Study On Algorithms For Several Classes Of Stochastic Optimization Problems

Stochastic optimization problems refer to the optimization problems with mathematical expectation or probability functions in the objective or constraints.There are two main methods of stochastic optimization:sample average approximation method and stochastic approximation method.This paper mainly studies the sample average approximation method for stochastic second-order cone and stochastic semidefinite programming problems,the stochastic approximation method for stochastic nonlinear equations,and the stochastic augmented Lagrange method for stochastic convex constrained optimization problems.The main results of this dissertation can be summarized as follows:1.Chapter 3 discusses the asymptotic convergence of the optimal value and optimal solutions of the stochastic approximate model generated under the sample average approximation method for two types of stochastic cone constrained optimization models(namely,second-order cone programming and semidefinite programming).The random variables are both in the objective function and constraints of the optimization problem.Firstly,we consider the stochastic second-order cone optimization problem.The asymptotic convergence of the approximation optimal value and optimal solution is different when the optimal solution set is single or not.And the properties are applied to a special kind of stochastic second-order cone programming—the problem of minimizing a sum of norms with weights.Secondly,for the stochastic semidefinite programming,we establish the asymptotic convergence of the approximation optimal solutions and the approximate optimal value.Finally,the properties of the optimal solution and the optimal value generated by the algorithm are verified on several cone constrained optimization problems.2.Chapter 4 establishes a stochastic Newton method for the stochastic nonlinear equation system.The stochastic approximation functions and the stochastic approximation Jacobian matrixes are constructed using the stochastic zero-and first-order oracles.The stochastic Newton method generates the iterative direction by the inexact Newton method,and the step size by the inexact line search.This paper achieves the almost sure global convergence and computational complexity of the algorithm.Further,if the sample size is appropriately selected,the algorithm can also establish a local superlinear convergence rate with a certain probability.Finally,on several large data sets,we verify the convergence of the algorithm and discuss the influence of algorithm parameters on the convergence rate.3.Chapter 5 proposes a stochastic augmented Lagrange method for the stochastic convex nonlinear programming.The stochastic method does not rely on the special selection of random approximate models.Firstly,it is proved that when the approximate model is sufficiently accurate with a sufficiently large but fixed probability,the sequence of multipliers generated by the algorithm converges with probability 1.Secondly,under the generalized Slater condition,the sequence of the iteration converges to the optimal solution of the stochastic nonlinear programming problem with probability 1.In addition,under different noises(biased and unbiased),we construct the corresponding approximation model and the selection of parameters of the algorithm.Finally,the preliminary numerical experiment of the algorithm is given.For different optimization models,we verify the effectiveness of the algorithm and discuss the influence of parameters on the convergence of the algorithm.