Expected Value Model And Sample Average Approximation Methods For Solving Stochastic Bilevel Programming Problems

The bilevel programming problem is a system optimization problem for a class of two-level hierarchical structures.Both the upper and lower level have their each objective function and constraint conditions.However,in the process of solving practical problems,it is often affected by some random factors,such as weather,demand,price,etc.If the decision makers ignore the existence of these factors in the process of solving practical problems,it will lead to decision-making mistakes,unable to get effective and reasonable results.Therefore,scholars gradually consider the bilevel programming problem with random factors.Due to the existence of random variables,the stochastic bilevel programming problem usually has no solutions in general.In order to meet the requirements of the solution with stochastic factors,we need to construct a reasonable deterministic model and the solution of the determined model is regarded as the solution of the stochastic bilevel programming problem.Therefore,the penalty function method is used to give the determined expectation value(EV)model of the stochastic bilevel programming problem,in order to get the solutions of the stochastic bilevel programming problem.Firstly,this paper considers the case where the lower level problem is convex programming,and uses the Karush-Kuhn-Tucker condition of the lower level problem to transform the stochastic bilevel programming problem into a single level optimization problem.Secondly,using the external penalty function,the single level random bilevel programming problem is transformed into the unconstrained optimization problem.Finally,since the EV model contains mathematical expectation,the mathematical expectation is not easy to solve.To solve this model,this paper applies the sample average approximation method to give the approximate problem of the EV model.In theory,this paper proves the convergence result of the global optimal solution of the EV model and the corresponding approximate model.Since the objective function in the EV model with penalty factor is complex,this paper first gives the corresponding smoothing method.The smoothing function of the objective function,and further using sample average approximation method,give the conclusion of the convergence of the stability point sequence.The content of this paper is summarized in detail,and the expected value model of SBLPP is further propose and prospects.