Theoretical Research And Application Of Distributionally Robust Least Squares Problem

Many practical issues such as curve fitting and model prediction can be solved through transforming into the least squares problem.Due to the uncertainty of parameters in these issues,it is possible to use the partial information of the historical data to construct the set of uncertain distributions.Two robust frameworks under uncertain sets defined by probabilistic uncertainty are presented in this paper.Specifically,Where X is the compact set in Rⁿ,A?R^m?nandb?R^mis the known matrix and vector,respectively,x_A?R^m?nandx_b?R^m are random errors,P is a distribution of A and b,which is controlled inl,the set of uncertain distributions.The uncertain set can be characterized by following two ways:?1?the uncertain set described by the moment constraint which the measure is bounded;?2?the uncertain set described by a Kantorovich distance which have a given reference measure.The uncertain set for the real distribution usually can be constructed by using the empirical distribution derived from the historical data as the reference distribution.The first uncertain set is defined by the first and second moments of the historical data,and the original problem can be transformed into a convex optimization problem at the same time.It can be solved by using the cutting plane method in finite steps when the sample space has the finite support.This algorithm can be realized by correlation solver of linear programming and linear cone programming.In addition,the optimal solution obtained by the discrete form can converge to the optimal solution of the original problem under certain conditions.The second uncertain set is constructed via defining the distance between the reference distribution and the true distribution by using the probability measure,and this method guarantees the convergence of the problem.Then the original problem is proved to be equivalent to a second-order cone model by using duality theories,and it can be solved by a cutting plane algorithm of support vector machine in the case of the sample space has the finite support.Finally,the application of distributionally robust least squares problem is given.