On Self-Adaptive Method for General Mixed Variational Inequalities

Abstract

We suggest and analyze a new self-adaptive method for solving general mixed variational inequalities, which can be viewed as an improvement of the method of (Noor 2003). Global convergence of the new method is proved under the same assumptions as Noor's method. Some preliminary computational results are given to illustrate the efficiency of the proposed method. Since the general mixed variational inequalities include general variational inequalities, quasivariational inequalities, and nonlinear (implicit) complementarity problems as special cases, results proved in this paper continue to hold for these problems.

1. Introduction

Variational inequalities introduced in the early sixties have played a critical and significant part in the study of several unrelated problems arising in finance, economics, network analysis, transportation, elasticity, and optimization. Variational inequalities theory has witnessed an explosive growth in theoretical advances, algorithmic development, and applications across all disciplines of pure and applied sciences, see [1–16]. A useful and important generalization of variational inequalities is the mixed variational inequality containing a nonlinear term 𝜑. But the applicability of the projection method is limited due to the fact that it is not easy to find the projection except in very special cases. Secondly, the projection method cannot be applied to suggest iterative algorithms for solving general mixed variational inequalities involving the nonlinear term 𝜑. This fact has motivated many authors to develop the auxiliary principle technique for solving the mixed variational inequalities. In recent years, several techniques have been developed to suggest and analyze various iterative methods for solving different types of variational inequalities. It is worth mentioning that if the nonlinear term in the variational inequalities is a proper, convex, and semilower continuous function, then it is well known that the variational inequalities involving the nonlinear term 𝜑 are equivalent to the fixed point problems and the resolvent equations. In [11], Noor solved the general mixed variational inequality problem by using the resolvent equations technique. Inspired and motivated by the results of Noor [11], we propose a new method for solving general mixed variational inequalities by using a new direction with a new step size 𝛼_𝑘. We prove the global convergence of the proposed method under the same assumptions as in [11]. An example is given to illustrate the efficiency and its comparison with the results of Noor [11, 14]. This shows that the method is robust and efficient. This new method can be viewed as an important and significant improvement of Noor and other methods.

2. Preliminaries

Let 𝐻 be a real Hilbert space, whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, let 𝐼 be the identity mapping on 𝐻, and 𝑇,𝑔∶𝐻→𝐻 be two operators. Let 𝜕𝜑 denotes the subdifferential of function 𝜑, where 𝜑∶𝐻→𝑅∪{+∞} is a proper convex lower semicontinuous function on 𝐻. It is well known that the subdifferential 𝜕𝜑 is a maximal monotone operator. We consider the problem of finding 𝑢^∗∈𝐻 such that 𝑇(𝑢^∗),𝑔(𝑣)−𝑔(𝑢^∗)+𝜑𝑔(𝑣)−𝜑𝑔(𝑢^∗)≥0,∀𝑔(𝑣)∈𝐻, (2.1)which is known as the mixed general variational inequality, see Noor [11]. We also note that the general variational inequality can be written in the equivalent form as find 𝑢^∗∈𝐻 such that0∈𝑇𝑢+𝜕𝜑𝑔(𝑢), (2.2)which is known as the problem of finding a zero of sum of two(more) monotone operators. It is well known that a wide class of linear and nonlinear problems arising in pure and applied sciences can be studied via the general mixed variational inequalities, see [1–16] and the references therein.

If 𝐾 is a closed convex set in 𝐻 and 𝜑(𝑣)≡𝐼_𝐾(𝑣),∀𝑣∈𝐻, where𝐼_𝐾(𝑣)=0if𝑣∈𝐾,+∞,otherwise (2.3)is the indicator function of 𝐾, then the problem (2.1) is equivalent to finding 𝑢^∗∈𝐻 such that 𝑔(𝑢^∗)∈𝐾 and 𝑇(𝑢^∗),𝑔(𝑣)−𝑔(𝑢^∗)≥0,∀𝑔(𝑣)∈𝐾. (2.4)Problem (2.4) is called the general variational inequality, which was first introduced and studied by Noor [9] in 1988. For the applications, formulation, and numerical methods of general variational inequalities (2.4), we refer the reader to the survey, see [1–3, 7, 12, 13, 16].

If 𝑔≡𝐼, then the problem (2.4) is equivalent to finding 𝑢^∗∈𝐾 such that 𝑇(𝑢^∗),𝑣−𝑢^∗≥0,∀𝑣∈𝐾, (2.5) which is known as the classical variational inequality introduced and studied by Stampacchia [17].

Lemma 2.1 (see [4]). For a given 𝑤∈𝐻,𝑧∈𝐻, and 𝜌>0, the inequality𝑤−𝑧,𝑧−𝑣+𝜌𝜑(𝑣)−𝜌𝜑(𝑧)≥0,∀𝑣∈𝐻 (2.6)holds if and only if 𝑧=𝐽_𝜑(𝑤), where 𝐽_𝜑=(𝐼+𝜌𝜕𝜑)⁻¹ is the resolvent operator.

It follows from Lemma 2.1 that 𝑤−𝐽_𝜑(𝑤),𝐽_𝜑(𝑤)−𝑣+𝜌𝜑(𝑣)−𝜌𝜑𝐽_𝜑(𝑤)≥0,∀𝑣,𝑤∈𝐻. (2.7)

If 𝜑 is the indicator function of a closed convex set 𝐾 in 𝐻, then the resolvent operator 𝐽_𝜑(⋅)=𝑃_𝐾, where 𝑃_𝐾 is the projection of 𝐻 onto the closed convex set 𝐾. It is well known that 𝐽_𝜑 is nonexpansive, that is,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑣)≤‖𝑢−𝑣‖,∀𝑢,𝑣∈𝐻. (2.8)

Lemma 2.2 (see [10]). 𝑢^∗ is solution of problem (2.1) if and only if 𝑢^∗∈𝐻 satisfies the relation:𝑔𝑢^∗=𝐽_𝜑𝑔𝑢^∗−𝜌𝑇𝑢^∗, (2.9) where 𝐽_𝜑=(𝐼+𝜌𝜕𝜑)⁻¹ is the resolvent operator .

From Lemma 2.2, it is clear that 𝑢^∗ is solution of (2.1) if and only if 𝑢^∗ is a zero point of the function𝑟𝑢^∗,𝜌∶=𝑔𝑢^∗−𝐽_𝜑𝑔𝑢^∗−𝜌𝑇𝑢^∗. (2.10)In [11], Noor used the fixed-point formulation (2.9) and the resolvent equations to suggest and analyze the following algorithm for solving problem (2.1).

Algorithm 2.3. For a given 𝑢⁰∈𝐻, compute the approximate solution 𝑢^𝑘+1 by the iterative schemes.
Predictor Step
𝑔𝑤^𝑘=𝐽_𝜑𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘, (2.11)where 𝜌_𝑘 satisfies𝜌_𝑘𝑇𝑢^𝑘−𝑇𝑔⁻¹𝐽_𝜑𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘,𝑟𝑢^𝑘,𝜌_𝑘≤𝛿𝑟(𝑢^𝑘,𝜌_𝑘)²,𝛿∈(0,1). (2.12)
Corrector Step
𝑔𝑢^𝑘+1=𝐽_𝜑𝑔𝑢^𝑘−𝛼_𝑘𝑑𝑢^𝑘,𝜌_𝑘, (2.13)where𝑑𝑢^𝑘,𝜌_𝑘=𝑟𝑢^𝑘,𝜌_𝑘−𝜌_𝑘𝑇𝑢^𝑘+𝜌_𝑘𝑇𝑔⁻¹𝐽_𝜑𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘,𝛼_𝑘=(1−𝛿)‖𝑟(𝑢^𝑘,𝜌_𝑘)‖²‖𝑑(𝑢^𝑘,𝜌_𝑘)‖² (2.14)is the corrector step size.
If 𝜑 is an indicator function of a closed convex set 𝐾 in 𝐻, then 𝐽_𝜑≡𝑃_𝐾 [10], the projection of 𝐻 onto 𝐾 and consequently Algorithm 2.3 collapses.

Algorithm 2.4 (see [11]). For a given 𝑢⁰∈𝐻, compute the approximate solution 𝑢^𝑘+1 by the iterative schemes.
Predictor Step
𝑔𝑤^𝑘=𝑃_𝐾𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘, (2.15)where 𝜌_𝑘 satisfies𝜌_𝑘𝑇𝑢^𝑘−𝑇𝑔⁻¹𝑃_𝐾𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘,𝑟𝑢^𝑘,𝜌_𝑘≤𝛿‖𝑟𝑢^𝑘,𝜌_𝑘‖²,𝛿∈(0,1). (2.16)
Corrector Step
𝑔𝑢^𝑘+1=𝑃_𝐾𝑔𝑢^𝑘−𝛼_𝑘𝑑𝑢^𝑘,𝜌_𝑘, (2.17)where𝑑𝑢^𝑘,𝜌_𝑘=𝑟𝑢^𝑘,𝜌_𝑘−𝜌_𝑘𝑇𝑢^𝑘+𝜌_𝑘𝑇𝑔⁻¹𝑃_𝐾𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘,𝛼_𝑘=(1−𝛿)‖𝑟(𝑢^𝑘,𝜌_𝑘)‖²‖𝑑(𝑢^𝑘,𝜌_𝑘)‖² (2.18)is the corrector step size.
Throughout this paper, we make following assumptions.

Assumptions
(i)𝐻 is finite dimension space.(ii)𝑔 is homeomorphism on 𝐻, that is, 𝑔 is bijective, continuous and 𝑔⁻¹ is continuous.(iii)𝑇 is continuous and g-pseudomonotone operator on 𝐻, that is,𝑇(𝑢),𝑔𝑢^′−𝑔(𝑢)≥0⟹𝑇𝑢^′,𝑔𝑢^′−𝑔(𝑢)≥0∀𝑢^′,𝑢∈𝐻. (2.19)(iv)The solution set of problem (2.1) denoted by 𝑆^∗ is nonempty.

3. Iterative Method and Basic Results

In this section, we suggest and analyze a new method for solving mixed general variational inequality (2.1) by using a new direction with a new step size 𝛼_𝑘, and this is the main motivation of this paper.

Algorithm 3.1 Step 1. Given 𝛾∈[1,2),𝜌₀>0,𝛿∈(0,1),𝑢⁰∈𝐻, and 𝑘=0.Step 2. Set 𝑤^𝑘=𝑔⁻¹𝐽_𝜑𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘. (3.1)Step 3. If 𝑟_𝑘∶=‖𝜌_𝑘(𝑇(𝑢^𝑘)−𝑇(𝑤^𝑘))‖‖𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)‖≤𝛿, (3.2) then set 𝜀^𝑘=𝜌_𝑘𝑇𝑤^𝑘−𝑇𝑢^𝑘,𝑑₁𝑢^𝑘,𝜌_𝑘∶=𝑔𝑢^𝑘−𝑔𝑤^𝑘+𝜀^𝑘,𝑑𝑢^𝑘,𝜌_𝑘∶=𝑔𝑢^𝑘−𝑔𝑤^𝑘+𝜌_𝑘𝑇𝑤^𝑘,𝜙𝑢^𝑘,𝜌_𝑘∶=𝑔𝑢^𝑘−𝑔𝑤^𝑘,𝑑₁𝑢^𝑘,𝜌_𝑘 (3.3)(3.4)(3.5)(3.6) the stepsize
𝛼_𝑘∶=𝜙(𝑢^𝑘,𝜌_𝑘)‖𝑑(𝑢^𝑘,𝜌_𝑘)‖² (3.7)and the next iterate𝑔𝑢^𝑘+1=𝐽_𝜑𝑔𝑢^𝑘−𝛾𝛼_𝑘𝑑𝑢^𝑘,𝜌_𝑘,𝜌_𝑘∶=1.5∗𝜌_𝑘if𝑟_𝑘𝜌≤0.5,_𝑘otherwise, (3.8)𝜌_𝑘+1=𝜌_𝑘 and 𝑘∶=𝑘+1; go to Step 2.Step 4. Reduce the value of 𝜌_𝑘 by 𝜌_𝑘∶=2/3∗𝜌_𝑘∗min{1,1/𝑟_𝑘};
set 𝑤^𝑘=𝑔⁻¹(𝑃_𝐾[𝑔(𝑢^𝑘)−𝜌_𝑘𝑇(𝑢^𝑘)]) and go to Step 3.
If 𝜑 is an indicator function of a closed convex set 𝐾 in 𝐻, then 𝐽_𝜑≡𝑃_𝐾 [10], the projection of 𝐻 onto 𝐾. Consequently, Algorithm 3.1 reduces to Algorithm 3.2 for solving the general variational inequalities (2.4).

Algorithm 3.2 Step 1. Given 𝛾∈[1,2),𝜌₀>0,𝛿∈(0,1),𝑢⁰∈𝐻, and 𝑘=0.Step 2. Set𝑤^𝑘=𝑔⁻¹𝑃_𝐾𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑢^𝑘. (3.9)Step 3. If𝑟_𝑘∶=‖𝜌_𝑘(𝑇(𝑢^𝑘)−𝑇(𝑤^𝑘))‖‖𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)‖≤𝛿, (3.10)
then set𝜀^𝑘∶=𝜌_𝑘𝑇𝑤^𝑘−𝑇𝑢^𝑘,𝑑₁𝑢^𝑘,𝜌_𝑘∶=𝑔𝑢^𝑘−𝑔𝑤^𝑘+𝜀^𝑘,𝑑𝑢^𝑘,𝜌_𝑘∶=𝑔𝑢^𝑘−𝑔𝑤^𝑘+𝜌_𝑘𝑇𝑤^𝑘,𝜙𝑢^𝑘,𝜌_𝑘∶=𝑔𝑢^𝑘−𝑔𝑤^𝑘,𝑑₁𝑢^𝑘,𝜌_𝑘 (3.11)
the stepsize𝛼_𝑘∶=𝜙(𝑢^𝑘,𝜌_𝑘)‖𝑑(𝑢^𝑘,𝜌_𝑘)‖² (3.12)
and the next iterate𝑔𝑢^𝑘+1=𝑃_𝐾𝑔𝑢^𝑘−𝛾𝛼_𝑘𝑑𝑢^𝑘,𝜌_𝑘,𝜌_𝑘∶=1.5∗𝜌_𝑘if𝑟_𝑘𝜌≤0.5,_𝑘otherwise, (3.13)
𝜌_𝑘+1=𝜌_𝑘 and 𝑘∶=𝑘+1; go to Step 2.Step 4. Reduce the value of 𝜌_𝑘 by 𝜌_𝑘∶=2/3∗𝜌_𝑘∗min{1,1/𝑟_𝑘};
set 𝑤^𝑘=𝑔⁻¹(𝐽_𝜑[𝑔(𝑢^𝑘)−𝜌_𝑘𝑇(𝑢^𝑘)]) and go to Step 3.

Remark 3.3. Equation (3.2) implies that
𝑔𝑢^𝑘−𝑔𝑤^𝑘,𝜀^𝑘≤𝛿𝑔𝑢^𝑘−𝑔𝑤^𝑘2,0<𝛿<1. (3.14)
The next lemma shows that 𝛼_𝑘 and 𝜙(𝑢^𝑘,𝜌_𝑘) are lower bounded away from zero, whenever 𝑢^𝑘≠𝑤^𝑘.

Lemma 3.4. For given 𝑢^𝑘∈𝐻 and 𝜌_𝑘>0, let 𝑤^𝑘 and 𝜀^𝑘 satisfy to (3.1) and (3.3), then 𝜙𝑢^𝑘,𝜌_𝑘≥1−𝛿𝑔𝑢^𝑘−𝑔𝑤^𝑘2,𝛼_𝑘≥(1−𝛿)‖𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)‖²‖𝑑(𝑢^𝑘,𝜌_𝑘)‖². (3.15)

Proof. It follows from (3.4) and (3.14) that𝜙𝑢^𝑘,𝜌_𝑘∶=𝑔𝑢^𝑘−𝑔𝑤^𝑘,𝑑₁𝑢^𝑘,𝜌_𝑘=𝑔𝑢^𝑘−𝑔𝑤^𝑘2+𝑔𝑢^𝑘−𝑔𝑤^𝑘,𝜀^𝑘≥(1−𝛿)𝑔𝑢^𝑘−𝑔𝑤^𝑘2. (3.16)
Otherwise, we have 𝛼_𝑘𝑔𝑢∶=^𝑘−𝑔𝑤^𝑘,𝑑₁𝑢^𝑘,𝜌_𝑘𝑑𝑢^𝑘,𝜌_𝑘²≥(1−𝛿)‖𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)‖²‖𝑑(𝑢^𝑘,𝜌_𝑘)‖², (3.17)we can get the assertion of this lemma.

The next lemma shows that ‖𝑟(𝑢,𝜌)‖ is a nondecreasing function with respect to 𝜌, which can be proved using the techniques as in [1].

Lemma 3.5. For all 𝑢∈𝐻 and 𝜌^′>𝜌>0, it holds that‖𝑟(𝑢,𝜌^′)‖≥‖𝑟(𝑢,𝜌)‖. (3.18)

Next lemma has already been studied in [11].

Lemma 3.6. For all 𝑢∈𝐻,𝑢^∗∈𝑆^∗, and 𝜌>0, we have
𝑔𝑢^𝑘−𝑔𝑢^∗,𝑑𝑢^𝑘,𝜌_𝑘≥𝜙𝑢^𝑘,𝜌_𝑘, (3.19) where 𝑑(𝑢^𝑘,𝜌_𝑘) and 𝜙(𝑢^𝑘,𝜌_𝑘) are defined in (3.5) and (3.6), respectively.

4. Convergence Analysis

In this section, we prove the global convergence of the proposed method. The following result plays a crucial role in the convergence analysis of the proposed method.

Theorem 4.1. Let 𝑢^∗∈𝐻 be a solution of problem (2.1) and let 𝑢^𝑘+1 be the sequence obtained from Algorithm 3.1. Then 𝑢^𝑘 is bounded and
𝑔𝑢^𝑘+1−𝑔𝑢^∗2≤𝑔𝑢^𝑘−𝑔𝑢^∗2−𝛾(2−𝛾)(1−𝛿)²‖𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)‖⁴‖𝑑(𝑢^𝑘,𝜌_𝑘)‖². (4.1)

Proof. Let 𝑢^∗∈𝐻 be a solution of problem (2.1). Then𝑔𝑢^𝑘+1−𝑔𝑢^∗2≤𝑔𝑢^𝑘−𝑔𝑢^∗−𝛾𝛼_𝑘𝑑𝑢^𝑘,𝜌_𝑘²=𝑔𝑢^𝑘−𝑔𝑢^∗2−2𝛾𝛼_𝑘𝑔𝑢^𝑘−𝑔𝑢^∗,𝑑𝑢^𝑘,𝜌_𝑘+𝛾²𝛼²_𝑘𝑑𝑢^𝑘,𝜌_𝑘²≤𝑔𝑢^𝑘−𝑔𝑢^∗2−2𝛾𝛼_𝑘𝜙𝑢^𝑘,𝜌_𝑘+𝛾²𝛼_𝑘𝜙𝑢^𝑘,𝜌_𝑘≤𝑔𝑢^𝑘−𝑔𝑢^∗2−𝛾(2−𝛾)(1−𝛿)²‖𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)‖⁴‖𝑑(𝑢^𝑘,𝜌_𝑘)‖², (4.2)where the first inequality follows from the nonexpansiveness of the resolvent operator, the second inequality follows from (3.7) and (3.19), and the third inequality follows from (3.15). Since 𝛾∈[1,2) and 𝛿∈(0,1), we have𝑔𝑢^𝑘+1−𝑔𝑢^∗≤𝑔𝑢^𝑘−𝑔𝑢^∗≤⋯≤𝑔𝑢⁰−𝑔𝑢^∗. (4.3)
Since 𝑔 is homeomorphism, it is easy to verify that the sequence 𝑢^𝑘 is bounded.

We now prove the convergence of Algorithm 3.1.

Theorem 4.2. The sequence {𝑢^𝑘} generated by the Algorithm 3.1 converges to a solution of problem (2.1).

Proof. It follows from (4.1) that^∞_𝑘=0‖𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)‖⁴‖𝑑(𝑢^𝑘,𝜌_𝑘)‖²<∞, (4.4)which means that lim_𝑘→∞𝑔𝑢^𝑘−𝑔𝑤^𝑘=0. (4.5) Since 𝑔 is homeomorphism, we havelim_𝑘→∞𝑢^𝑘−𝑤^𝑘=0. (4.6)This implies that {𝑤^𝑘} is bounded. Since ‖𝑟(𝑢^𝑘,𝜌)‖ is a nondecreasing function of 𝜌, it follows from 𝜌_𝑘≥𝜌_min that𝑟𝑤^𝑘,𝜌_min≤𝑟𝑤^𝑘,𝜌_𝑘=𝑔𝑤^𝑘−𝐽_𝜑𝑔𝑤^𝑘−𝜌_𝑘𝑇𝑤^𝑘(using(3.1)and(3.3))=𝐽_𝜑[𝑔𝑢^𝑘−𝜌_𝑘𝑇𝑤^𝑘+𝜀^𝑘]−𝐽_𝜑[𝑔𝑤^𝑘−𝜌_𝑘𝑇𝑤^𝑘]≤𝑔𝑢^𝑘−𝑔𝑤^𝑘+𝜀^𝑘(using(3.2))≤(1+𝛿)𝑔𝑢^𝑘−𝑔𝑤^𝑘 (4.7)and from (4.5), we getlim_𝑘→∞𝑟𝑤^𝑘,𝜌_min=0. (4.8) Let 𝑢 be a cluster point of {𝑤^𝑘} and the subsequence {𝑤^𝑘𝑗} converges to 𝑢. Since 𝑟(𝑢,𝜌) is a continuous function of 𝑢, it follows from (4.8) that𝑟𝑢,𝜌_min=lim_𝑗→∞𝑟𝑤^𝑘𝑗,𝜌_min=0. (4.9)From Lemma 2.2, it follows that 𝑢 is a solution point of problem (2.1). Note that inequality (4.1) is true for all solution point of problem (2.1), hence we have 𝑔𝑢^𝑘+1−𝑔𝑢≤𝑔𝑢^𝑘−𝑔𝑢,∀𝑘≥0. (4.10)Since {𝑔(𝑤^𝑘𝑗)}→𝑔(𝑢) and 𝑔(𝑢^𝑘)−𝑔(𝑤^𝑘)→0, for any given 𝜀>0, there is an 𝑙>0, such that
𝑔𝑤^𝑘𝑙−𝑔𝑢<𝜀2,𝑔𝑢^𝑘𝑙−𝑔𝑤^𝑘𝑙<𝜀2. (4.11)Therefore, for any 𝑘≥𝑘_𝑙, it follows from (4.10) and (4.11) that𝑔𝑢^𝑘−𝑔𝑢≤𝑔𝑢^𝑘𝑙−𝑔𝑢≤𝑔𝑢^𝑘𝑙−𝑔𝑤^𝑘𝑙+𝑔𝑤^𝑘𝑙−𝑔𝑢<𝜀 (4.12)and thus the sequence {𝑔(𝑢^𝑘)} converges to 𝑔(𝑢). Using 𝑔 is homeomorphism, we see that the sequence {𝑢^𝑘} converges to 𝑢.
We now prove that the sequence {𝑢^𝑘} has exactly one cluster point. Assume that 𝑢 is another cluster point and satisfies𝛿∶=𝑔𝑢−𝑔𝑢>0. (4.13)Since 𝑢 is a cluster point of the sequence {𝑢^𝑘} and 𝑔 is homeomorphism, there is a 𝑘₀>0 such that𝑔𝑢^𝑘0−𝑔𝑢≤𝛿2. (4.14)On the other hand, since 𝑢∈𝑆^∗ and from (4.1), we have𝑔𝑢^𝑘−𝑔𝑢≤𝑔𝑢^𝑘0−𝑔𝑢∀𝑘≥𝑘₀, (4.15)it follows that𝑔𝑢^𝑘−𝑔𝑢≥𝑔𝑢−𝑔𝑢−𝑔𝑢^𝑘−𝑔𝑢≥𝛿2∀𝑘≥𝑘₀. (4.16)This contradicts the assumption that 𝑢 is cluster point of {𝑢^𝑘}. Thus the sequence {𝑢^𝑘} converges to 𝑢∈𝑆^∗.

5. Numerical Results

In this section, we present some numerical results for the proposed method. In order to verify the theoretical assertions, we consider the following problems:minℎ(𝑢)=∑^𝑛_𝑗=1𝑢_𝑗𝑢log^𝑗𝑝_𝑗,s.t.𝐴𝑢∈Π,𝑢≥0, (5.1)where 𝐴 is an 𝑛×𝑛 matrix, Π is a simple closed convex set in 𝑅^𝑛,0<𝑝∈𝑅^𝑛 is a parameter vector. Here, the statement that the set Π is simple means that the projection onto Π is simple to carry out. For the same reason given in Fletcher (see [5, page 222]), each element of the optimal solution of problem (5.1) is positive. Thus the bounds 𝑢≥0 are inactive and can be ignored, therefore problem (5.1) can be written as minℎ(𝑢)=∑^𝑛_𝑗=1𝑢_𝑗𝑢log^𝑗𝑝_𝑗,s.t.𝐴𝑢−𝑣=0,𝑣∈Π. (5.2)By attaching the Lagrange multiplier 𝑦∈𝑅^𝑛 to the equality constraints 𝐴𝑢−𝑣=0, the Lagrange function of problem (5.2) is𝐿(𝑢,𝑣,𝑦)=ℎ(𝑢)−𝑦^𝑇(𝐴𝑢−𝑣), (5.3)which is defined on 𝑅^𝑛×Π×𝑅^𝑛. If (𝑢^∗,𝑣^∗,𝑦^∗)∈𝑅^𝑛×Π×𝑅^𝑛 is a KKT point of problem (5.2), then we have ∇ℎ𝑢^∗−𝐴^𝑇𝑦^∗=0,𝑣−𝑣^∗𝑇𝑦^∗≥0,∀𝑣∈Π,𝐴𝑢^∗−𝑣^∗=0. (5.4) Note that problem (5.1) is invariant under multiplication ℎ by some positive scalar 𝜌. Denoting 𝑓(𝑢)=∇ℎ(𝑢) and eliminating 𝑣 and 𝑣^∗ in (5.4), we see that problem (5.1) is equivalent to a general variational inequality problem. Find 𝑤^∗ such that

𝑄𝑤^∗∈Ω,𝑄𝑣₁−𝑄𝑤^∗𝑇𝐹𝑤^∗≥0,∀𝑄𝑣₁∈Ω, (5.5)where 𝑢𝑦𝑢𝑤=,𝑄(𝑤)=𝐴𝑢,𝐹(𝑤)=𝜌𝑓(𝑢)−𝐴^𝑇𝑦𝑦,Ω=𝑅^𝑛×Π. (5.6)It is well known (see [7, Theorem 1]) that solving (5.5)-(5.6) is equivalent to finding a zero point of the function𝑒(𝑤)∶=𝑄(𝑤)−𝑃_Ω𝑄(𝑤)−𝐹(𝑤)=𝜌𝑓(𝑢)−𝐴^𝑇𝑦𝐴𝑢−𝑃_Π[𝐴𝑢−𝑦]. (5.7)Then, solving (5.5)-(5.6) is equivalent to find a pair (𝑢^∗,𝑦^∗), such that

𝜌𝑓(𝑢^∗)=𝐴^𝑇𝑦^∗,𝑔𝑢^∗∈Π,𝑔(𝑣)−𝑔𝑢^∗𝑇𝑦^∗≥0,∀𝑔(𝑣)∈Π, (5.8)(5.9) where𝑔(𝑢)=𝐴𝑢. (5.10)In this case Algorithms 2.3 and 3.1 collapse to Algorithms 2.4 and 3.2, respectively.

In the test, we let 𝑣^′∈𝑅^𝑛 be a randomly generated vector, 𝑣^′_𝑗∈(−0.5,0.5), and 𝐴=𝐼−2(𝑣^′𝑣^′𝑇)/(𝑣^′𝑇𝑣^′) be an 𝑛×𝑛 Householder matrix. Let𝑢^∗_𝑗∈(0.1,1.1),𝑦^∗_𝑖∈(−0.5,0.5). (5.11)Note that𝑓_𝑗𝑢^∗=∇ℎ𝑢^∗_𝑗=log𝑢^∗_𝑗−log𝑝_𝑗+1. (5.12)Since𝑓𝑢^∗=𝐴^𝑇𝑦^∗, (5.13)we set𝑝_𝑗=𝑢^∗_𝑗exp1−𝑒^𝑇_𝑗𝐴^𝑇𝑦^∗,Π={𝑧∣𝑙_𝐵≤𝑧≤𝑢_𝐵}, (5.14)where𝑙_𝐵𝑖=𝐴𝑢^∗_𝑖if𝑦^∗_𝑖≥0,𝐴𝑢^∗_𝑖+𝑦^∗_𝑖𝑢otherwise,_𝐵𝑖=𝐴𝑢^∗_𝑖if𝑦^∗_𝑖<0,𝐴𝑢^∗_𝑖+𝑦^∗_𝑖otherwise. (5.15) In this way, we have𝐴𝑢^∗∈Π,𝐴𝑢^∗=𝑃_Π𝐴𝑢^∗−𝑦^∗. (5.16)

In all the tests, we take 𝛿=0.95 and 𝛾=1.95. The calculations are started with a vector 𝑢⁰, whose elements are randomly chosen in (0,1) and stopped, whenever ‖𝑟(𝑢,𝜌)‖_∞≤10⁻⁷.

Since 𝑢^∗ is known, we also report the distance ‖𝑢^𝑘−𝑢^∗‖ after ‖𝑟(𝑢,𝜌)‖_∞≤10⁻⁷. All codes are written in Matlab and run on a P4-2.00G note book computer. We test the problem with dimensions 𝑛=200 and 𝑛=300. The iteration numbers and the computational time for Algorithms 2.4 and 3.2 with different dimensions and initial parameter 𝜌 are given in the Tables 1-2, and for Algorithm 3.2 and the method of Noor [14] in Tables 3-4

Table 1: Numerical results for problem (5.9) with 𝑛=200.

Table 2: Numerical results for problem (5.9) with 𝑛=300.

Table 3: Numerical results for problem (5.9) with 𝑛=200.

Table 4: Numerical results for problem (5.9) with 𝑛=300.

From Tables 1–4, we could see that Algorithms 2.4 and the method in [14] work well, if 𝜌 is sufficient large. If the parameter 𝜌 is too small, then the iteration numbers and the computational time can increase significantly. Also these tables show that Algorithm 3.2 is very efficient for the problem tested. In addition, for our method, it seems that the computational time and the iteration numbers are not very sensitive to the problem size.

Acknowledgments

The authors would like to thank the referees and Professor Dr. Alois Steindl for their very constructive comments and suggestions. Abdellah Bnouhachem was supported by china Postdoctoral science Foundation Grant no. 20060390915 and the National Key Technology R&D Program (2006BAH02A06).