Abstract
The proximal point method (PPM) for solving maximal monotone operator inclusion problem is a highly powerful tool for algorithm design, analysis and interpretation. To accelerate convergence of the PPM, inertial PPM (iPPM) was proposed in the literature. In this note, we point out that some of the attractive properties of the PPM, e.g., the generated sequence is contractive with the set of solutions, do not hold anymore for iPPM. To partially inherit the advantages of the PPM and meanwhile incorporate inertial extrapolation steps, we propose an iPPM with alternating inertial steps. Our analyses show that the even subsequence generated by the proposed iPPM is contractive with the set of solutions. Moreover, we establish global convergence result under much relaxed conditions on the inertial extrapolation stepsizes, e.g., monotonicity is no longer needed and the stepsizes are significantly enlarged compared to existing methods. Furthermore, we establish certain $o(1/k)$ convergence rate results, where $k$ denotes the iteration counter. These features are new to inertial type PPMs.
Highlights
Let T : Rn ⇒ Rn be a set-valued maximal monotone operator
We propose an inertial PPM (iPPM) with alternating inertial steps, which inherits the contractive property of the proximal point method (PPM) to some extent
(iii) For iPPM, if xk+1 = xk ∈ T −1(0), all subsequent points will be equal to xk+1 ∈ T −1(0) because ∥xk+2 − x∗∥ ≤ ∥xk+1 − x∗∥ holds for any x∗ ∈ T −1(0)
Summary
Let T : Rn ⇒ Rn be a set-valued maximal monotone operator. We consider the following operator inclusion problem find x∗ ∈ Rn such that 0 ∈ T (x∗). Its efficient solution is of practical interests in many situations. The proximal point method (PPM, [25, 24, 31]) converts (1) to a fixed point problem of a firmly nonexpansive resolvent operator. The resolvent operator of T is defined by JλT := (I + λT )−1, i.e., for any x ∈ Rn, JλT (x) is the unique solution of 0 ∈ x + λT (x). Initialized at any x0 ∈ Rn, the PPM iterates for k ≥ 0 as xk+1 = JλT (xk)
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