| Optimization theory and methods is a widely used branch of mathematics.The optimization problems it researches generally exist in practical application fields such as engineering design,resource allocation and production planning.With the development of optimization theory,there are more and more studies on optimization methods and their convergence that can be used to solve optimization problems defined on Riemannian manifolds.The necessity of these studies is that many problems in practical application fields can not be reduced to optimization problems defined on linear space,but need to be defined and studied under the structure of Riemannian manifolds.In recent years,great progress has been made in the studies of optimization methods with subgradients on Riemannian manifolds and their convergence properties,which has attracted extensive attention from scholars at home and abroad.In this thesis,the incremental subgradient methods on Riemannian manifolds and their convergence analysis are studied.These methods are effective for solving large-scale optimization problems composed of the sum of several component functions on Riemannian manifolds.The main content of this thesis includes the following aspects.Firstly,an incremental subgradient method for solving large-scale unconstrained optimization problems consisting of the sum of several component functions on Riemannian manifolds is established.Each iteration of the method can be viewed as a cycle of subgradient iterations of each component functions in a fixed order.On this basis,the convergence properties of the method under the fixed stepsize rules are further studied.The fixed stepsize rules includes constant stepsize rule and diminishing stepsize rule.At the same time,some typical application examples are given.Secondly,the convergence properties of the incremental subgradient method on Riemannian manifolds under the dynamic stepsize rules are further studied.Several dynamic stepsizes are defined under the conditions that the optimal value is known and unknown,and the convergence properties of the method under these stepsize rules are given too.In addition,an incremental subgradient method for large-scale constrained optimization problems on Riemannian manifolds is defined,and some convergence results with theoretical significance and application value are obtained.Finally,the randomized incremental subgradient methods for large-scale unconstrained and constrained optimization problems on Riemannian manifolds are constructed respectively.The randomness of these methods is reflected in that on the one hand,each iteration of the methods randomly selects component function for subgradient iteration,on the other hand,each iteration uses the subgradient with random error as the search direction.The convergence properties of these methods under the fixed stepsize rules and the dynamic stepsize rules are also proved. |
