A Regularized Gradient Projection Method for the Minimization Problem

Abstract

We investigate the following regularized gradient projection algorithm 𝑥_𝑛+1=𝑃_𝑐(𝐼−𝛾_𝑛(∇𝑓+𝛼_𝑛𝐼))𝑥_𝑛, 𝑛≥0. Under some different control conditions, we prove that this gradient projection algorithm strongly converges to the minimum norm solution of the minimization problem min_𝑥∈𝐶𝑓(𝑥).

1. Introduction

Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐻→𝑅 be a real-valued convex function.

Consider the following constrained convex minimization problem:min_𝑥∈𝐶𝑓(𝑥). (1.1) Assume that (1.1) is consistent, that is, it has a solution and we use Ω to denote its solution set. If 𝑓 is Fréchet differentiable, then 𝑥^∗∈𝐶 solves (1.1) if and only if 𝑥^∗∈𝐶 satisfies the following optimality condition:∇𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (1.2) where ∇𝑓 denotes the gradient of 𝑓. Note that (1.2) can be rewritten as𝑥^∗−𝑥^∗−∇𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (1.3) This shows that the minimization (1.1) is equivalent to the fixed point problem𝑃_𝐶𝑥^∗−𝛾∇𝑓𝑥^∗=𝑥^∗, (1.4) where 𝛾>0 is any constant and 𝑃_𝐶 is the nearest point projection from 𝐻 onto 𝐶. By using this relationship, the gradient-projection algorithm is usually applied to solve the minimization problem (1.1). This algorithm generates a sequence {𝑥_𝑛} through the recursion:𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛,𝑛≥0, (1.5) where the initial guess 𝑥₀∈𝐶 is chosen arbitrarily and {𝛾_𝑛} is a sequence of stepsizes which may be chosen in different ways. The gradient-projection algorithm (1.5) is a powerful tool for solving constrained convex optimization problems and has well been studied in the case of constant stepsizes 𝛾_𝑛=𝛾 for all 𝑛. The reader can refer to [1–9] and the references therein. It is known [3] that if 𝑓 has a Lipschitz continuous and strongly monotone gradient, then the sequence {𝑥_𝑛} can be strongly convergent to a minimizer of 𝑓 in 𝐶. If the gradient of 𝑓 is only assumed to be Lipschitz continuous, then {𝑥_𝑛} can only be weakly convergent if 𝐻 is infinite dimensional. In order to get the strong convergence, Xu [10] studied the following regularized method:𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛,𝑛≥0, (1.6) where the sequences {𝜃_𝑛}⊂(0,1) and {𝛾_𝑛}⊂(0,∞) satisfy the following conditions:(1)0<𝛾_𝑛≤𝛼_𝑛/(𝐿+𝛼_𝑛)² for all 𝑛;(2)lim_𝑛→∞𝛼_𝑛=0; (3)∑^∞_𝑛=0𝛼_𝑛𝛾_𝑛=∞; (4)lim_𝑛→∞(|𝛾_𝑛−𝛾_𝑛−1|+|𝛼_𝑛𝛾_𝑛−𝛼_𝑛−1𝛾_𝑛−1|/(𝛼_𝑛𝛾_𝑛)²)=0.

Xu [10] proved that the sequence {𝑥_𝑛} converges strongly to a minimizer of (1.1).

Motivated by Xu’s work, in the present paper, we further investigate the gradient projection method (1.6). Under some different control conditions, we prove that this gradient projection algorithm strongly converges to the minimum norm solution of the minimization problem (1.1).

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.1) We will use Fix(𝑇) to denote the set of fixed points of 𝑇, that is, Fix(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}. A mapping 𝑇∶𝐶→𝐶 is said to be 𝜈-inverse strongly monotone (𝜈-ism), if there exists a constant 𝜈>0 such that𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐶. (2.2) Recall that the (nearest point or metric) projection from 𝐻 onto 𝐶, denoted 𝑃_𝐶, assigns, to each 𝑥∈𝐻, the unique point 𝑃_𝐶(𝑥)∈𝐶 with the property𝑥−𝑃_𝐶(𝑥)=inf𝑥−𝑦∶𝑦∈𝐶. (2.3) It is well known that the metric projection 𝑃_𝐶 of 𝐻 onto 𝐶 has the following basic properties:(a)‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻;(b)⟨𝑥−𝑦,𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)⟩≥‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖² for every 𝑥,𝑦∈𝐻;(c)⟨𝑥−𝑃_𝐶(𝑥),𝑦−𝑃_𝐶(𝑥)⟩≤0 for all 𝑥∈𝐻,𝑦∈𝐶.

Next we adopt the following notation:(i)𝑥_𝑛→𝑥 means that 𝑥_𝑛 converges strongly to 𝑥;(ii)𝑥_𝑛⇀𝑥 means that 𝑥_𝑛 converges weakly to 𝑥;(iii)𝜔_𝑤(𝑥_𝑛)∶={𝑥∶∃𝑥_𝑛𝑗⇀𝑥} is the weak 𝜔-limit set of the sequence {𝑥_𝑛}.

Lemma 2.1 (see [11]). Given 𝑇∶𝐻→𝐻 and letting 𝑉=𝐼−𝑇 be the complement of 𝑇, given also 𝑆∶𝐻→𝐻, (a)𝑇 is nonexpansive if and only if 𝑉 is (1/2)-ism;(b)if 𝑆 is 𝜈-ism, then, for 𝛾>0, 𝛾𝑆 is (𝜈/𝛾)-ism;(c)𝑆 is averaged if and only if the complement 𝐼−𝑆 is 𝜈-ism for some 𝜈>1/2.

Lemma 2.2 (see [12], (demiclosedness principle)). Let 𝐶 be a closed and convex subset of a Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦. (2.4) In particular, if 𝑦=0, then 𝑥∈Fix(𝑇).

Lemma 2.3 (see [13]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋, and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. (2.5) Suppose that 𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛 (2.6) for all 𝑛≥0 and limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (2.7) Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.4 (see [14]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (2.8) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(1)∑^∞_𝑛=1𝛾_𝑛=∞; (2)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞. Then, lim_𝑛→∞𝑎_𝑛=0.

3. Main Result

In this section, we will state and prove our main result.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝑅 be a real-valued Fréchet differentiable convex function. Assume Ω≠∅. Assume that the gradient ∇𝑓 is 𝐿-Lipschitzian. Let {𝑥_𝑛} be a sequence generated by the following hybrid gradient projection algorithm: 𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛,𝑛≥0, (3.1) where the sequences {𝜃_𝑛}⊂(0,1) and {𝛾_𝑛}⊂(0,2/(𝐿+2𝛼_𝑛)) satisfy the following conditions:(1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(2)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿 and lim_𝑛→∞(𝛾_𝑛+1−𝛾_𝑛)=0. Then, the sequence {𝑥_𝑛} generated by (3.1) converges to a minimizer 𝑥 of (1.1).

Proof. Note that the Lipschitz condition implies that the gradient ∇𝑓 is (1/𝐿)-ism [10]. Then, we have 𝑃_𝐶𝐼−𝛾∇𝑓+𝛼𝐼𝑥−𝑃_𝐶𝐼−𝛾∇𝑓+𝛼𝐼𝑦²≤𝐼−𝛾∇𝑓+𝛼𝐼𝑥−𝐼−𝛾∇𝑓+𝛼𝐼𝑦²=1−𝛼𝛾𝑥−𝑦−𝛾∇𝑓(𝑥)−∇𝑓𝑦²=1−𝛼𝛾²𝑥−𝑦²−21−𝛼𝛾𝛾𝑥−𝑦,∇𝑓(𝑥)−∇𝑓𝑦+𝛾²∇𝑓(𝑥)−∇𝑓𝑦²≤1−𝛼𝛾²𝑥−𝑦²𝛾1−21−𝛼𝛾𝐿∇𝑓(𝑥)−∇𝑓𝑦²+𝛾²∇𝑓(𝑥)−∇𝑓(𝑦)². (3.2) If 𝛾∈[0,2/(𝐿+2𝛼)], then 2(1−𝛼𝛾)𝛾(1/𝐿)≥𝛾². It follows that 𝑃_𝐶𝐼−𝛾∇𝑓+𝛼𝐼𝑥−𝑃_𝐶𝐼−𝛾∇𝑓+𝛼𝐼𝑦²≤1−𝛼𝛾²𝑥−𝑦². (3.3) Thus, 𝑃_𝐶𝐼−𝛾∇𝑓+𝛼𝐼𝑥−𝑃_𝐶𝐼−𝛾∇𝑓+𝛼𝐼𝑦≤1−𝛼𝛾𝑥−𝑦, (3.4) for all 𝑥,𝑦∈𝐶.
Take any 𝑥^∗∈𝑆. Since 𝑥^∗∈𝐶 solves the minimization problem (1.1) if and only if 𝑥^∗ solves the fixed-point equation 𝑥^∗=𝑃_𝐶(𝐼−𝛾∇𝑓)𝑥^∗ for any fixed positive number 𝛾, so we have 𝑥^∗=𝑃_𝐶(𝐼−𝛾_𝑛∇𝑓)𝑥^∗ for all 𝑛≥0. From (3.1) and (3.4), we get 𝑥_𝑛+1−𝑥^∗=𝑃_𝐶𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛−𝑃_𝐶𝐼−𝛾_𝑛∇𝑓𝑥^∗=𝑃_𝐶𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛−𝑃_𝐶𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥^∗+𝑃_𝐶𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥^∗−𝑃_𝐶𝐼−𝛾_𝑛∇𝑓𝑥^∗≤1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾_𝑛𝑥^∗≤max𝑥_𝑛−𝑥^∗,𝑥^∗. (3.5) Thus, we deduce by induction that 𝑥_𝑛−𝑥^∗≤max𝑥₀−𝑥^∗,𝑥^∗. (3.6) This indicates that the sequence {𝑥_𝑛} is bounded.
Since the gradient ∇𝑓 is (1/𝐿)-ism, 𝛾∇𝑓 is (1/𝛾𝐿)-ism. So by Lemma 2.1, 𝐼−𝛾_𝑛∇𝑓 is (𝛾_𝑛𝐿/2)-averaged; that is, 𝐼−𝛾_𝑛∇𝑓=(1−(𝛾_𝑛𝐿/2))𝐼+(𝛾_𝑛𝐿/2)𝑇 for some nonexpansive mapping 𝑇. Since 𝑃_𝐶 is (1/2)-averaged, 𝑃_𝐶=(𝐼+𝑆)/2 for some nonexpansive mapping 𝑆. Then, we can rewrite 𝑥_𝑛+1 as 𝑥_𝑛+1=12𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛=12𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛=2−𝛾_𝑛𝐿4𝑥_𝑛+𝛾_𝑛𝐿4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛=2−𝛾_𝑛𝐿4𝑥_𝑛+2+𝛾_𝑛𝐿4𝑦_𝑛, (3.7) where 𝑦_𝑛=42+𝛾_𝑛𝐿𝛾_𝑛𝐿4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛. (3.8) It follows that ‖𝑦_𝑛+1−𝑦_𝑛‖=42+𝛾_𝑛+1𝐿𝛾_𝑛+1𝐿4𝑇𝑥_𝑛+1−12𝛾_𝑛+1𝛼_𝑛+1𝑥_𝑛+1+12𝑆𝐼−𝛾_𝑛+1∇𝑓+𝛼_𝑛+1𝐼𝑥_𝑛+1−42+𝛾_𝑛𝐿𝛾_𝑛𝐿4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛≤42+𝛾_𝑛+1𝐿𝛾_𝑛+1𝐿4𝑇𝑥_𝑛+1−12𝛾_𝑛+1𝛼_𝑛+1𝑥_𝑛+1+12𝑆𝐼−𝛾_𝑛+1∇𝑓+𝛼_𝑛+1𝐼𝑥_𝑛+1−𝛾_𝑛𝐿4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛+42+𝛾_𝑛+1𝐿−42+𝛾_𝑛𝐿𝛾_𝑛𝐿4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛≤42+𝛾_𝑛+1𝐿𝛾_𝑛+1𝐿4𝑇𝑥_𝑛+1−𝛾_𝑛𝐿4𝑇𝑥_𝑛+12𝛾_𝑛+1𝛼_𝑛+1𝑥_𝑛+1+12𝛾_𝑛𝛼_𝑛𝑥_𝑛+22+𝛾_𝑛+1𝐿𝐼−𝛾_𝑛+1∇𝑓+𝛼_𝑛+1𝐼𝑥_𝑛+1−𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛+42+𝛾_𝑛+1𝐿−42+𝛾_𝑛𝐿𝛾_𝑛𝐿4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛. (3.9) Now we choose a constant 𝑀 such that sup_𝑛𝑥_𝑛,𝐿𝑇𝑥_𝑛,∇𝑓𝑥_𝑛,𝛾_𝑛𝐿4𝑇𝑥_𝑛−12𝛾_𝑛𝛼_𝑛𝑥_𝑛+12𝑆𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛≤𝑀. (3.10) We have the following estimates: 𝛾_𝑛+1𝐿4𝑇𝑥_𝑛+1−𝛾_𝑛𝐿4𝑇𝑥_𝑛=𝛾_𝑛+1𝐿4𝑇𝑥_𝑛+1−𝑇𝑥_𝑛+𝛾_𝑛+1𝐿4−𝛾_𝑛𝐿4𝑇𝑥_𝑛≤𝛾_𝑛+1𝐿4𝑇𝑥_𝑛+1−𝑇𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑇𝑥_𝑛4≤𝛾_𝑛+1𝐿4𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑀,𝐼−𝛾_𝑛+1∇𝑓+𝛼_𝑛+1𝐼𝑥_𝑛+1−𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛≤𝐼−𝛾_𝑛+1∇𝑓𝑥_𝑛+1−𝐼−𝛾_𝑛+1∇𝑓𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛∇𝑓𝑥_𝑛+𝛾_𝑛+1𝛼_𝑛+1‖𝑥_𝑛+1‖+𝛾_𝑛𝛼_𝑛‖𝑥_𝑛‖≤𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀. (3.11) Thus, we deduce 𝑦_𝑛+1−𝑦_𝑛≤42+𝛾_𝑛+1𝐿𝛾_𝑛+1𝐿4𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑀+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀+22+𝛾_𝑛+1𝐿𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀+42+𝛾_𝑛+1𝐿−42+𝛾_𝑛𝐿𝑀≤𝑥_𝑛+1−𝑥_𝑛+62+𝛾_𝑛+1𝐿𝛾_𝑛+1−𝛾_𝑛+𝛾_𝑛+1𝛼_𝑛+1+𝛾_𝑛𝛼_𝑛𝑀+4𝐿2+𝛾_𝑛+1𝐿2+𝛾_𝑛𝐿𝛾_𝑛+1−𝛾_𝑛𝑀. (3.12) Note that 𝛼_𝑛→0 and 𝛾_𝑛+1−𝛾_𝑛→0. Hence, by Lemma 2.3, we get limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.13) It follows that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.14) Consequently, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=lim_𝑛→∞2+𝛾_𝑛𝐿4‖𝑦_𝑛−𝑥_𝑛‖=0. (3.15) Now we show that the weak limit set 𝜔_𝑤(𝑥_𝑛)⊂Ω. Choose any 𝑥∈𝜔_𝑤(𝑥_𝑛). Since {𝑥_𝑛} is bounded, there must exist a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥. At the same time, the real number sequence {𝛾_𝑛𝑗} is bounded. Thus, there exists a subsequence {𝛾_𝑛𝑗𝑖} of {𝛾_𝑛𝑗} which converges to 𝛾. Without loss of generality, we may assume that 𝛾_𝑛𝑗→𝛾. Note that 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿. So, 𝛾∈(0,2/𝐿); that is, 𝛾_𝑛𝑗→𝛾∈(0,2/𝐿) as 𝑗→∞. Next, we only need to show that 𝑥∈Ω. First, from (3.15) we have that ‖𝑥_𝑛𝑗+1−𝑥_𝑛𝑗‖→0. Then, we have 𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾∇𝑓𝑥_𝑛𝑗≤𝑥_𝑛𝑗−𝑥_𝑛𝑗+1+𝑥_𝑛𝑗+1−𝑃_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗+𝑃_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾∇𝑓𝑥_𝑛𝑗=𝑃_𝐶𝐼−𝛾_𝑛𝑗∇𝑓+𝛼_𝑛𝑗𝐼𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗+𝑃_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝛾∇𝑓𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥_𝑛𝑗+1≤𝛼_𝑛𝑗𝛾_𝑛𝑗𝑥_𝑛𝑗+𝛾_𝑛𝑗−𝛾∇𝑓𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥_𝑛𝑗+1⟶0. (3.16) Since 𝛾∈(0,2/𝐿), 𝑃_𝐶(𝐼−𝛾∇𝑓) is nonexpansive. It then follows from Lemma 2.2 (demiclosedness principle) that 𝑥∈Fix(𝑃_𝐶(𝐼−𝛾∇𝑓)). Hence, 𝑥∈Ω because of Ω=Fix(𝑃_𝐶(𝐼−𝛾∇𝑓)). So, 𝜔_𝑤(𝑥_𝑛)⊂Ω.
Finally, we prove that 𝑥_𝑛→𝑥, where 𝑥 is the minimum norm solution of (1.1). First, we show that limsup_𝑛→∞⟨𝑥,𝑥_𝑛−𝑥⟩≥0. Observe that there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} satisfying limsup_𝑛→∞𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝑥,𝑥_𝑛𝑗−𝑥. (3.17) Since {𝑥_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑖} of {𝑥_𝑛𝑗} such that 𝑥_𝑛𝑗𝑖⇀𝑥. Without loss of generality, we assume that 𝑥_𝑛𝑗⇀𝑥. Then, we obtain limsup_𝑛→∞𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝑥,𝑥_𝑛𝑗−𝑥=⟨𝑥,𝑥−𝑥⟩≥0. (3.18) Since 𝛾_𝑛<2/(𝐿+2𝛼_𝑛), 𝛾_𝑛/(1−𝛼_𝑛𝛾_𝑛)<2/𝐿. So, 𝐼−(𝛾_𝑛/(1−𝛼_𝑛𝛾_𝑛))∇𝑓 is nonexpansive. By using the property (b) of 𝑃_𝐶, we have 𝑥_𝑛+1−𝑥²=𝑃_𝐶𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛−𝑃_𝐶𝑥−𝛾_𝑛∇𝑓(𝑥)²≤𝐼−𝛾_𝑛∇𝑓+𝛼_𝑛𝐼𝑥_𝑛−𝑥−𝛾_𝑛∇𝑓(𝑥),𝑥_𝑛+1=−𝑥1−𝛼_𝑛𝛾_𝑛𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛∇𝑓𝑥_𝑛−𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛∇𝑓𝑥,𝑥_𝑛+1−𝑥−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1≤−𝑥1−𝛼_𝑛𝛾_𝑛𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛∇𝑓𝑥_𝑛−𝛾𝐼−^𝑛1−𝛼_𝑛𝛾_𝑛∇𝑓𝑥𝑥_𝑛+1−𝑥−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1≤−𝑥1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥𝑥_𝑛+1−𝑥−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1≤−𝑥1−𝛼_𝑛𝛾_𝑛2𝑥_𝑛−𝑥²+12𝑥_𝑛+1−𝑥²−𝛼_𝑛𝛾_𝑛𝑥,𝑥_𝑛+1−𝑥. (3.19) It follows that 𝑥_𝑛+1−𝑥²≤1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥²+𝛼_𝑛𝛾_𝑛−𝑥,𝑥_𝑛+1−𝑥. (3.20) From Lemma 2.4, (3.18) and (3.20), we deduce that 𝑥_𝑛→𝑥. This completes the proof.

Remark 3.2. We obtain the strong convergence of the regularized gradient projection method (3.1) under some different control conditions.

Remark 3.3. From the proof of result, we observe that our algorithm (3.1) converges to a special solution 𝑥 of the minimization (1.1). As a matter of fact, this special solution 𝑥 is the minimum-norm solution of the minimization (1.1). Finding the minimum-norm solution of practical problem is an interesting work due to its applications. A typical example is the least-squares solution to the constrained linear inverse problem; see, for example, [15]. For some related works on the minimum-norm solution and the minimization problems, please see [16–22].

Acknowledgments

Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279, and NSFC 71161001-G0105. W. Jigang is supported in part by NSFC 61173032.