Constructing Mixed Algorithms on the Basis of Some Bundle Method

Abstract

© 2020, Springer Nature Switzerland AG. In the paper, a method is proposed for minimizing a nondifferentiable convex function. This method belongs to a class of bundle methods. In the developed method it is possible to periodically produce discarding all previously constructed cutting planes that form the model of the objective function. These discards are applied when approximation of the epigraph of the objective function is sufficiently good in the a neighborhood of the current iteration point, and the quality of this approximation is estimated by using the model of the objective function. It is proposed an approach for constructing mixed minimization algorithms on the basis of the developed bundle method with involving any relaxation methods. The opportunity to mix the developed bundle method with other methods is provided as follows. In the proposed method during discarding the cutting planes the main iteration points are fixed with the relaxation condition. Any relaxation minimization method can be used to build these points. Moreover, the convergence of all such mixed algorithms will be guaranteed by the convergence of the developed bundle method. It is important to note that the procedures for updating cutting planes introduced in the bundle method will be transferred to mixed algorithms. The convergence of the proposed method is investigated, its properties are discussed, an estimate of the accuracy of the solution and estimation of the complexity of finding an approximate solution are obtained.