Totally Convex Functions And The Proximal Point Algorithm In Banach Spaces

In the present researches of algorithms,Bregman's optimization method is an important subject,which promotes the development of the theories of algorithms.The method has been exploited in each aspect such as in the researches of optimization algorithms and computing fixed points of nonlinear mappings,etc..Bregman's optimization method is based on the theories of convex analysis and optimization,and the combination of the method with these theories promotes the deeper researches for convex functions.In the paper,we studied the notions related to Bregman's optimization method and their properties,and in the context of these knowledge applied the method to the optimization algorithms of some types of operators(the operators with Bregman's monotonicity type properties)in Banach spaces.The notion of totally convex functions is one of bases for Bregman's optimization method,and applying total convexity of functions in designing and analyzing iterative algorithms is a vital key to the class of methods.We considered the properties of totally convex functions and other classes of functions(uniformly convex functions and uniformly smooth functions),and investigated the properties of operators(the operators satisfying certain monotonicity type properties,resolvents of which have nonexpansivity type properties conditioned by Bregman functions,and Bregman projection operator,etc.)under the notion of total convexity.We applied the properties of these functions and operators to designing Bregman's optimization algorithms and discussing theirs convergence analyses.First,by using total convexity of functions,we studied the existence of the continuous selection of set-valued maps,and discussed the equivalence of the total convexity and uniform convexity of functions in some weaker sense.We gave the characteristics of uniform smoothness on bounded sets of functions.We introduced the notion of locally uniformly total convexity of convex functions,discussed its properties and applied those to algorithms interesting in some optimization problems.We introduced locally uniformly totally convex Banach spaces,and gave the characteristics of locally uniform total convexity of Banach spaces.Also,the existence of zeros and fixed points of some multivalued operators and single-valued operators in optimization problems and the existence of solutions of variational inequality problems were discussed.Second,in Banach spaces with a specific geometric structure,we used the properties of the function ||·||² to study Bregman optimization algorithms.In a uniformly convex and uniformly smooth Banach space X,for finding solutions to O∈T(u),where T:X(?)X^* is a maximal monotone operator,we studied a modified proximal point algorithm,discussed the strong or weak convergence of the algorithm and estimated the rate of convergence of the algorithm under different conditions on data,and applied the algorithm to a convex minimization problem.Additionally,considering a hybrid problem with a constraint set, i.e.,for finding a u∈T^-1O∩F^-1O∩C in the above Banach space X,where T is set-valued maximal monotone,C(?)X is a nonempty closed convex subset,and F is single-valued inverse monotone or its resolvent F_αis strongly nonexpansive,we studied a hybrid method of the extragradient method for F and the proximal point algorithm for T.We proved the weak convergence of the hybrid iterations under some assumptions on these operators and parameters,i.e.,the iteration sequences generated by converge weakly to a point of the above intersection.