On the complexity of a hybrid proximal extragradient projective method for solving monotone inclusion problems

Abstract

In a series of papers (Solodov and Svaiter in J Convex Anal 6(1):59–70, 1999; Set-Valued Anal 7(4):323–345, 1999; Numer Funct Anal Optim 22(7–8):1013–1035, 2001) Solodov and Svaiter introduced new inexact variants of the proximal point method with relative error tolerances. Point-wise and ergodic iteration-complexity bounds for one of these methods, namely the hybrid proximal extragradient method (1999) were established by Monteiro and Svaiter (SIAM J Optim 20(6):2755–2787, 2010). Here, we extend these results to a more general framework, by establishing point-wise and ergodic iteration-complexity bounds for the inexact proximal point method studied by Solodov and Svaiter (2001). Using this framework we derive global convergence results and iteration-complexity bounds for a family of projective splitting methods for solving monotone inclusion problems, which generalize the projective splitting methods introduced and studied by Eckstein and Svaiter (SIAM J Control Optim 48(2):787–811, 2009).

Appendix

The next result is a technical lemma borrowed from [16]. Given $L\in M(n)$, we define the associated linear mapping $L:\mathcal {H}^n\rightarrow \mathcal {H}^n$ via

$$\begin{aligned} Lu = L(u_1,\ldots ,u_n) = (v_1,\ldots ,v_n)\ \text{ with } v_i = \sum _{j=1}^n l_{ij} u_j \end{aligned}$$

(65)

and $l_{ij}$ denoting the elements of L for $i,j=1,\ldots ,n$.

Lemma 3

LetLbe any real$n\times n$matrix. For all$u=(u_1,\ldots ,u_n)\in \mathcal {H}^n$,

• 1. $\Vert Lu\Vert \le \Vert L\Vert \Vert u\Vert$,

• 2. $\langle {u},{Lu}\rangle \ge \kappa (L)\Vert u\Vert ^2$,

• 3. $\kappa (L)\le \Vert L\Vert$,

whereLuis defined by (65).

Proof

For the proof of items (1) and (2), see [16, Lemma 4.1]. Item (3) follows trivially from items (1) and (2) and Cauchy-Schwartz inequality. $\square$