The Convergence Theory Study Of Bi-level Stochastic Programming

There are a lot of hierarchical decision-making systems in the field of economic management, which can be abstracted into multi-level optimization models. Bi-level optimization problem is a basic form of hierarchical multi-level optimization problems. Because of large numbers of random phenomena in the process of decision-making, multi-level stochastic programming models with comprehensive practical background have been put forward. So far, multi-level stochastic programming has not been deeply researched, and the calculation methods are not effective. Calculation methods for the bi-level stochastic programming methods are researched based on summarizing relative theory in the paper, the main content as follows:1) This part proposed the convergence of approximate solutions for Bi-level stochastic programming and proved that any optimum solution sequence of corresponding problems will converge to one of the optimum solutions of the original problem if random vector sequence {ξ^(k)(ω)} converges toξ(ω) in distribution. These results provide the theoretical foundation for constructing approximate algorithms.2) This part discussed a method for Bi-level compensated stochastic programming in which the objective function is replaced by its empirical mean. This method converts a stochastic optimization problem into a deterministic one for which many methods are available. The advantage of the method is that there is no requirement on the distribution of the random variables involved.3) This part studied the stability in Bi-level probabilistic constrained programming firstly, and discusses a method for Bi-level probabilistic constrained stochastic programming in which the subjective function is replaced by its empirical mean. This method converts a stochastic optimization problem into a deterministic one for which many methods are available.