Asymptotic Property Of Two Stage Linear Second-order Conic Stochastic Programs

The linear two-stage stochastic program is a class of important optimization problems which includes two stages of problems.It has many applications in our life,such as the newsboy problem and task assignment problem.In this paper,we carry on research in two aspects.On the one hand,we study the qualitative and quantitative stability analysis for the linear two-stage stochastic program with second-order cone constraint.Moreover we obtain Hadamard direction-ally differentiability,statistical inference and empirical approximation estimation for the optimal function of the second stage problem.On the other hand,we pay attention to the relationship between two-stage problem and the bilevel programming problem.Under a relatively weak,we use smoothing Lagrangian method to solve a nonsmooth and nonconvex bilevel program with abstract constraints.Then we prove the convergence condition for the algorithm and use numerical experiments to verify the effectiveness of the algorithm.In chapter 3,we study the stability analysis of the linear two stage second-order cone s-tochastic programming problem in which all parameters are random variables.Firstly we prove that the Slater condition of the perturbed problem and its dual problem both hold.Secondly we obtain the continuity and level boundedness for the feasible sets of the perturbed problem and its dual problem.Finally we demonstrate the upper semi-continuity of solution mappings for the perturbed problem and the Lagrangian dual problem.In chapter 4,we demonstrate that the optimal value function of the two stage problem can be expressed as a min-max optimization problem over two compact convex sets,and we use the Lagrangian dual theorem to prove that the optimal function is Lipschitz continuous and Hadamard directionally differentiable.Then we present the asymptotic distribution of a SAA(sample average approximation)estimator of the optimal value function for the second stage program.In chapter 5,we consider quantitative stability for full random two stage linear second-order conic stochastic programs when the probability distribution of random variables is per-turbed.We first investigate locally Lipschitz continuity for the feasible solution mappings of the primal problem and dual problem in the sense of Hausdorff distance which can derive the Lipschitz continuity for the objective function of the first stage problem.Then we establish the quantitative stability results for the optimal value function and the optimal solution mapping of the perturbation problem.Finally,we apply the results to the convergence analysis of optimal value function and optimal solution set for empirical approximations of the stochastic problems.In chapter 6,we use the augmented Lagrangian method to deal with a class of nonsmooth and nonconvex optimization problem with an abstract constraint.When the penalty parameter is bounded,we prove that any accumulation point of the iteration sequence generated by the algo-rithm is a feasible stationary point and the boundedness of the penalty parameter can be ensured by WNNAMCQ.Finally,we use the algorithm to solve the bilevel programming problem and show numerical experiments.