizer strategy.