The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Abstract

We establish the strong convergence of prediction-correction and relaxed hybrid steepest-descent method (PRH method) for variational inequalities under some suitable conditions that simplify the proof. And it is to be noted that the proof is different from the previous results and also is not similar to the previous results. More importantly, we design a set of practical numerical experiments. The results demonstrate that the PRH method under some descent directions is more slightly efficient than that of the modified and relaxed hybrid steepest-descent method, and the PRH Method under some new conditions is more efficient than that under some old conditions.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, let 𝐾 be a nonempty closed convex subset of 𝐻, and let 𝐹∶𝐻→𝐻 be an operator. Then the variational inequality problem VI(𝐹,𝐾) [1] is to find 𝑥^∗∈𝐾 such that 𝑥^∗∈𝐾,𝑥−𝑥^∗,𝐹𝑥^∗≥0,∀𝑥∈𝐾. (1)

The literature contains many methods for solving variational inequality problems; see [2–25] and references therein. According to the relationship between the variational inequality problems and a fixed point problem, we can obtain 𝑥^∗isthesolutionofVI(𝐹,𝐾)⟺𝑥^∗=𝑃_𝐾𝑥^∗−𝛽𝐹𝑥^∗,𝛽>0, (2) where the projection operator 𝑃_𝐾 is the projection from 𝐻 onto 𝐾, that is, 𝑃_𝐾(𝑥)=argmin_𝑦∈𝐾𝑥−𝑦,∀𝑥∈𝐻. (3) In this paper, 𝐹∶𝐻→𝐻 is an operator with 𝐹∶𝜅-Lipschtz and 𝜂-strongly monotone; that is, 𝐹 satisfies the following conditions: 𝐹(𝑥)𝑦,𝐹(𝑥)𝑦−𝐹≤𝜅𝑥−𝑦−𝐹,𝑥−𝑦≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐾. (4) If 𝛽 is small enough, then 𝑃_𝐾 is a contraction. Naturally, the convergence of Picard iterates generated by the right-hand side of (2) is obtained by Banach’s fixed point theorem. Such a method is called the projection method or more results about the projection method see [6, 8, 20] and so forth.

In fact, the projection 𝑃_𝐾 in the contraction methods may not be easy to compute, and a great effort is to compute the projection 𝑃_𝐾 in each iteration. Yamada and Deutsch have provided a hybrid steepest-descent method for solving the VI(𝐹,𝐾) [2, 3] in order to reduce the difficulty and complexity of computing the projection 𝑃_𝐾. Subsequently, the convergence of hybrid steepest-descent methods was given out by Xu and Kim [4] and Zeng et al. [5]. Naturally, by analyzing several three-step iterative methods in each iteration by the fixed pointed equation, we can obtain the Noor iterations. Recently, Ding et al. [7] proposed a three-step relaxed hybrid steepest-descent method for variational inequalities, and the simple proof of three-step relaxed hybrid steepest-descent methods under different conditions was introduced by Yao et al. [24]. The literature [14, 16] described a modified and relaxed hybrid steepest-descent (MRHSD) method and the different convergence of the MRHSD method under the different conditions. A set of practical numerical experiments in the literature [16] demonstrated that the MRHSD method has different efficiency under different conditions. Subsequently, the prediction-correction and relaxed hybrid steepest-descent method (PRH method) [15] makes more use of the history information and less decreases the loss of information than the methods [7, 14]. The PRH method introduced more descent directions than the MRHSD method [14, 16], and computing these descent directions only needs the history information.

In this paper, we will prove the strong convergence of PRH method under different and suitable restrictions imposed on parameters (Condition 12), which differs from that of [15]. Moreover, the proof of strong convergence is different from the previous proof in [15], which is not similar to that in [7] in Step 2. And more importantly, numerical experiments verify that the PRH method under Condition 12 is more efficient than that under Condition 10, and the PRH method under some descent directions is more slightly efficient than that of the MRHSD method [14, 16]. Furthermore, it is easy to obtain these descent directions.

The remainder of the paper is organized as follows. In Section 2, we review several lemmas and preliminaries. We prove the convergence theorem under Condition 12 in Section 3. In Section 4, we give out a series of numerical experiments, which demonstrated that the PRH method under Condition 12 is more efficient than under Condition 10. Section 5 concludes the paper.

2. Preliminaries

In order to proof the later convergence theorem, we introduce several lemmas and the main results in the following.

Lemma 1. In a real Hilbert space H, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝐻. (5)

The lemma is a basic result of a Hilbert space with the inner product.

Lemma 2 (demiclosedness principle). Assume that 𝑇 is a nonexpansive self-mapping on a nonempty closed convex subset 𝐾 of a Hilbert space 𝐻. If 𝑇 has a fixed point, then (𝐼−𝑇) is demiclosed. That is, whenever 𝑥_𝑛 is a sequence in 𝐾 weakly converging to some 𝑥∈𝐾 and the sequence (𝐼−𝑇)𝑥_𝑛 strongly converges to some 𝑦∈𝐻, it follows that (𝐼−𝑇)𝑥=𝑦. Here 𝐼 is the identity operator of 𝐻.

The following lemma is an immediate result of a projection mapping onto a closed convex subset of a Hilbert space.

Lemma 3. Let 𝐾 be a nonempty closed convex subset of 𝐻. For all𝑥,𝑦∈𝐻 and 𝑧∈𝐾, then(1)⟨𝑃_𝐾(𝑥)−𝑥,𝑧−𝑃_𝐾(𝑦)⟩≥0, (2)‖𝑃_𝐾(𝑥)−𝑃_𝐾(𝑦)‖²≤‖𝑥−𝑦‖²−‖𝑃_𝐾(𝑥)−𝑥+𝑦−𝑃_𝐾(𝑦)‖².

Lemma 4 (see [13]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequence in a Banach space X and let {𝜁_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝜁_𝑛≤limsup_𝑛→∞𝜁_𝑛<1. Suppose 𝑥_𝑛+1=(1−𝜁_𝑛)𝑦_𝑛+𝜁_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then limsup_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 5 ([5, 7]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying the inequality 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝜏_𝑛+𝛾_𝑛,∀𝑛≥0, (6) where 𝛼_𝑛, 𝜏_𝑛, and 𝛾_𝑛 satisfy the following conditions: (1)𝛼_𝑛⊂[0,1],∑^∞_𝑛=0𝛼_𝑛=∞,or∏^∞_𝑛=0(1−𝛼_𝑛)=0, (2)lim_𝑛→∞sup𝜏_𝑛≤0, (3)  𝛾_𝑛⊂[0,∞),∑^∞_𝑛=0𝛾_𝑛<∞. Then lim_𝑛→∞𝑠_𝑛=0.

Since 𝐹 is 𝜂-strongly monotone, VI(𝐹,𝐾) has a unique solution 𝑥^∗∈𝐾 [5]. Assume that 𝑇∶𝐻→𝐻 is a nonexpansive mapping with the fixed point set Fix(𝑇)=𝐾. Obviously Fix(𝑃_𝐾)=𝐾.

For any given numbers 𝜆∈(0,1) and 𝜇∈(0,2𝜂/𝜅²), we define the mapping 𝑇^𝜆_𝜇∶𝐻→𝐻 by𝑇^𝜆_𝜇𝑥∶𝑇𝑥−𝜆𝜇𝐹(𝑇𝑥),∀𝑥∈𝐻. (7)

Lemma 6 (see [5]). If 0<𝜇<2𝜂/𝜅² and 0<𝜆<1, then 𝑇^𝜆_𝜇 is a contraction. In fact, 𝑇^𝜆_𝜇𝑥−𝑇^𝜆_𝜇𝑦≤(1−𝜆𝛿)𝑥−𝑦, (8) where 𝛿=1−1−𝜇(2𝜂−𝜇𝜅²), for all 𝑥,𝑦∈𝐻.

Lemma 7 (see [7]). Let {𝛼_𝑛} be a sequence of nonnegative numbers with limsup_𝑛→∞𝛼_𝑛<∞ and let {𝛽_𝑛} be sequence of real numbers with limsup_𝑛→∞𝛽_𝑛≤0. Thenlimsup_𝑛→∞𝛼_𝑛𝛽_𝑛≤0. (9)

3. Convergence Theorem

Before analyzing the convergence theorem, we first review the PRH method and related results [15].

Algorithm 8 (see [15]). Take three fixed numbers 𝑡,𝜌,𝛾∈(0,2𝜂/𝜅²), starting with arbitrarily chosen initial points 𝑥₀∈𝐻, compute the sequences {𝑥_𝑛},{𝑥_𝑛},{𝑥_𝑛},{𝑥_𝑛} such that;  Prediction  Step  1: 𝑥_𝑛=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)[𝑇𝑥_𝑛−𝜆^′′_𝑛+1𝛾𝐹(𝑇𝑥_𝑛)], Step  2: 𝑥_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)[𝑇𝑥_𝑛−𝜆^′_𝑛+1𝜌𝐹(𝑇𝑥_𝑛)], Step  3: 𝑥_𝑛=𝜃_𝑛𝑥_𝑛+(1−𝜃_𝑛)𝑥_𝑛,0≤𝜃_𝑛≤1, Correction  Step  4: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)[𝑇𝑥_𝑛     −𝜆_𝑛+1𝑡𝐹(𝑇𝑥_𝑛)], where 𝑇∶𝐻→𝐻 is a nonexpansive mapping.

Let {𝛼_𝑛}⊂[0,1),{𝛽_𝑛}⊂[0,1] and {𝛾_𝑛}⊂[0,1], {𝜆_𝑛},{𝜆^′_𝑛},{𝜆_𝑛^′′}⊂(0,1) satisfy the following conditions.

Remark 9. In fact, the PRH method is the MRHSD method when 𝜃_𝑛≡0,forall𝑛.

Condition 10. One has(1)^∞₁𝛼_𝑛−𝛼_𝑛−1<∞,^∞₁𝛽_𝑛−𝛽_𝑛−1<∞,_∞1𝛾_𝑛−𝛾_𝑛−1(2)<∞,lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞𝛽_𝑛=1,lim_𝑛→∞𝛾_𝑛(3)=1,lim_𝑛→∞𝜆_𝑛=0,lim_𝑛→∞𝜆_𝑛𝜆_𝑛+1=1,^∞₁𝜆_𝑛(4)𝜆=∞,_𝑛≥max𝜆^′_𝑛,𝜆_𝑛^′′,∀𝑛≥1. (10)

Theorem 11 (see [15]). In Condition 10, the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈𝐾, and 𝑥^∗ is the unique solution of the 𝑉𝐼(𝐹,𝐾).

We obtain the strong convergence theorem of PRH method for variational inequalities under different assumptions.

Condition 12. One has (1)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1,lim_𝑛→∞𝛽_𝑛=1,lim_𝑛→∞𝛾_𝑛(2)=1,lim_𝑛→∞𝜆_𝑛=0,^∞₁𝜆_𝑛(3)𝜆=∞,_𝑛≥max𝜆^′_𝑛,𝜆_𝑛^′′,∀𝑛≥1. (11)

Theorem 13. The sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈𝐾, and 𝑥^∗ is the unique solution of the 𝑉𝐼(𝐹,𝐾). Assume that {𝛼_𝑛},{𝛽_𝑛} and {𝛾_𝑛}, {𝜆_𝑛},{𝜆^′_𝑛},{𝜆_𝑛^′′} satisfy Condition 12.

Proof. We divide the proof into several steps.
Step 1. {𝑥_𝑛},{𝑥_𝑛},{𝑥_𝑛}, and {𝑥_𝑛} are bounded. Since 𝐹 is 𝜂-strongly monotone, VI(𝐹,𝐾) (1) has a unique solution 𝑥^∗∈𝐾, and 𝑇^𝜆𝑛+1_𝑡𝑥^∗=𝑥^∗−𝜆_𝑛+1𝑡𝐹(𝑥^∗), 𝑇^{𝜆′𝑛+1}_𝜌𝑥^∗=𝑥^∗−𝜆_𝑛+1𝜌𝐹(𝑥^∗), 𝑇^{𝜆′′𝑛+1}_𝛾𝑥^∗=𝑥^∗−𝜆_𝑛+1𝛾𝐹(𝑥^∗).
A series of computations yields 𝑥_𝑛+1−𝑥^∗=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝜆𝑛+1_𝑡𝑥−𝑥^∗≤𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑇^𝜆𝑛+1_𝑡𝑥−𝑥^∗≤𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛼_𝑛×𝑇^𝜆𝑛+1_𝑡𝑥−𝑇^𝜆𝑛+1_𝑡𝑥^∗+𝑇^𝜆𝑛+1_𝑡𝑥^∗−𝑥^∗≤𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛼_𝑛×1−𝜆_𝑛+1𝜏𝑥_𝑛−𝑥^∗+𝜆_𝑛+1𝑡𝐹𝑥^∗, (12) where 𝜏=1−1−𝑡(2𝜂−𝑡𝜅²)∈(0,1), 𝑥_𝑛−𝑥^∗=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇^{𝜆′𝑛+1}_𝜌𝑥_𝑛−𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝑇^{𝜆′𝑛+1}_𝜌𝑥_𝑛−𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛×𝑇^{𝜆′𝑛+1}_𝜌𝑥_𝑛−𝑇^{𝜆′𝑛+1}_𝜌𝑥^∗+𝑇^{𝜆′𝑛+1}_𝜌𝑥^∗−𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛×1−𝜆^′_𝑛+1𝜏^′𝑥_𝑛−𝑥^∗+𝜆^′_𝑛+1𝜌𝐹𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝜆^′_𝑛+1𝜌𝐹𝑥^∗, (13) where 𝜏^′=1−1−𝜌(2𝜂−𝑡𝜅²)∈(0,1).
Moreover, we also obtain 𝑥_𝑛−𝑥^∗=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑇^{𝜆′′𝑛+1}_𝛾𝑥_𝑛−𝑥^∗≤𝛾_𝑛𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝑇^{𝜆′′𝑛+1}_𝛾𝑥_𝑛−𝑥^∗≤𝛾_𝑛𝑥_𝑛−𝑥^∗+1−𝛾_𝑛×𝑇^{𝜆′′𝑛+1}_𝛾𝑥_𝑛−𝑇^{𝜆′′𝑛+1}_𝛾𝑥^∗+𝑇^{𝜆′′𝑛+1}_𝛾𝑥^∗−𝑥^∗≤𝛾_𝑛𝑥_𝑛−𝑥^∗+1−𝛾_𝑛×1−𝜆^′′_𝑛+1𝜏^′′𝑥_𝑛−𝑥^∗+𝜆^′′_𝑛+1𝛾𝐹𝑥^∗≤𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗, (14) where 𝜏^′′=1−1−𝛾(2𝜂−𝑡𝜅²)∈(0,1), subtituting; (14) into (13) and (14) into (12), we immediately obtain 𝑥_𝑛−𝑥^∗=𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛×1−𝜆^′_𝑛+1𝜏^′𝑥_𝑛−𝑥^∗+𝜆^′_𝑛+1𝜌𝐹𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛×1−𝜆^′_𝑛+1𝜏^′𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗+𝜆^′_𝑛+1𝜌𝐹𝑥^∗≤𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝜆_𝑛+1𝛾+𝜌𝐹𝑥^∗. (15) Furthermore, 𝑥_𝑛−𝑥^∗=𝜃_𝑛𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛−𝑥^∗≤𝜃_𝑛𝑥_𝑛−𝑥^∗+1−𝜃_𝑛𝑥_𝑛−𝑥^∗≤𝜃_𝑛𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗+1−𝜃_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝜆_𝑛+1×𝛾+𝜌𝐹𝑥^∗≤𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗+1−𝛽_𝑛𝜆_𝑛+1𝛾+𝜌𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛼_𝑛×1−𝜆_𝑛+1𝜏𝑥_𝑛−𝑥^∗+𝜆_𝑛+1𝑡𝐹𝑥^∗≤𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗+1−𝛼_𝑛1−𝜆_𝑛+1𝜏×𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗+1−𝛽_𝑛𝜆_𝑛+1𝛾+𝜌𝐹𝑥^∗+𝜆_𝑛+1𝑡𝐹𝑥^∗≤𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛1−𝛾_𝑛𝜆_𝑛+1𝛾𝐹𝑥^∗+1−𝛼_𝑛×1−𝜆_𝑛+1𝜏𝑥_𝑛−𝑥^∗+𝜆_𝑛+12𝛾+𝜌+𝑡𝐹𝑥^∗. (16) It is easy to obtain the following by induction: 𝑥_𝑛−𝑥^∗≤𝑀₀,∀𝑛≥0, (17) where 𝑀₀=max{3‖𝑥₀−𝑥^∗‖,3(𝜌+𝛾+𝑡)‖𝐹(𝑥^∗)‖/𝜏}, 𝑥_𝑛−𝑥^∗≤𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝜆_𝑛+1𝛾+𝜌𝐹𝑥^∗≤𝜏1+3𝑀₀,𝑥_𝑛−𝑥^∗≤𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗≤𝜏1+3𝑀₀,𝑥_𝑛−𝑥^∗≤𝜃_𝑛𝑥_𝑛−𝑥^∗+1−𝜃_𝑛𝑥_𝑛−𝑥^∗𝜏≤21+3𝑀₀. (18) Hence𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐹𝑇𝑥_𝑛,𝐹𝑇𝑥_𝑛,𝐹𝑇𝑥_𝑛,𝐹𝑇𝑥_𝑛 (19) are also bounded.
Step 2. Consider ‖𝑥_𝑛+1−𝑥_𝑛‖→0.
Indeed, by a series of computations, we have 𝑥_𝑛−𝑥_𝑛−1=𝛾_𝑛𝑥_𝑛−𝛾_𝑛−1𝑥_𝑛−1+1−𝛾_𝑛𝑇^{𝜆′′𝑛+1}_𝛾𝑥_𝑛−1−𝛾_𝑛−1𝑇^{𝜆𝑛′′}_𝛾𝑥_𝑛−1≤𝛾_𝑛𝑥_𝑛−𝛾_𝑛−1𝑥_𝑛−1+1−𝛾_𝑛𝑇^{𝜆′′𝑛+1}_𝛾𝑥_𝑛−1−𝛾_𝑛−1𝑇^{𝜆𝑛′′}_𝛾𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1−𝛾_𝑛𝜆^′′_𝑛+1−1−𝛾_𝑛−1𝜆_𝑛^′′×𝛾𝐹𝑇𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝑇𝑥_𝑛−1. (20) According to (20) and the prediction step of Algorithm 8, we also obtain 𝑥_𝑛−𝑥_𝑛−1=𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1+1−𝛽_𝑛𝑇^{𝜆′𝑛+1}_𝜌𝑥_𝑛−1−𝛽_𝑛−1𝑇^𝜆′𝑛_𝜌𝑥_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1+1−𝛽_𝑛𝑇^{𝜆′𝑛+1}_𝜌𝑥_𝑛−1−𝛽_𝑛−1𝑇^𝜆′𝑛_𝜌𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝜆^′_𝑛+1−1−𝛽_𝑛−1𝜆^′_𝑛×𝜌𝐹𝑇𝑥_𝑛−1+1−𝛽_𝑛1−𝜆^′_𝑛+1𝜏^′×𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝑇𝑥_𝑛−1+1−𝛽_𝑛1−𝜆^′_𝑛+1𝜏^′1−𝛾_𝑛𝜆^′′_𝑛+1−𝛾_𝑛−1𝜆_𝑛^′′×𝛾𝐹𝑇𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1×𝑥_𝑛−1+𝑇𝑥_𝑛−1+𝑇𝑥_𝑛−1. (21) Also by the prediction step of Algorithm 8 and (20), (21), we have 𝑥_𝑛−𝑥_𝑛−1≤𝜃_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝜃_𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1−𝛾_𝑛𝜆^′′_𝑛+1−1−𝛾_𝑛−1𝜆_𝑛^′′𝛾𝐹𝑇𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝑇𝑥_𝑛−1+1−𝛽_𝑛𝜆^′_𝑛+1−1−𝛽_𝑛−1𝜆^′_𝑛𝜌𝐹𝑇𝑥_𝑛−1+1−𝛽_𝑛1−𝜆^′_𝑛+1𝜏^′𝛾_𝑛−𝛾_𝑛−1×𝑥_𝑛−1+𝑇𝑥_𝑛−1+1−𝛽_𝑛1−𝜆^′_𝑛+1𝜏^′×1−𝛾_𝑛𝜆^′′_𝑛+1−𝛾_𝑛−1𝜆_𝑛^′′𝛾𝐹𝑇𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1+𝑇𝑥_𝑛−1+𝑇𝑥_𝑛−1. (22)
Let 𝑦_𝑛=𝑇^𝜆𝑛+1_𝑡𝑥_𝑛=𝑇𝑥_𝑛−𝜆_𝑛+1𝑡𝐹𝑇𝑥_𝑛, (23)
so we get 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛. (24) Furthermore, 𝑦_𝑛−𝑦_𝑛−1=𝑇𝑥_𝑛−𝑇𝑥_𝑛−1+𝜆_𝑛𝑡𝐹𝑇𝑥_𝑛−1−𝜆_𝑛+1𝑡𝐹𝑇𝑥_𝑛≤𝑇𝑥_𝑛−𝑇𝑥_𝑛−1+𝜆_𝑛𝑡𝐹𝑇𝑥_𝑛−1+𝜆_𝑛+1𝑡𝐹𝑇𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛−1+𝜆_𝑛𝑡𝐹𝑇𝑥_𝑛−1+𝜆_𝑛+1𝑡𝐹𝑇𝑥_𝑛. (25) Apply lim_𝑛→∞𝛽_𝑛=1,lim_𝑛→∞𝜆_𝑛=0, and lim_𝑛→∞𝛾_𝑛=1 and (22), (25) to get 𝑦_𝑛−𝑦_𝑛−1−𝑥_𝑛−𝑥_𝑛−1≤1−𝛾_𝑛𝜆^′′_𝑛+1−1−𝛾_𝑛−1𝜆_𝑛^′′𝛾𝐹𝑇𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝑇𝑥_𝑛−1+1−𝛽_𝑛𝜆^′_𝑛+1−1−𝛽_𝑛−1𝜆^′_𝑛𝜌𝐹𝑇𝑥_𝑛−1+1−𝛽_𝑛1−𝜆^′_𝑛+1𝜏^′𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝑇𝑥_𝑛−1+1−𝛽_𝑛1−𝜆^′_𝑛+1𝜏^′×1−𝛾_𝑛𝜆^′′_𝑛+1−𝛾_𝑛−1𝜆_𝑛^′′𝛾𝐹𝑇𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1+𝑇𝑥_𝑛−1+𝑇𝑥_𝑛−1+𝜆_𝑛𝑡𝐹𝑇𝑥_𝑛−1+𝜆_𝑛+1𝑡𝐹𝑇𝑥_𝑛⟶0. (26) According to Lemma 4, we obtain lim_𝑛→∞𝑦_𝑛−1−𝑥_𝑛−1=0. (27) Furthermore, by lim_𝑛→∞𝛾_𝑛=1, we also get 𝑥_𝑛−𝑥_𝑛=−1−𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑇𝑥_𝑛−𝜆^′_𝑛+1𝛾𝐹𝑇𝑥_𝑛≤1−𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑇𝑥_𝑛+𝜆^′_𝑛+1𝛾𝐹𝑇𝑥_𝑛⟶0. (28) By (27), (28) and the correction step of Algorithm 8, we immediately conclude that 𝑥_𝑛−𝑥_𝑛−1=𝛼_𝑛−1𝑥_𝑛−1+1−𝛼_𝑛−1𝑦_𝑛−1−𝑥_𝑛−1≤𝛼_𝑛−1𝑥_𝑛−1−𝑥_𝑛−1+1−𝛼_𝑛−1𝑦_𝑛−1−𝑥_𝑛−1⟶0, (29) so we get 𝑥_𝑛+1−𝑥_𝑛⟶0. (30)
Step 3. Consider‖𝑥_𝑛+1−𝑇𝑥_𝑛‖→0.
Indeed, by the prediction step of Algorithm 8, we have 𝑥_𝑛−𝑥_𝑛=−1−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛−𝜆^′_𝑛+1𝜌𝐹𝑇𝑥_𝑛≤1−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛+𝜆^′_𝑛+1𝜌𝐹𝑇𝑥_𝑛. (31) According to the assumption lim_𝑛→∞𝛽_𝑛=1 and lim_𝑛→∞𝜆_𝑛=0, then 𝑥_𝑛−𝑥_𝑛⟶0. (32) By (32), we immediately obtain 𝑥_𝑛−𝑥_𝑛≤𝜃_𝑛𝑥_𝑛−𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛−𝑥_𝑛⟶0. (33)
By a series of computations, we can get 𝑥_𝑛+1−𝑇𝑥_𝑛=𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛+1−𝛼_𝑛𝑇^𝜆𝑛+1_𝑡𝑥−𝑇𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛+1−𝛼_𝑛𝜆_𝑛+1𝑡𝐹𝑇𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+𝜆_𝑛+1𝑡‖𝐹(𝑇𝑥)‖≤𝛼_𝑛𝑥_𝑛+1−𝑇𝑥_𝑛+𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛−𝑥_𝑛+𝜆_𝑛+1𝑡‖𝐹(𝑇𝑥)‖. (34) Hence, by (28), (33), and (34), we also obtain 𝑥_𝑛+1−𝑇𝑥_𝑛≤𝛼_𝑛1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛−𝑥_𝑛1−𝛼_𝑛+𝜆_𝑛+1𝑡‖𝐹(𝑇𝑥)‖1−𝛼_𝑛⟶0. (35) Using Steps 2 and 3, it is easy to obtain the following corollary.
Corollary 14. Consider ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0.
Applying Steps   2 and 3 , one gets𝑥_𝑛+1−𝑇𝑥_𝑛⟶0,𝑥_𝑛+1−𝑥_𝑛⟶0, (36)so it is easy to see that𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛+1−𝑇𝑥_𝑛+𝑥_𝑛+1−𝑥_𝑛⟶0. (37)
Step 4. Consider lim_𝑛→∞sup⟨−𝐹(𝑥^∗),𝑇𝑥_𝑛−𝑥^∗⟩≤0.
For some 𝑥∈𝐻, here exits {𝑇𝑥_𝑛𝑖}→𝑥 weakly and such that lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗=lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛𝑖−𝑥^∗. (38) According to {𝑇𝑥_𝑛𝑖}→𝑥, we have 𝑥∈Fix(𝑇)=𝐾. (39) By 𝑥^∗ being the unique solution of VI(𝐹,𝐾), we can obtain lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗=lim_𝑛→∞sup−𝐹𝑥^∗,𝑥−𝑥^∗≤0. (40) Since ‖𝑇𝑥_𝑛−𝑇𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛‖→0, we immediately conclude that lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗≤lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑇𝑥_𝑛+lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗≤lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗≤0. (41)
Step 5. By Step 1 and Lemma 1, we have 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑇^𝜆𝑛+1_𝑡𝑥_𝑛−𝑥^∗2≤𝛼_𝑛𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛×𝑇^𝜆𝑛+1_𝑡𝑥_𝑛−𝑇^𝜆𝑛+1_𝑡𝑥^∗+𝑇^𝜆𝑛+1_𝑡𝑥^∗−𝑥^∗2≤𝛼_𝑛𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛×𝑇^𝜆𝑛+1_𝑡𝑥_𝑛−𝑇^𝜆𝑛+1_𝑡𝑥^∗2+2𝑇^𝜆𝑛+1_𝑡𝑥^∗−𝑥^∗,𝑇^𝜆𝑛+1_𝑡𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗2+1−𝛼_𝑛1−𝜆_𝑛+1𝜏²×𝑥_𝑛−𝑥^∗+1−𝛾_𝑛𝜆^′′_𝑛+1𝛾𝐹𝑥^∗+1−𝛽_𝑛𝜆_𝑛+1𝛾+𝜌𝐹𝑥^∗2+2𝑡𝜆_𝑛+1−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗−𝑡𝜆_𝑛+1𝐹𝑇𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑥^∗2+1−𝛾_𝑛𝜆_𝑛+1+𝛾𝑀1−𝛼_𝑛1−𝜆_𝑛+1𝜏²𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛1−𝜆_𝑛+1𝜏²1−𝛽_𝑛𝜆_𝑛+1𝑀+1−𝛼_𝑛1−𝜆_𝑛+1𝜏²1−𝛾_𝑛𝜆_𝑛+1𝛾𝑀+2𝑡𝜆_𝑛+1−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗−𝑡𝜆_𝑛+1𝐹𝑇𝑥_𝑛≤1−1−𝛼_𝑛𝜆_𝑛+1𝜏𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛𝜆_𝑛+1𝜏𝑤^′_𝑛+1, (42) where 𝑤^′_𝑛+1=2𝑡−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗−𝑡𝜆_𝑛+1𝐹𝑇𝑥_𝑛𝜏1−𝛼_𝑛+𝜑_𝑛𝜏1−𝛼_𝑛+𝜉_𝑛𝜏1−𝛼_𝑛,𝜑_𝑛=1−𝛾_𝑛𝜉𝛾𝑀,_𝑛=1−𝛼_𝑛1−𝜆_𝑛+1𝜏²1−𝛽_𝑛𝑀+1−𝛼_𝑛1−𝜆_𝑛+1𝜏²1−𝛾_𝑛𝜆_𝑛+1𝛾𝑀, (43) and 𝑀₀≪𝑀<∞.
Denote 𝑠^′_𝑛+1=𝑥_𝑛+1−𝑥^∗,𝑢_𝑛=1−𝛼_𝑛𝜆_𝑛+1𝜏. (44) We can rewrite (42) as𝑠^′_𝑛+1≤1−𝑢_𝑛𝑠^′_𝑛+𝑢_𝑛𝑤^′_𝑛+0. (45) In fact, 𝑢_𝑛,𝑤^′_𝑛 satisfies Lemma 5; according to lim_𝑛→∞𝛽_𝑛=1,lim_𝑛→∞𝛾_𝑛=1,lim_𝑛→∞𝜆_𝑛=0, (46) we obtain 𝜑_𝑛𝜏1−𝛼_𝑛𝜉⟶0,_𝑛𝜏1−𝛼_𝑛⟶0. (47) Moreover, by Step 4, we also obtain lim_𝑛→∞2𝑡−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗−𝑡𝜆_𝑛+1𝐹𝑇𝑥_𝑛𝜏1−𝛼_𝑛≤2𝑡𝜏lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗+𝜆_𝑛+1−𝐹𝑥^∗,−𝑡𝐹𝑇𝑥_𝑛≤2𝑡𝜏lim_𝑛→∞sup−𝐹𝑥^∗,𝑇𝑥_𝑛−𝑥^∗+lim_𝑛→∞sup𝜆_𝑛+1−𝐹𝑥^∗,−𝑡𝐹𝑇𝑥_𝑛≤0+0=0. (48)
Furthermore, by (43), (47), and (48), it is easy to obtain lim_𝑛→∞sup𝑤^′_𝑛≤0. (49) Consequently apply Lemma 5 to obtain 𝑥_𝑛−𝑥^∗⟶0. (50)

4. Numerical Experiments

The problem considered in this section is 1min2‖𝑋−𝐶‖²_𝐹∣𝑋∈𝐾, (51) where ‖⋅‖_𝐹 is the matrix Fröbenis norm; that is, ‖𝐶‖_𝐹=^∞∞_𝑖=1𝑗=1𝐶_𝑖𝑗^21/2. (52)

Note that the matrix Fröbenis norm is induced by the inner product ⟨𝐴,𝐵⟩=Trace𝐴^𝑇𝐵. (53) The problems arise from finance and statistics, and we form the test problems similarly as in [9, 21].

Let 𝐾=𝑆^𝑛₊∩ß, where𝑆^𝑛₊=𝐻∈ℝ^𝑛×𝑛∣𝐻^𝑇=𝐻,𝐻≽0,ß=𝐻∈ℝ^𝑛×𝑛∣𝐻^𝑇=𝐻,𝐻_𝐿≤𝐻≤𝐻_𝑈. (54) Let 𝐻_𝐿,𝐻_𝑈 be given 𝑛×𝑛 symmetric matrices, and 𝐶 asymmetric which differs from previous approaches [9, 21], and it is to be noted that the extended contraction method (EC method) [9] has much difficulty in computing the examples when 𝐶 is asymmetric, where 𝐻_𝐿≤𝐻_𝑈 in element wise:𝐻_𝐿≤𝐻_𝑈∶𝐻_𝐿𝑖𝑗≤𝐻_𝑈𝑖𝑗,∀𝑖,𝑗∈1,…,𝑛. (55)

Then (51) is equivalent to the following variational inequality: 𝑋^′1−𝑋,∇2‖𝑋−𝐶‖²≥0,∀𝑋^′∈𝐾. (56) So we get 𝑋^′−𝑋,𝑋−𝐶≥0,∀𝑋^′∈𝐾. (57)

According to Condition 10, we take the following parameter sequences, and let Condition 10 denote the parameter sequences: 𝛼_𝑛=1,𝜆ln𝑛_𝑛=𝜆^′_𝑛=𝜆_𝑛^′′=1(),𝛽ln𝑛+1_𝑛=𝛾_𝑛1=1−,ln𝑛𝛾=𝜌=𝑡=𝑐₀>0. (58) According to Condition 12, we take the following parameter sequences, and let Condition 12 denote the parameter sequences: 𝛼_𝑛1=0.8−()𝛼10∗ln𝑛,𝑛=2𝑘,_𝑛1=0.3−()𝜆10∗ln𝑛,𝑛=2𝑘−1,_𝑛=𝜆^′_𝑛=𝜆_𝑛^′′=1(),𝛽ln𝑛+1_𝑛1=1−𝛽ln𝑛,𝑛=2𝑘,_𝑛1=1−𝛾ln𝑛,𝑛=2𝑘−1,_𝑛1=1−𝛾ln𝑛,𝑛=2𝑘,_𝑛1=1−ln(2𝑛),𝑛=2𝑘−1,𝛾=𝜌=𝑡=𝑐₀>0. (59) Obviously, we have much difficulty in computing the projection of 𝑃_𝐾[𝑋],forall𝑥∈𝑆^𝑛. In order to reduce the difficulty and complexity of computing the projection 𝑃_𝐾, we define 𝑇𝑋 by 𝑇𝑋=𝐻(𝐺(𝑋)), (60) where 𝐺(𝑋)=min𝐻_𝑈,max𝑋,𝐻_𝐿,𝐻(𝑋)=𝑃_𝑆^𝑛₊(𝑋), (61) which can be computed without difficulty and the fixed point set of Fix(𝑇)=𝐾. According to Theorems 11 and 13, the sequences generated by Algorithm 8 under Conditions 10 and 12 are convergent.

The computation begins with ones (𝑛,𝑛) in MATLAB and stops as soon as ‖𝑥_𝑘+1−𝑥_𝑘‖≤10⁻³or10⁻². All codes were implemented in MATLAB 7.1 and ran at a Pentium R 1.70G processor, 2G Acer note computer.

We test the problems with 𝑛=100, 200, 300, 400, 500, 1000, and 2000. The test results with the PRH method under different conditions are reported in Tables 1, 2, 3, and 4. And the CPU time is in seconds. It is to be noted that the results of extended contraction method are only given out when the iteration step (It) is less than or equal to 100.

Table 1 

Numerical results for the PRH method and the EC method.

Table 2 

Numerical results for tolerance 10⁻⁴.

Table 3 

Numerical results for tolerance 10⁻³.

Table 4 

Numerical results for tolerance 10⁻⁴.

Test Examples 1. In this example we generate the data in a similar manner as in [9]. The entries of diagonal elements of 𝐶 are randomly generated in the interval (0,2); the entries of off-diagonal elements of 𝐶 are randomly generated in the interval (−1,1) (Algorithm 1): 𝐻_𝑈𝑗𝑗=𝐻_𝐿𝑗𝑗𝐻=1,_𝑈𝑖𝑗=−𝐻_𝐿𝑖𝑗=0.1,∀𝑖≠𝑗,𝑖,𝑗=1,2,…,𝑛. (62) When 𝑛≥1000 and tolerance 10⁻⁴, the computation time of the proposed method is too long, so the results of the PRH method give out approximate solution with 𝑛≥1000 and tolerance 10⁻³ in the following. And the extended contraction method (EC method) has much difficulty in computing the examples when 𝐶 is asymmetric. Furthermore, by introducing auxiliary variable, the certain projection method or relaxed-PPA method [10] can be implemented by these tests.

Matlab code: 𝐶 = zeros(𝑛,𝑛); HU = ones(𝑛,𝑛)∗0.1; HL = −HU; for 𝑖=1∶𝑛  for 𝑗=1∶𝑛   𝑡=mod(𝑡∗42108+13846,46273);   𝐶(𝑖,𝑗)=𝑡∗2/46273−1;  end; end; for 𝑖=1∶𝑛  𝐶(𝑖,𝑖) = abs(𝐶(𝑖,𝑖))∗2; HU(𝑖,𝑖)=1; HL(𝑖,𝑖)=1; end;

Algorithm 1  

Test Examples 2. We form the data of the second problems similarly as in the first test examples. The entries of diagonal elements of 𝐶 are randomly generated in the interval (0,2); the entries of off-diagonal elements of 𝐶 are generated from a uniform distribution in the same interval (Algorithm 2): 𝐻_𝑈𝑗𝑗=𝐻_𝐿𝑗𝑗𝐻=1,_𝑈𝑖𝑗=−𝐻_𝐿𝑖𝑗=0.1,∀𝑖≠𝑗,𝑖,𝑗=1,2,…,𝑛. (63)

Matlab code: 𝐶 = zeros(𝑛,𝑛); HU = ones(𝑛,𝑛)∗0.1; HL = −HU; for 𝑖=1∶𝑛  for 𝑗=1∶𝑛   𝐶=−1+2∗rand(𝑛);  end; end; for 𝑖=1∶𝑛  𝐶(𝑖,𝑖) = abs(𝐶(𝑖,𝑖))∗2; HU(𝑖,𝑖)=1; HL(𝑖,𝑖)=1; end;

Algorithm 2  

From Tables 1 to 3, we found that the iteration numbers and CPU time of PRH under Condition 12 are more efficient than that under Condition 10. In Table 4 of our method, the tests’ results give out that the PRH method under some descent directions is more slightly efficient than those of the MRHSD method [14, 16], and it is easy to obtain these descent directions. Furthermore, it is important to find 𝛾,𝜌,and𝑡 by Tables 2 and 4.

5. Conclusions

We have proved the strong convergence of PRH method under Condition 12, which differs from Condition 10. The result can be considered as an improvement and refinement of the previous results [14]. And more importantly, numerical experiments demonstrated that the PRH method under Condition 12 is more efficient than that under Condition 10, and the PRH method under some descent directions is more slightly efficient than that of the MRHSD method. How to select parameters of the PRH method for solving variational inequalities is worthy of further investigations in the future.

Acknowledgments

This research was supported by National Science and Technology Support Program (Grant no. 2011BAH24B06), Joint Fund of National Natural Science Foundation of China and Civil Aviation Administration of China (Grant no. U1233105), and Science Foundation of the Civil Aviation Flight University of China (Grant no. J2010-45).