The Approximate Proximal Algorithms For Maximal Monotone Operators In Banach Spaces

In this paper we are concerned with the approximate proximal algorithms of maximal monotone operator in Banach space. Let B be a reflexive Banach space, T:B→P(B⁺) is a maximal monotone operator. For solving problem (1): 0∈T(x). People have even adopted the method named proximal point algorithm (PPA), three methods have been established as follows:1. Approximate proximal point algorithm (PAAP).(Rockafellar used it first in 1976.)2. Projection approximate proximal point algorithm (PAPPA).3. Approximate proximal point algorithm using Bregman function. In Hilbert space, it is sufficient to adopt the first two methods. The third was used in reflexive space these years, this kind of algorithms are organized as follows:1. Choose x⁰∈B,2. Given x^k,, find (x|^^k,e^k)∈B×B such that:λ_k(f¹(x^k)-f¹(x|^^k)]-e^k∈T(x|^^k),where an inequality restriction about x|^^k,e^k,λ_k,x^k is satisfies.3. If x|^^k=x^k, then stop; Otherwise:define x^k+1=g(x|^^k), iterative. WhereD_f(x,y)=f(x)-f(y)-(f'(y),x-y) is the Bregman distance between x and y,fis a proper convex, lower semicontinuous and Gateaux differentiable function in B,f' is the G- differential coefficient of f; g is one function of x|^^k; λ_k satisfies some scope demmands.Different from the existing approximate proximal point algorithm using Bregman function, we give Algorithm â… in chapter 3, and substitute for ( was approximated by ). And set=.In order to obtain the convergence analysis of algorithmâ… , we list some preparative knowledge. The main conclusion of Chapter 3 is summarized in Theorem3.2.Theorem 3.2 (convergence) Let, satisfies ,, and , dom()=; Let {}be the sequence generated by Algorithmâ… ; If problem (1) has solutions, then:{} has weak accumulation points and all of them are solutions of (1).If also satisfies , then the whole sequence {} is weakly convergent to a solution of (1).From Theorem3.2 we find out that our new algorithm has the same convergence prorerties as the traditional approximate proximal alrigothms using function. It's a exciting result really.In Chapter 4,we establish a projection proximal algorithm in space: Algorithmâ…¡. This chapter has generaled He's algorithm in [11].We substitute the error rule:≤+,,≥0,=<1 ,=<1 ,<+.for the error rule in [11]:≤,=<1.Same as [11], we adopt the projection steps here. The proofs of Theorem 4.1, 4.2,4.3 lead to the convergence Theorem:Theorem 4.4 Let {},{},{} be the sequence generated by Algorithmâ…¡, then:1. {}has one weak accumulation ∈ at least.2. when =, ∈.In Chapter 5, after reviewing the conception: monotone (monotone) in [12], strictly monotone(strictly monotone) will be raised.In Theorem 5.2 a sufficient and necessary condition of convergence is listed which is about the sequence {} generated by Algorithmâ… :Theorem 5.2 Let {} be the sequence generated by Algorithmâ… , satisfies -, then: Be illumed by proposition 5.4, 5.5, we establish two significant inequalities:(31) (,)+(,)≤(,) , (>0) (32) (,)+(,)≥(1+ ) (,). (>0,>0)Meanwhile, Algorithmâ… has been extended to a general algorithm: Algorithm â…¢. We give the weak convergence analysis of Algorithmâ…¢ from Theorem 5.3, 5.4.Fortunately, we also find a case when the sequence {} generated by Algorithmâ…¢ is strongly convergent: Theorem 5.5 Let {} be the sequence generated by Algorithm â…¢; (0); and there is some a>0, >0 satisfies (31) and (32),then{} is strictly monotone about , If <<:(,)≤(,)=(,). (0<<1=...