Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

Abstract

Consider the variational inequality VI(𝐶,𝐹) of finding a point 𝑥^∗∈𝐶 satisfying the property ⟨𝐹𝑥^∗,𝑥−𝑥^∗⟩≥0, for all 𝑥∈𝐶, where 𝐶 is the intersection of finite level sets of convex functions defined on a real Hilbert space 𝐻 and 𝐹∶𝐻→𝐻 is an 𝐿-Lipschitzian and 𝜂-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of VI(𝐶,𝐹). Since our algorithm avoids calculating the projection 𝑃_𝐶 (calculating 𝑃_𝐶 by computing several sequences of projections onto half-spaces containing the original domain 𝐶) directly and has no need to know any information of the constants 𝐿 and 𝜂, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

1. Introduction

The variational inequality problem can mathematically be formulated as the problem of finding a point 𝑥^∗∈𝐶 with the property 𝐹𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (1) where 𝐻 is a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, 𝐶 is a nonempty closed convex subset of 𝐻, and 𝐹∶𝐶→𝐻 is a nonlinear operator. Since its inception by Stampacchia [1] in 1964, the variational inequality problem VI(𝐶,𝐹) has received much attention due to its applications in a large variety of problems arising in structural analysis, economics, optimization, operations research and engineering sciences; see [1–23] and the references therein. Using the projection technique, one can easily show that VI(𝐶,𝐹) is equivalent to the fixed-point problem (see, for example, [15]).

Lemma 1. 𝑥^∗∈𝐶 is a solution of 𝑉𝐼(𝐶,𝐹) if and only if 𝑥^∗∈𝐶 satisfies the fixed-point relation: 𝑥^∗=𝑃_𝐶(𝐼−𝜆𝐹)𝑥^∗, (2) where 𝜆>0 is an arbitrary constant, 𝑃_𝐶 is the orthogonal projection onto 𝐶, and 𝐼 is the identity operator on 𝐻.

Recall that an operator 𝐹∶𝐶→𝐻 is called monotone, if 𝐹𝑥−𝐹𝑦,𝑥−𝑦≥0∀𝑥,𝑦∈𝐶. (3) Moreover, a monotone operator 𝐹 is called strictly monotone if the equality “=” holds only when 𝑥=𝑦 in the last relation. It is easy to see that VI(𝐶,𝐹) (1) has at most one solution if 𝐹 is strictly monotone.

For variational inequality (1), 𝐹 is generally assumed to be Lipschitzian and strongly monotone on 𝐶; that is, for some constants 𝐿,𝜂>0,  𝐹 satisfies the conditions 𝐹𝑥−𝐹𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶,⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (4) In this case, 𝐹 is also called an 𝐿-Lipschitzian and 𝜂-strongly monotone operator. It is quite easy to show the simple result as follows.

Lemma 2. Assume that 𝐹 satisfies conditions (4) and 𝜆 and 𝜇 are constants such that 𝜆∈(0,1) and 𝜇∈(0,2𝜂/𝐿²), respectively. Let 𝑇^𝜇=𝑃_𝐶(𝐼−𝜇𝐹) (or 𝐼−𝜇𝐹) and 𝑇^𝜆,𝜇=𝑃_𝐶(𝐼−𝜆𝜇𝐹) (or 𝐼−𝜆𝜇𝐹). Then 𝑇^𝜇 and 𝑇^𝜆,𝜇 are all contractions with coefficients 1−𝜏 and 1−𝜆𝜏, respectively, where 𝜏=(1/2)𝜇(2𝜂−𝜇𝐿²).

Using Banach's contraction mapping principle, the following well-known result can be obtained easily from Lemmas 1 and 2.

Theorem 3. Assume that 𝐹 satisfies the conditions (4). Then 𝑉𝐼(𝐶,𝐹) has a unique solution. Moreover, for any 0<𝜆<2𝜂/𝐿², the sequence {𝑥_𝑛} with initial guess 𝑥₀∈𝐶 and defined recursively by 𝑥_𝑛+1=𝑃_𝐶(𝐼−𝜆𝐹)𝑥_𝑛,𝑛≥0, (5) converges strongly to the unique solution of 𝑉𝐼(𝐶,𝐹).

However, Algorithm (5) has two evident weaknesses. On one hand, Algorithm (5) involves calculating the mapping 𝑃_𝐶, while the computation of a projection onto a closed convex subset is generally difficult. If 𝐶 is the intersection of finite closed convex subsets of 𝐻, that is, 𝐶=⋂^𝑚_𝑖=1𝐶_𝑖(≠∅), where 𝐶_𝑖(𝑖=1,…,𝑚) is a closed convex subset of 𝐻, then the computation of 𝑃_𝐶 is much more difficult. On the other hand, the determination of the stepsize 𝜆 depends on the constants 𝐿 and 𝜂. This means that in order to implement Algorithm (5), one has first to compute (or estimate) the constants 𝐿 and 𝜂, which is sometimes not an easy work in practice.

In order to overcome the above weaknesses of the algorithm (5), a new relaxed and self-adaptive algorithm is proposed in this paper to solve VI(𝐶,𝐹), where 𝐶 is the intersection of finite level sets of convex functions defined on 𝐻 and 𝐹∶𝐻→𝐻 is an 𝐿-Lipschitzian and 𝜂-strongly monotone operator. Our method calculates 𝑃_𝐶 by computing finite sequences of projections onto half-spaces containing the original set 𝐶 and selects the stepsizes through a self-adaptive way. The implementation of our algorithm avoids computing 𝑃_𝐶 directly and has no need to know any information about 𝐿 and 𝜂.

The rest of this paper is organized as follows. Some useful lemmas are listed in the next section; in particular, a new lemma is established in order to prove strong convergence theorems of our algorithms, which can also be used as a fundamental tool for solving some nonlinear problems relating to fixed point. In the last section, a relaxed algorithm (for the case where 𝐿 and 𝜂 are known) and a relaxed self-adaptive algorithm (for the case where 𝐿 and 𝜂 are not known) are proposed, respectively. The strong convergence theorems of our algorithms are proved.

2. Preliminaries

Throughout the rest of this paper, we denote by 𝐻 a real Hilbert space and by 𝐼 the identity operator on 𝐻. If 𝑓∶𝐻→ℝ is a differentiable functional, then we denote by ∇𝑓 the gradient of 𝑓. We will also use the following notations: (i)→ denotes strong convergence.(ii)⇀ denotes weak convergence.(iii)𝜔_𝑤(𝑥_𝑛)={𝑥∣∃{𝑥_𝑛𝑘}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑘⇀𝑥} denotes the weak 𝜔-limit set of {𝑥_𝑛}.

Recall a trivial inequality, which is well known and in common use.

Lemma 4. For all 𝑥,𝑦∈𝐻, there holds the following relation: 𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩. (6)

Recall that a mapping 𝑇∶𝐻→𝐻 is said to be nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,𝑥,𝑦∈𝐻. (7)𝑇∶𝐻→𝐻 is said to be firmly nonexpansive if, for 𝑥,𝑦∈𝐻, 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²−(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦². (8)

The following are characterizations of firmly nonexpansive mappings (see [7] or [24]).

Lemma 5. Let 𝑇∶𝐻→𝐻 be an operator. The following statements are equivalent.(i)𝑇 is firmly nonexpansive. (ii)𝐼−𝑇 is firmly nonexpansive. (iii)‖𝑇𝑥−𝑇𝑦‖²≤⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩,  𝑥,𝑦∈𝐻.

We know that the orthogonal projection 𝑃_𝐶 from 𝐻 onto a nonempty closed convex subset 𝐶⊂𝐻 is a typical example of a firmly nonexpansive mapping [7], which is defined by 𝑃_𝐶𝑥∶=argmin_𝑦∈𝐶𝑥−𝑦²,𝑥∈𝐻. (9) It is well known that 𝑃_𝐶𝑥 is characterized [7] by the inequality (for 𝑥∈𝐻) 𝑃_𝐶𝑥∈𝐶,𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥≤0,∀𝑦∈𝐶. (10)

It is well known that the following lemma [25] is often used when we analyze the strong convergence of some algorithms for solving some nonlinear problems, such as fixed points of nonlinear mappings, variational inequalities, and split feasibility problems. In fact, this lemma has been regarded as a fundamental tool for solving some nonlinear problems relating to fixed point.

Lemma 6 (see [25]). Assume (𝑎_𝑛) is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥0, (11) where (𝛾_𝑛) is a sequence in (0,1) and (𝛿_𝑛) is a sequence in ℝ such that(i)∑^∞_𝑛=0𝛾_𝑛=∞, (ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=0|𝛾_𝑛𝛿_𝑛|<∞.
Then lim_𝑛→∞𝑎_𝑛=0.

In this paper, inspired and encouraged by an idea in [26], we obtain the following lemma. Its key effect on the proofs of our main results will be illustrated in the next section and this may show that this lemma is likely to become a new fundamental tool for solving some nonlinear problems relating to fixed point.

Lemma 7. Assume (𝑠_𝑛) is a sequence of nonnegative real numbers such that 𝑠_𝑛+1≤1−𝛾_𝑛𝑠_𝑛+𝛾_𝑛𝛿_𝑛+𝛽_𝑛𝑠,𝑛≥0,_𝑛+1≤𝑠_𝑛−𝜂_𝑛+𝛼_𝑛,𝑛≥0, (12)(13) where (𝛾_𝑛) is a sequence in (0,1), (𝜂_𝑛) is a sequence of nonnegative real numbers and (𝛿_𝑛), (𝛼_𝑛), and (𝛽_𝑛) are three sequences in ℝ such that(i)∑^∞_𝑛=0𝛾_𝑛=∞, (ii)lim_𝑛→∞𝛼_𝑛=0, (iii)lim_𝑘→∞𝜂_𝑛𝑘=0 implies limsup_𝑘→∞𝛿_𝑛𝑘≤0 for any subsequence (𝑛_𝑘)⊂(𝑛), (iv)limsup_𝑛→∞(𝛽_𝑛/𝛾_𝑛)≤0.
Then lim_𝑛→∞𝑠_𝑛=0.

Proof. Following and generalizing an idea in [26], we distinguish two cases to prove 𝑠_𝑛→0 as 𝑛→0.
Case  1. (𝑠_𝑛) is eventually decreasing (i.e., there exists 𝑘≥0 such that 𝑠_𝑛>𝑠_𝑛+1 holds for all 𝑛≥𝑘). In this case, (𝑠_𝑛) must be convergent, and from (13) it follows that 𝜂_𝑛≤𝑠_𝑛−𝑠_𝑛+1+𝛼_𝑛. (14) Noting condition (ii), letting 𝑛→∞ in (14) yields 𝜂_𝑛→0 as 𝑛→∞. Using condition (iii), we get that limsup_𝑛→∞𝛿_𝑛≤0. Noting this together with conditions (i) and (iv), we obtain 𝑠_𝑛→0 by applying Lemma 6 to (12).
Case  2. (𝑠_𝑛) is not eventually decreasing. Hence, we can find an integer 𝑛₀ such that 𝑠_𝑛0≤𝑠_𝑛0+1. Let us now define 𝐽_𝑛∶=𝑛₀≤𝑘≤𝑛∶𝑠_𝑘≤𝑠_𝑘+1,𝑛>𝑛₀. (15) Obviously, 𝐽_𝑛 is nonempty and satisfies 𝐽_𝑛⊆𝐽_𝑛+1. Let 𝜏(𝑛)∶=max𝐽_𝑛,𝑛>𝑛₀. (16) It is clear that 𝜏(𝑛)→∞ as 𝑛→∞ (otherwise, (𝑠_𝑛) is eventually decreasing). It is also clear that 𝑠_𝜏(𝑛)≤𝑠_𝜏(𝑛)+1 for all 𝑛>𝑛₀. Moreover, 𝑠_𝑛≤𝑠_𝜏(𝑛)+1,∀𝑛>𝑛₀. (17) In fact, if 𝜏_𝑛=𝑛, then inequity (17) is trivial; if 𝜏(𝑛)=𝑛−1, then 𝜏(𝑛)+1=𝑛, and (17) is also trivial. If 𝜏(𝑛)<𝑛−1, then there exists an integer 𝑖≥2 such that 𝜏(𝑛)+𝑖=𝑛. Thus we deduce from the definition of 𝜏(𝑛) that 𝑠_𝜏(𝑛)+1>𝑠_𝜏(𝑛)+2>⋯>𝑠_{𝜏(𝑛)+𝑖}=𝑠_𝑛, (18) and inequity (17) holds again. Since 𝑠_𝜏(𝑛)≤𝑠_𝜏(𝑛)+1 for all 𝑛>𝑛₀, it follows from (14) that 0≤𝜂_𝜏(𝑛)≤𝛼_𝜏(𝑛)⟶0, (19) so that 𝜂_𝜏(𝑛)→0 as 𝑛→∞ using condition (ii). Due to the condition (iii), this implies that limsup_𝑛→∞𝛿_𝜏(𝑛)≤0. (20) Noting 𝑠_𝜏(𝑛)≤𝑠_𝜏(𝑛)+1 for all 𝑛>𝑛₀ again, it follows from (12) that 𝑠_𝜏(𝑛)≤𝛿_𝜏(𝑛)+𝛽_𝜏(𝑛)𝛾_𝜏(𝑛). (21) Combining (20), (21), and condition (iv) yields limsup_𝑛→∞𝑠_𝜏(𝑛)≤0, (22) and hence 𝑠_𝜏(𝑛)→0 as 𝑛→∞. This together with (13) implies that 𝑠_𝜏(𝑛)+1≤𝑠_𝜏(𝑛)−𝜂_𝜏(𝑛)+𝛼_𝜏(𝑛)⟶0, (23) which together with (17), in turn, implies that 𝑠_𝑛→0 as 𝑛→∞.

The following result is just a special case of Lemma 7, that is, the case where 𝛽_𝑛=0 for all 𝑛≥0.

Lemma 8. Assume (𝑠_𝑛) is a sequence of nonnegative real numbers such that 𝑠_𝑛+1≤1−𝛾_𝑛𝑠_𝑛+𝛾_𝑛𝛿_𝑛𝑠,𝑛≥0,_𝑛+1≤𝑠_𝑛−𝜂_𝑛+𝛼_𝑛,𝑛≥0, (24) where (𝛾_𝑛) is a sequence in (0,1), (𝜂_𝑛) is a sequence of nonnegative real numbers, and (𝛿_𝑛) and (𝛼_𝑛) are two sequences in ℝ such that(i)∑^∞_𝑛=0𝛾_𝑛=∞, (ii)lim_𝑛→∞𝛼_𝑛=0, (iii)lim_𝑘→∞𝜂_𝑛𝑘=0 implies limsup_𝑘→∞𝛿_𝑛𝑘≤0 for any subsequence (𝑛_𝑘)⊂(𝑛).
Then lim_𝑛→∞𝑠_𝑛=0.

Recall that a function 𝑓∶𝐻→ℝ is called convex if 𝑓()𝑦(𝑥)+()𝑓𝑦,()𝜆𝑥+1−𝜆≤𝜆𝑓1−𝜆∀𝜆∈0,1,∀𝑥,𝑦∈𝐻. (25) A differentiable function 𝑓 is convex if and only if there holds the following relation: 𝑓(𝑧)≥𝑓(𝑥)+⟨∇𝑓(𝑥),𝑧−𝑥⟩,∀𝑧∈𝐻. (26) Recall that an element 𝑔∈𝐻 is said to be a subgradient of 𝑓∶𝐻→ℝ at 𝑥 if 𝑓(𝑧)≥𝑓(𝑥)+⟨𝑔,𝑧−𝑥⟩,∀𝑧∈𝐻. (27)

A function 𝑓∶𝐻→ℝ is said to be subdifferentiable at 𝑥, if it has at least one subgradient at 𝑥. The set of subgradients of 𝑓 at the point 𝑥 is called the subdifferential of 𝑓 at 𝑥 and is denoted by 𝜕𝑓(𝑥). The last relation above is called the subdifferential inequality of 𝑓 at 𝑥. A function 𝑓 is called subdifferentiable, if it is subdifferentiable at all 𝑥∈𝐻. If a function 𝑓 is differentiable and convex, then its gradient and subgradient coincide.

Recall that a function 𝑓∶𝐻→ℝ is said to be weakly lower semicontinuous (𝑤-lsc) at 𝑥 if 𝑥_𝑛⇀𝑥 implies 𝑓(𝑥)≤liminf_𝑛→∞𝑓𝑥_𝑛. (28)

3. Iterative Algorithms

In this section, we consider the iterative algorithms for solving a particular kind of variational inequality (1) in which the closed convex subset 𝐶 is of the particular structure, that is the intersection of finite level sets of convex functions given as follows: 𝐶=^𝑚_𝑖=1𝑥∈𝐻∶𝑐_𝑖(𝑥)≤0, (29) where 𝑚 is a positive integer and 𝑐_𝑖∶𝐻→ℝ(𝑖=1,…,𝑚) is a convex function. We always assume that 𝑐_𝑖(𝑖=1,…,𝑚) is subdifferentiable on 𝐻 and 𝜕𝑐_𝑖(𝑖=1,…,𝑚) is a bounded operator (i.e., bounded on bounded sets). It is worth noting that every convex function defined on a finite-dimensional Hilbert space is subdifferentiable and its subdifferential operator is a bounded operator (see [27, Corollary 7.9]). We also assume that 𝐹∶𝐻→𝐻 is an 𝐿-Lipschitzian and 𝜂-strongly monotone operator. It is well known that in this case VI(𝐶,𝐹) has a unique solution, henceforth, which is denoted by 𝑥^∗.

Without loss of the generality, we will consider only the case 𝑚=2; that is, 𝐶=𝐶¹⋂𝐶², where 𝐶¹=𝑥∈𝐻∶𝑐₁(𝑥),𝐶≤0₂=𝑥∈𝐻∶𝑐₂(𝑥)≤0. (30) All of our results can be extended easily to the general case.

The computation of a projection onto a closed convex subset is generally difficult. To overcome this difficulty, Fukushima [21] suggested a way to calculate the projection onto a level set of a convex function by computing a sequence of projections onto half-spaces containing the original level set. This idea is followed by Yang [28] and López et al. [29], respectively, who introduced the relaxed 𝐶𝑄 algorithms for solving the split feasibility problem in finite-dimensional and infinite-dimensional Hilbert spaces, respectively. This idea is also used by Censor et al. [30] in the subgradient extragradient method for solving variational inequalities in a Hilbert space.

We are now in a position to introduce a relaxed algorithm for computing the unique solution 𝑥^∗ of VI(𝐶,𝐹), where 𝐶=𝐶¹⋂𝐶² and 𝐶^𝑖(𝑖=1,2) is given as in (30). This scheme applies to the case where 𝐿 and 𝜂 are easy to be determined.

Algorithm 1. Choose an arbitrary initial guess 𝑥₀∈𝐻. The sequence (𝑥_𝑛) is constructed via the formula 𝑥_𝑛+1=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛,𝑛⩾0, (31) where 𝐶¹_𝑛=𝑥∈𝐻∶𝑐₁𝑥_𝑛≤⟨𝜉¹_𝑛,𝑥_𝑛,𝐶−𝑥⟩_2𝑛=𝑥∈𝐻∶𝑐₂𝑃_𝐶¹_𝑛𝑥_𝑛≤⟨𝜉²_𝑛,𝑃_𝐶¹_𝑛𝑥_𝑛−𝑥⟩, (32) where 𝜉¹_𝑛∈𝜕𝑐₁(𝑥_𝑛),  𝜉²_𝑛∈𝜕𝑐₂(𝑃_𝐶¹_𝑛𝑥_𝑛), the sequence (𝜆_𝑛) is in (0,1), and 𝜇 is a constant such that 𝜇∈(0,2𝜂/𝐿²).

We now analyze strong convergence of Algorithm 1, which also illustrates the application of Lemma 7 (or Lemma 8).

Theorem 9. Assume that 𝜆_𝑛→0(𝑛→∞) and ∑^+∞_𝑛=1𝜆_𝑛=+∞. Then the sequence (𝑥_𝑛) generated by Algorithm 1 converges strongly to the unique solution 𝑥^∗ of 𝑉𝐼(𝐶,𝐹).

Proof. Firstly, we verify that (𝑥_𝑛) is bounded. Indeed, it is easy to see from the subdifferential inequality and the definitions of 𝐶¹_𝑛 and 𝐶²_𝑛 that 𝐶¹_𝑛⊃𝐶¹ and 𝐶²_𝑛⊃𝐶² hold for all 𝑛≥0, and hence it follows that 𝐶¹_𝑛⋂𝐶²_𝑛⊃𝐶¹⋂𝐶²=𝐶. Since the projection operators 𝑃_𝐶¹_𝑛 and 𝑃_𝐶²_𝑛 are nonexpansive, we obtain from (31), Lemmas 2 and 4 that 𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛(𝐼−𝜆_𝑛𝜇𝐹)𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2≤𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑥^∗−𝜆_𝑛𝜇𝐹𝑥^∗2≤1−𝜏𝜆_𝑛𝑥_𝑛−𝑥^∗2−2𝜆_𝑛𝜇𝐹𝑥^∗,𝑥_𝑛−𝑥^∗−𝜆_𝑛𝜇𝐹𝑥_𝑛≤1−𝜏𝜆_𝑛𝑥_𝑛−𝑥^∗2−2𝜆_𝑛𝜇𝐹𝑥^∗,𝑥_𝑛−𝑥^∗+2𝜆²_𝑛𝜇²𝐹𝑥^∗𝐹𝑥_𝑛,𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥^∗+𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛(𝐼−𝜆_𝑛𝜇𝐹)𝑥^∗−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2≤1−𝜏𝜆_𝑛𝑥_𝑛−𝑥^∗2+2𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥^∗−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝜏𝜆_𝑛𝑥_𝑛−𝑥^∗2+2𝜆_𝑛𝜇𝐹𝑥^∗𝑥_𝑛+1−𝑥^∗≤1−𝜏𝜆_𝑛𝑥_𝑛−𝑥^∗2+14𝜏𝜆_𝑛𝑥_𝑛+1−𝑥^∗2+4𝜆_𝑛𝜇²𝜏𝐹𝑥^∗2, (33)(34) where 𝜏=(1/2)𝜇(2𝜂−𝜇𝐿²).
Consequently 𝑥_𝑛+1−𝑥^∗2≤1−𝜏𝜆_𝑛1−(1/4)𝜏𝜆_𝑛𝑥_𝑛−𝑥^∗2+(3/4)𝜏𝜆_𝑛1−(1/4)𝜏𝜆_𝑛16𝜇²3𝜏²𝐹𝑥^∗2. (35) It turns out that 𝑥_𝑛+1−𝑥^∗≤max𝑥_𝑛−𝑥^∗,4𝜇√3𝜏𝐹𝑥^∗, (36) inductively 𝑥_𝑛−𝑥^∗≤max𝑥₀−𝑥^∗,4𝜇√3𝜏𝐹𝑥^∗, (37) and this means that (𝑥_𝑛) is bounded. Obviously, (𝐹𝑥_𝑛) is also bounded.
Secondly, since a projection is firmly nonexpansive, we obtain 𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2≤𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶¹_𝑛𝑥^∗2−𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛²≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑃_𝐶¹_𝑛𝑥_𝑛²−𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛²; (38) thus we also have 𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛+𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2≤𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2+2𝜆_𝑛𝜇𝐹𝑥_𝑛⋅𝑥_𝑛−𝑥^∗+𝜆²_𝑛𝜇²𝐹𝑥_𝑛²≤𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2+𝜆_𝑛𝑀, (39) where 𝑀 is a positive constant such that 𝑀≥sup_𝑛{2𝜇‖𝐹𝑥_𝑛‖⋅‖𝑥_𝑛−𝑥^∗‖+𝜆_𝑛𝜇²‖𝐹𝑥_𝑛‖²}. The combination of (38) and (39) leads to 𝑥_𝑛+1−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑃_𝐶¹_𝑛𝑥_𝑛²−𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛²+𝜆_𝑛𝑀. (40) Setting 𝑠_𝑛=𝑥_𝑛−𝑥^∗2,𝛾_𝑛=𝜏𝜆_𝑛,𝛼_𝑛=𝑀𝜆_𝑛,𝛿_𝑛=−2𝜇𝜏𝐹𝑥^∗,𝑥_𝑛−𝑥^∗+2𝜆_𝑛𝜇²𝜏𝐹𝑥^∗𝐹𝑥_𝑛,𝜂_𝑛=𝑥_𝑛−𝑃_𝐶¹_𝑛𝑥_𝑛²+𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛², (41) then (33) and (40) can be rewritten as the following forms, respectively: 𝑠_𝑛+1≤1−𝛾_𝑛𝑠_𝑛+𝛾_𝑛𝛿_𝑛,𝑠_𝑛+1≤𝑠_𝑛−𝜂_𝑛+𝛼_𝑛. (42) Finally, observing that the conditions 𝜆_𝑛→0 and ∑^∞_𝑛=1𝜆_𝑛=∞ imply 𝛼_𝑛→0 and ∑^∞_𝑛=1𝛾_𝑛=∞, respectively, in order to complete the proof using Lemma 7 (or Lemma 8), it suffices to verify that lim_𝑘→∞𝜂_𝑛𝑘=0 (43) implies limsup_𝑘→∞𝛿_𝑛𝑘≤0 (44) for any subsequence (𝑛_𝑘)⊂(𝑛). In fact, if 𝜂_𝑛𝑘→0 as 𝑘→∞, then ‖𝑥_𝑛𝑘−𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘‖→0 and ‖𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘−𝑃_𝐶²_𝑛𝑘𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘‖→0 hold. Since 𝜕𝑐₁ and 𝜕𝑐₂ are bounded on bounded sets, we have two positive constants 𝜅₁ and 𝜅₂ such that ‖𝜉¹_𝑛𝑘‖≤𝜅₁ and ‖𝜉²_𝑛𝑘‖≤𝜅₂ for all 𝑘≥0 (noting that (𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘) is also bounded due to the fact that ‖𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘−𝑥^∗‖=‖𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘−𝑃_𝐶¹_𝑛𝑘𝑥^∗‖≤‖𝑥_𝑛𝑘−𝑥^∗‖). From (32) and the trivial fact that 𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘∈𝐶¹_𝑛𝑘 and 𝑃_𝐶²_𝑛𝑘𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘∈𝐶²_𝑛𝑘, it follows that 𝑐₁𝑥_𝑛𝑘≤𝜉¹_𝑛𝑘,𝑥_𝑛𝑘−𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘≤𝜅₁𝑥_𝑛𝑘−𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘,𝑐₂𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘≤⟨𝜉²_𝑛𝑘,𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘−𝑃_𝐶²_𝑛𝑘𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘⟩≤𝜅₂𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘−𝑃_𝐶²_𝑛𝑘𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘. (45)(46) Now if 𝑥^′∈𝜔_𝑤(𝑥_𝑛𝑘), and (𝑥_𝑛𝑘) such that 𝑥_𝑛𝑘⇀𝑥^′ without loss of the generality, then the 𝑤-lsc and (45) imply that 𝑐₁𝑥^′≤liminf_𝑘→∞𝑐₁𝑥_𝑛𝑘≤0. (47) This means that 𝑥^′∈𝐶¹ holds. On the other hand, noting ‖𝑥_𝑛𝑘−𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘‖→0, we can assert that 𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘⇀𝑥^′ and have from the 𝑤-lsc and (46) that 𝑐₂𝑥^′≤liminf_𝑘→∞𝑐₂𝑃_𝐶¹_𝑛𝑘𝑥_𝑛𝑘≤0. (48) This, in turn, implies that 𝑥^′∈𝐶². Moreover, we obtain that 𝑥^′∈𝐶¹⋂𝐶² and hence 𝜔_𝑤(𝑥_𝑛𝑘)⊂𝐶¹⋂𝐶²=𝐶.
Noting 𝑥^∗ is the unique solution of VI(𝐶,𝐹), it turns out that limsup_𝑘→∞−2𝜇𝜏𝐹𝑥^∗,𝑥_𝑛𝑘−𝑥^∗=−2𝜇𝜏liminf_𝑘→∞𝐹𝑥^∗,𝑥_𝑛𝑘−𝑥^∗=−2𝜇𝜏inf_{𝑤∈𝜔𝑤𝑥𝑛𝑘}⟨𝐹𝑥^∗,𝑤−𝑥^∗⟩≤0. (49) Since 𝜆_𝑛→0 and (𝐹𝑥_𝑛) is bounded, it is easy to see that limsup_𝑘→∞𝛿_𝑛𝑘≤0.

Observing that in Algorithm 1 the determination of the stepsize 𝜇 still depends on the constants 𝐿 and 𝜂; this means that in order to implement Algorithm 1, one has first to estimate the constants 𝐿 and 𝜂, which is sometimes not an easy work in practice.

To overcome this difficulty, we furthermore introduce a so-called relaxed and self-adaptive algorithm, that is, a modification of Algorithm 1, in which the stepsize is selected through a self-adaptive way that has no connection with the constants 𝐿 and 𝜂.

Algorithm 2. Choose an arbitrary initial guess 𝑥₀∈𝐻 and an arbitrary element 𝑥₁∈𝐻 such that 𝑥₁≠𝑥₀. Assume that the 𝑛th iterate 𝑥_𝑛  (𝑛≥1) has been constructed. Continue and calculate the (𝑛+1)th iterate 𝑥_𝑛+1 via the following formula: 𝑥_𝑛+1=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝜆_𝑛𝜇_𝑛𝐹𝑥_𝑛,𝑛≥1, (50) where 𝐶¹_𝑛 and 𝐶²_𝑛 are given as in (32), the sequence (𝜆_𝑛) is in (0,1), and the sequence (𝜇_𝑛) is determined via the following relation:

𝜇_𝑛=𝐹𝑥_𝑛−𝐹𝑥_𝑛−1,𝑥_𝑛−𝑥_𝑛−1𝐹𝑥_𝑛−𝐹𝑥_𝑛−1²,if𝑥_𝑛≠𝑥_𝑛−1,𝜇_𝑛−1,if𝑥_𝑛=𝑥_𝑛−1,𝑛≥1. (51)

Firstly, we show that the sequence (𝑥_𝑛) is well defined. Noting strong monotonicity of 𝐹, 𝑥₁≠𝑥₀ implies that 𝐹𝑥₁≠𝐹𝑥₀ and 𝜇₁ is well defined via the first formula of (51). Consequently, 𝜇_𝑛(𝑛≥2) is well defined inductively according to (51) and thus the sequence (𝑥_𝑛) is also well defined.

Next, we estimate (𝜇_𝑛) roughly. If 𝑥_𝑛≠𝑥_𝑛−1 (that is, 𝐹𝑥_𝑛≠𝐹𝑥_𝑛−1), set 𝜂_𝑛=𝐹𝑥_𝑛−𝐹𝑥_𝑛−1,𝑥_𝑛−𝑥_𝑛−1𝑥_𝑛−𝑥_𝑛−1²,𝐿_𝑛=𝐹𝑥_𝑛−𝐹𝑥_𝑛−1𝑥_𝑛−𝑥_𝑛−1,𝑛≥1. (52) Obviously, it turns out that 𝜂≤𝜂_𝑛=𝐹𝑥_𝑛−𝐹𝑥_𝑛−1,𝑥_𝑛−𝑥_𝑛−1𝑥_𝑛−𝑥_𝑛−1²≤𝐹𝑥_𝑛−𝐹𝑥_𝑛−1𝑥_𝑛−𝑥_𝑛−1=𝐿_𝑛≤𝐿. (53) Consequently 𝜂𝐿²≤𝜇_𝑛=𝜂_𝑛𝐿²_𝑛≤1𝜂_𝑛≤1𝜂. (54) By the definition of (𝜇_𝑛), we can assert that (54) holds for all 𝑛≥1.

Lemma 7 (or Lemma 8) is also important for the proof of the strong convergence of Algorithm 2.

Theorem 10. Assume that 𝜆_𝑛→0(𝑛→∞) and ∑^+∞_𝑛=1𝜆_𝑛=+∞. Then the sequence (𝑥_𝑛) generated by Algorithm 2 converges strongly to the unique solution 𝑥^∗ of 𝑉𝐼(𝐶,𝐹).

Proof. Setting 𝛾_𝑛=𝜆_𝑛𝜇_𝑛 and 𝛽_𝑛=(1/2)(2𝜂−𝛾_𝑛𝐿²), it concludes observing 𝜆_𝑛→0 and (54) that there exists some positive integer 𝑛₀ such that 0<𝛾_𝑛<𝜂𝐿²,𝑛≥𝑛₀, (55) and consequently 𝛽_𝑛≥12𝜂,𝑛≥𝑛₀. (56) Using Lemma 2, we have from (55) that 𝑃_𝐶𝑛(𝐼−𝛾_𝑛𝐹) (so is 𝐼−𝛾_𝑛𝐹) is a contraction with coefficient 1−𝛾_𝑛𝛽_𝑛. This concludes that, for all 𝑛≥𝑛₀, 𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛(𝐼−𝛾_𝑛𝐹)𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2≤(𝐼−𝛾_𝑛𝐹)𝑥_𝑛−(𝐼−𝛾_𝑛𝐹)𝑥^∗−𝛾_𝑛𝐹𝑥^∗2≤1−𝛾_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2−2𝛾_𝑛𝐹𝑥^∗,𝑥_𝑛−𝑥^∗−𝛾_𝑛𝐹𝑥_𝑛,≤1−𝛾_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2−2𝛾_𝑛⟨𝐹𝑥^∗,𝑥_𝑛−𝑥^∗⟩+2𝛾²_𝑛𝐹𝑥^∗𝐹𝑥_𝑛,𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝛾_𝑛𝐹𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝛾_𝑛𝐹𝑥^∗+𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛(𝐼−𝛾_𝑛𝐹)𝑥^∗−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗2≤1−𝛾_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+2𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝐼−𝛾_𝑛𝐹𝑥^∗−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛾_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+2𝛾_𝑛𝐹𝑥^∗𝑥_𝑛+1−𝑥^∗≤1−𝛾_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+𝛾_𝑛𝛽_𝑛4𝑥_𝑛+1−𝑥^∗2+4𝛾_𝑛𝛽_𝑛𝐹𝑥^∗2, (57)(58) and hence 𝑥_𝑛+1−𝑥^∗2≤1−𝛾_𝑛𝛽_𝑛1−(1/4)𝛾_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+(3/4)𝛾_𝑛𝛽_𝑛1−(1/4)𝛾_𝑛𝛽_𝑛163𝛽²_𝑛𝐹𝑥^∗2. (59) Using (56), it turns out that 𝑥_𝑛+1−𝑥^∗≤max𝑥_𝑛−𝑥^∗,8√3𝜂𝐹𝑥^∗,𝑛≥𝑛₀, (60) inductively 𝑥_𝑛−𝑥^∗≤max𝑥_𝑛0−𝑥^∗,8√3𝜂𝐹𝑥^∗,𝑛≥𝑛₀, (61) and this means that (𝑥_𝑛) is bounded, so is (𝐹𝑥_𝑛).
By an argument similar to getting (38)–(40), we have 𝑥_𝑛+1−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑃_𝐶¹_𝑛𝑥_𝑛²−𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛²+𝛾_𝑛𝑀, (62) where 𝑀 is a positive constant. Setting 𝑠_𝑛=𝑥_𝑛−𝑥^∗2,𝛿_𝑛2=−𝛽_𝑛⟨𝐹𝑥^∗,𝑥_𝑛−𝑥^∗⟩+2𝛾_𝑛𝛽_𝑛𝐹𝑥^∗𝐹𝑥_𝑛,𝛼_𝑛=𝑀𝛾_𝑛,𝜎_𝑛=𝑥_𝑛−𝑃_𝐶¹_𝑛𝑥_𝑛²+𝑃_𝐶¹_𝑛𝑥_𝑛−𝑃_𝐶²_𝑛𝑃_𝐶¹_𝑛𝑥_𝑛², (63) then (57) and (62) can be rewritten as the following forms, respectively: 𝑠_𝑛+1≤1−𝛾_𝑛𝛽_𝑛𝑠_𝑛+𝛾_𝑛𝛽_𝑛𝛿_𝑛,𝑠_𝑛+1≤𝑠_𝑛−𝜎_𝑛+𝛼_𝑛. (64) Clearly, 𝜆_𝑛→0 and ∑^∞_𝑛=1𝜆_𝑛=∞, together with (54) and (56), imply that 𝛼_𝑛→0 and ∑^∞_𝑛=1𝛾_𝑛𝛽_𝑛=∞.
By an argument very similar to the proof of Theorem 9, it is not difficult to verify that lim_𝑘→∞𝜎_𝑛𝑘=0 (65) implies limsup_𝑘→∞𝛿_𝑛𝑘≤0 (66) for any subsequence (𝑛_𝑘)⊂(𝑛). Thus we can complete the proof by using Lemma 7 (or Lemma 8).

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant no. 11201476) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.