Numerical Study On Optimal Control Problems And Stochastic Programming

In this thesis,some novel numerical methods are developed to solve a class of optimal control problems and stochastic optimization problems.In the last decade,optimal control problems have been developed rapidly in the fields of applied and computational mathematics.Optimal control problems play a key role involved in the implementation of scientific research,industry and engineering,financial investment,etc.,including geophysics,climate science,materials science,medical imaging,shape design,machinery manufacturing,option pricing,and so on.As a consequence,researchers have been motivated in the design and analysis of efficient computational methods for solving optimal control problems.The ultimate aim of optimal control is to manipulate or control the systems described by partial differential equations in a desired way,to find the optimal "control" which satisfies the requirements in the system,and to make the "state" adequately close to or reach the target state.We focus on the numerical methods for Poisson constrained optimal control problem,which can be expressed as a convex optimization problem with linear constraint in the Banach space as follows(?)J(y,u)=?(y)+?(u)s.t.Ay+Bu=p,u?Uad.Here,functions y and u represent the "state" and "control" in the systems,respectively.The linear constraint Ay+Bu=p can be specified by the variational forms of different types of Poisson equations.With the finite element method approximation,we will obtain a class of linear con-strained optimization problems in Rn,which is given by(?)Jh(yh,uh)=?h(yh)+?h(uh)s.t.Ayh+Buh=ph,uh?Uadh.Here,the matrices A and B are the well-known finite element stiffness and mass matrices,and yh,uh,Ph ?Rn.It is obvious that the above problem has a separable structure,and we will adopt the alternating direction method of multipliers(ADMM)to solve this problem.ADMM is an effective algorithm and is a benchmark tool for various convex minimization models with separable objective functions and linear constraints,which re-sults in subproblems solvable at each iteration.For the discretized Poisson constrained optimal control problem with different control types,we design different iterative schemes of ADMM for solving the corresponding problems.The error analyses of our numerical method consist of two parts:the approximation error of finite element discretization and the iterative error of ADMM.Besides,we present the total error estimates in the form of the objective function value as|J(y*,u*)-Jh(yhK,uhK)|,where(y*,u*)and(yhK,uhK)denote the optimal solution of Poisson constrained optimal control problem and the iterative of the discretized problem,respectively.Here,K repre-sents the number of iterations.Numerical experiments are performed to verify the efficiencies of our methods.Stochastic optimization problems are widely used in the fields of artificial intelligence.Statistical learning,machine learning,deep learning,and the intelligent systems are all rely on stochastic optimization problems.Such as search engines,recommendation platforms,and speech and image recognition software,have become an indispensable part of modern society.The success of stochastic optimization methods for variety learning models has inspired researchers to tackle more challenging stochastic optimization problems and to design new methods.In this work,a block mirror stochastic gradient method is developed to solve stochas-tic optimization problems as(?)f(x)=E[F(x,?)],(1)and the corresponding composite problems,(?)(2)where the feasible set X and the variables x are treated as multiple blocks.Taking the advantage of the coordinate structure,the proposed method combines the feature of classic mirror descent stochastic approximation method and the block coordinate gradient descent method.The former can solve stochastic optimization problems or large-scale optimization problems efficiently,and the latter has significant efficiencies in solving optimization problems with multiple blocks.Our method update all the blocks of variables by Gauss-Seidel type iteration with reshuffled order of each block.We established the convergence of our numerical method for problem(1)in convex case,and for problem(2)in both convex and nonconvex cases.For both problems in convex cases,we show that it has the same order of convergence rate as that of the classic stochastic gradient method.For nonconvex case of problems problem(2),we establish the estimate in terms of the expectation violation of first-order optimality conditions.The analysis of our method is challenging since we need more specific assumption,which need more analysis technique.Numerical simulations verify that our methods are promising and comparable with other existing methods.