Approximation To Solutions For Functional Equations Via Bundle Methods

Bundle methods have shown a very high effectiveness in solving nonsmooth optimization problems.In view of the complexity and variability of practical problems,the extension and application of the related theoretical results of the bundle methods often have a very high practical value.For functional equations,regardless of the types of dynamic programming that appear in the multistage decision process,or in many other scientific fields,they have played a very important role in all aspects of the field of scientific research.In this paper,by combining the bundle idea with the common iterative process of searching for the solutions for a special kind of functional equations,we try to search for a new method to approximate the unique solution of functional equations,it is also a new application of bundle method of nonsmooth optimization.We assume that the function arising in functional equation u:S×D→R⁺ is a nonsmooth,lower semi continuous,normal convex function,then we can construct a piecewise linear approximation function u_n（x,y）,for u（x,y） and present a new method to approximate the unique solution of functional equations under certain conditions.In addition,the innovation of this paper lies in the fact that the condition which assures the convergence of the proposed iterative sequence is weaker than the existing one [1],and we only require the piecewise linear convex approximate functions satisfy uniformly the boundedness with respect to y.This allows us to reduce a lot of unnecessary restrictions on the implementation process.Our main results extend the results due to [1] and open a new way to study the solutions of functional equations.The main structure of this paper is as follows: Firstly,some basic concepts and results are presented.We unify some symbols and cite the corresponding definitions and lemmas for the convergence analysis.Note that we use the idea of bundle methods to approximate the objective function by a series of functions which are easier to deal with in this paper,and next this paper describes the relevant knowledge of the bundle methods for nonsmooth optimization.Secondly,in the main part of this thesis,we put forward specific methods and measures of approximation to solution for functional equation（1）,not only give the specific iterative approximate sequence{w_n}_n≥0 which converges to the unique solution of functional equation（1）,but also prove the convergence of iterative sequence.Finally,we compare the results obtained with the existing conclusions,and find that the construction of iterative sequence {w_n}_n≥0 is not only dependent on the previous iteration sequence w_n-1,but also onthe approximate function u_n-1,and we relax the strong uniform boundedness conditions imposed on u itself.