Algorithm Of Stochastic Quadratic Programming With Recourse Under LPI

As an optimization tool,stochastic programming has been widely used in economics,industry and other fields.In addition to the traditional stochastic linear programming,the most frequently studied is the stochastic quadratic programming problem,mainly because it can be better applied to practical problems.For the solution of the stochastic quadratic programming problem,the most widely used is the L-shape algorithm,which can solve large-scale problems.However,due to the complexity of the calculation,the improvement of the L-shape algorithm is also concerned.In many known studies,most of them solve the stochastic programming problem based on the fact that the probability distribution of stochastic variables is completely known.However,when solving practical problems,it is impossible to obtain complete information of the probability distribution,and can only get partial information in fact.This paper introduces LPI theory and solves stochastic quadratic programming problems when stochastic variables satisfy LPI.Two methods are proposed in this paper.The first one is the improved Nelder-Mead method,which achieves the optimal solution by transforming the polyhedron.Compared to the traditional Nelder-Mead method,an adaptive random search(ARS)is used instead of compression to avoid the algorithm converge to a local solution.The second is the sub-gradient algorithm.Since the two-stage function is not differential after applying LPI,it can't be solved by analytical method.Through a theorem,the formula for solving the sub-gradient is obtained,and the sub-gradient algorithm is designed.The quadratic programming problem is solved and the convergence proof of the algorithm is given.In the last part,the paper solves the specific examples by two methods,and the final results also show the effectiveness of the algorithm.