On Penalty and Gap Function Methods for Bilevel Equilibrium Problems

Abstract

We consider bilevel pseudomonotone equilibrium problems. We use a penalty function to convert a bilevel problem into one-level ones. We generalize a pseudo-∇-monotonicity concept from ∇-monotonicity and prove that under pseudo-∇-monotonicity property any stationary point of a regularized gap function is a solution of the penalized equilibrium problem. As an application, we discuss a special case that arises from the Tikhonov regularization method for pseudomonotone equilibrium problems.

1. Introduction

Let 𝐶 be a nonempty closed-convex subset in ℝ^𝑛, and let 𝑓,𝑔∶𝐶×𝐶→ℝ be two bifunctions satisfying 𝑓(𝑥,𝑥)=𝑔(𝑥,𝑥)=0 for every 𝑥∈𝐶. Such a bifunction is called an equilibrium bifunction. We consider the following bilevel equilibrium problem (BEP for short):find𝑥∈𝑆_𝑔suchthat𝑓𝑥,𝑦≥0,∀𝑦∈𝑆_𝑔,(1.1) where 𝑆_𝑔={𝑢∈𝐶∶𝑔(𝑢,𝑦)≥0,∀𝑦∈𝐶}, that is, 𝑆_𝑔 is the solution set of the equilibrium problemsfind𝑢∈𝐶suchthat𝑔𝑢,𝑦≥0,∀𝑦∈𝐶.(1.2) As usual, we call problem (1.1) the upper problem and (1.2) the lower one. BEPs are special cases of mathematical programs with equilibrium constraints. Sources for such problems can be found in [1–3]. Bilevel monotone variational inequality, which is a special case of problem (1.1), was considered in [4, 5]. Moudafi in [6] suggested the use of the proximal point method for monotone BEPs. Recently, Ding in [7] used the auxiliary problem principle to BEPs. In both papers, the bifunctions 𝑓 and 𝑔 are required to be monotone on 𝐶. It should be noticed that under the pseudomonotonicity assumption on 𝑔 the solution-set 𝑆_𝑔 of the lower problem (1.2) is a closed-convex set whenever 𝑔(𝑥,⋅) is lower semicontinuous and convex on 𝐶 for each 𝑥. However, the main difficulty is that, even the constrained set 𝑆_𝑔 is convex, it is not given explicitly as in a standard mathematical programming problem, and therefore the available methods (see, e.g., [8–14] and the references therein) cannot be applied directly.

In this paper, first, we propose a penalty function method for problem (1.1). Next, we use a regularized gap function for solving the penalized problems. Under certain pseudo-∇-monotonicity properties of the regularized bifunction, we show that any stationary point of the gap function on the convex set 𝐶 is a solution to the penalized subproblem. Finally, we apply the proposed method to the Tikhonov regularization method for pseudomonotone equilibrium problems.

2. A Penalty Function Method

Penalty function method is a fundamental tool widely used in optimization to convert a constrained problem into unconstrained (or easier constrained) ones. This method was used to monotone variational inequalities in [5] and equilibrium problems in [15]. In this section, we use the penalty function method in the bilevel problem (1.1). First, let us recall some well-known concepts on monotonicity and continuity (see, e.g., [16]) that will be used in the sequel.

Definition 2.1. The bifunction 𝜙∶𝐶×𝐶→ℝ is said to be as follows:(a)strongly monotone on 𝐶 with modulus 𝛽>0 if 𝜙𝑥,𝑦+𝜙𝑦,𝑥≤−𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐶,(2.1)(b)monotone on 𝐶 if 𝜙𝑥,𝑦+𝜙𝑦,𝑥≤0,∀𝑥,𝑦∈𝐶,(2.2)(c)pseudomonotone on 𝐶 if ∀𝑥,𝑦∈𝐶∶𝜙𝑥,𝑦≥0⟹𝜙𝑦,𝑥≤0,(2.3)(d)upper semicontinuous at 𝑥 with respect to the first argument on 𝐶 if lim_𝑧→𝑥𝜙𝑧,𝑦≤𝜙𝑥,𝑦,∀𝑦∈𝐶,(2.4)(e)lower semicontinuous at 𝑦 with respect to the second argument on 𝐶 if lim_𝑤→𝑦𝜙(𝑥,𝑤)≥𝜙𝑥,𝑦,∀𝑥∈𝐶.(2.5)

Clearly, (a)⇒(b)⇒(c).

Definition 2.2 (see [17]). The bifunction 𝜙∶𝐶×𝐶→ℝ is said to be coercive on 𝐶 if there exists a compact subset 𝐵⊂ℝ^𝑛 and a vector 𝑦₀∈𝐵∩𝐶 such that 𝜙𝑥,𝑦₀<0,∀𝑥∈𝐶⧵𝐵.(2.6)

Theorem 2.3 (see [18, Proposition 2.1.14]). Let 𝜙∶𝐶×𝐶→ℝ be a equilibrium bifunction such that 𝜙(⋅,𝑦) is upper semicontinuous on 𝐶 for each 𝑦∈𝐶 and 𝜙(𝑥,⋅) is lower semicontnous, convex on 𝐶 for each 𝑥∈𝐶. Suppose that 𝐶 is compact or 𝜙 is coercive on 𝐶, then there exists at least one 𝑥^∗∈𝐶 such that 𝜙(𝑥^∗,𝑦)≥0 for every 𝑦∈𝐶.

The following proposition tells us about a relationship between the coercivity and the strong monotonicity.

Proposition 2.4. Suppose that the equilibrium bifunction 𝜙 is strongly monotone on 𝐶, and 𝜙(𝑥,⋅) is convex, lower semicontinuous with respect to the second argument for all 𝑥∈𝐶, then for each 𝑦∈𝐶, there exists a compact set 𝐵 such that 𝑦∈𝐵 and 𝜙(𝑥,𝑦)<0forall𝑥∈𝐶⧵𝐵.

Proof. Suppose by contradiction that the conclusion does not hold, then there exists an element 𝑦₀∈𝐶 such that for every compact set 𝐵 there is an element 𝑥_𝐵∈𝐶⧵𝐵 such that 𝜙(𝑥_𝐵,𝑦₀)≥0. Take 𝐵∶=𝐵_𝑟 as the closed ball centered at 𝑦₀ with radius 𝑟>1. Then there exists 𝑥_𝑟∈𝐶⧵𝐵_𝑟 such that 𝜙(𝑥^𝑟,𝑦₀)≥0. Let 𝑥 be the intersection of the line segment [𝑦₀,𝑥_𝑟] with the unit sphere 𝑆(𝑦₀;1) centered at 𝑦₀ and radius 1. Hence, 𝑥_𝑟=𝑦₀+𝑡(𝑟)(𝑥−𝑦₀), where 𝑡(𝑟)>𝑟. By the strong monotonicity of 𝜙, we have 𝜙𝑦₀,𝑥_𝑟≤−𝜙𝑥_𝑟,𝑦₀−𝛽𝑥_𝑟−𝑦₀²≤−𝜙𝑥_𝑟,𝑦₀−𝛽𝑡(𝑟)²𝑥−𝑦₀².(2.7) Since 𝜙(𝑦₀,⋅) is convex on 𝐶, it follows that 𝜙𝑦₀≤1,𝑥𝑡(𝑟)𝜙𝑦₀,𝑥_𝑟+𝑡(𝑟)−1𝑡(𝑟)𝜙𝑦₀,𝑦₀,(2.8) which implies that 𝜙(𝑦₀,𝑥)≤−𝛽𝑡(𝑟)‖𝑥−𝑦₀‖²≤−𝛽𝑟. Thus, 𝜙𝑦₀,𝑥⟶−∞as𝑟⟶∞.(2.9) However, since 𝜙(𝑦₀,⋅) is lower semicontinuous on 𝐶, by the well-known Weierstrass Theorem, 𝜙(𝑦₀,⋅) attains its minimum on the compact set 𝑆(𝑦₀;1)∩𝐶. This fact contradicts (2.9).

From this proposition, we can derive the following corollaries.

Corollary 2.5 (see [18]). If the bifunction 𝜙 is strongly monotone on 𝐶, and 𝜙(𝑥,⋅) is convex, lower semicontinuous with respect to the second argument for all 𝑥∈𝐶, then 𝜙 is coercive on C.

Corollary 2.6. Suppose that the bifunction 𝑓 is strongly monotone on 𝐶, and 𝑓(𝑥,⋅) is convex, lower semicontinuous with respect to the second argument for all 𝑥∈𝐶. If the bifunction 𝑔 is coercive on 𝐶 then, for every 𝜖>0, the bifunction 𝑔+𝜖𝑓 is uniformly coercive on 𝐶, for example, there exists a point 𝑦₀∈𝐶 and a compact set 𝐵 both independent of 𝜖 such that 𝑔𝑥,𝑦₀+𝜖𝑓𝑥,𝑦₀<0,∀𝑥∈𝐶⧵𝐵.(2.10)

Proof. From the coercivity of 𝑔, we conclude that there exists a compact 𝐵₁ and 𝑦₀∈𝐶 such that 𝑔(𝑥,𝑦₀)<0forall𝑥∈𝐶⧵𝐵₁. Since 𝑓 is strongly monotone, convex, lower semicontinuous on 𝐶, by choosing 𝑦=𝑦₀, from Proposition 2.4, there exists a compact 𝐵₂ such that 𝑓(𝑥,𝑦₀)<0forall𝑥∈𝐶⧵𝐵₂. Set 𝐵=𝐵₁∪𝐵₂, then 𝐵 is compact and 𝑔(𝑥,𝑦₀)+𝜖𝑓(𝑥,𝑦₀)<0forall𝑥∈𝐶⧵𝐵.

Remark 2.7. It is worth to note that if both 𝑓, 𝑔 are coercive and pseudomonotone on 𝐶, then the function 𝑓+𝑔 is not necessary coercive or pseudomonotone on 𝐶.

To see this, let us consider the following bifunctions.

Example 2.8. Let 𝑓(𝑥,𝑦)∶=(𝑥₁𝑦₂−𝑥₂𝑦₁)𝑒^𝑥1, 𝑔(𝑥,𝑦)∶=(𝑥₂𝑦₁−𝑥₁𝑦₂)𝑒^𝑥2, and 𝐶={(𝑥₁,𝑥₂)∶𝑥₁≥−1,(1/10)(𝑥₁−9)≤𝑥₂≤10𝑥₁+9} then we have(i)𝑓(𝑥,𝑦),𝑔(𝑥,𝑦) are pseudomonotone and coercive on C,(ii)for all𝜖>0 the bifunctions 𝑓_𝜖(𝑥,𝑦)=𝑔(𝑥,𝑦)+𝜖𝑓(𝑥,𝑦) are neither pseudomonotone nor coercive on 𝐶.
Indeed, (i)if 𝑓(𝑥,𝑦)≤0, then 𝑓(𝑦,𝑥)≥0, thus 𝑓 is pseudomonotone on 𝐶. By choosing 𝑦⁰=(𝑦⁰₁,0),(0<𝑦⁰₁≤1) and 𝐵={(𝑥₁,𝑥₂)∶𝑥²₁+𝑥²₂≤𝑟}(𝑟>1), we have 𝑓(𝑥,𝑦⁰)=−𝑥₂𝑦⁰₁𝑒^𝑥1<0forall𝑦∈𝐶⧵𝐵, which means that 𝑓 is coercive on 𝐶. Similarly, we can see that 𝑔 is coercive on 𝐶,(ii)by definition of 𝑓, we have that𝑓_𝜖𝑥,𝑦=𝑥₂𝑦₁−𝑥₁𝑦₂𝑒^𝑥2−𝜖𝑒^𝑥1,∀𝜖>0.(2.11)Take 𝑥(𝑡)=(𝑡,2𝑡),forall𝑦(𝑡)=(2𝑡,𝑡), then 𝑓_𝜖(𝑥(𝑡),𝑦(𝑡))=3𝑡²(𝑒^2𝑡−𝜖𝑒^𝑡)>0, whereas 𝑓_𝜖(𝑦(𝑡),𝑥(𝑡))=−3𝑡²(𝑒^𝑡−𝜖𝑒^2𝑡)>0 for 𝑡 is sufficiently large. So 𝑓_𝜖 is not pseudomonotone on 𝐶.
Now, we show that the bifunction 𝑓_𝜖(𝑥,𝑦)=(𝑥₂𝑦₁−𝑥₁𝑦₂)(𝑒^𝑥2−𝜖𝑒^𝑥1) is not coercive on 𝐶. Suppose, by contradiction, that there exist a compact set 𝐵 and 𝑦⁰=(𝑦⁰₁,𝑦⁰₂)∈𝐵∩𝐶 such that 𝑓_𝜖(𝑥,𝑦⁰)<0forall𝑥∈𝐶⧵𝐵, then, by coercivity of 𝑓_𝜖, it follows, 𝑦⁰₁,𝑦⁰₂>0 and 𝑦⁰₁≠𝑦²₀. With 𝑥(𝑡)=(𝑡,𝑘𝑡),(𝑡>0), we have 𝑓_𝜖(𝑥(𝑡),𝑦⁰)=𝑡(𝑘𝑦⁰₁−𝑦⁰₂)(𝑒^𝑘𝑡−𝜖𝑒^𝑡). However(i)if 𝑦⁰₁>𝑦⁰₂, then from 1<𝑘<10 it follows that 𝑥(𝑡)∈𝐶 and 𝑓_𝜖(𝑥(𝑡),𝑦⁰)>0 for 𝑡 is sufficiently large, which contradicts with coercivity,(ii)if 𝑦⁰₁<𝑦⁰₂, then, by choosing 1/10<𝑘<1, we obtain 𝑥(𝑡)∈𝐶 and 𝑓_𝜖(𝑥(𝑡),𝑦⁰)>0 for 𝑡 is large enough. But this cannot happen because of the coercivity of 𝑓_𝜖.
Now, for each fixed 𝜖>0, we consider the penalized equilibrium problem PEP(𝐶,𝑓_𝜖) defined as find𝑥_𝜖∈𝐶suchthat𝑓_𝜖𝑥_𝜖,𝑦∶=𝑔𝑥_𝜖,𝑦+𝜖𝑓𝑥_𝜖,𝑦≥0,∀𝑦∈𝐶.(2.12) By SOL(𝐶,𝑓_𝜖), we denote the solution set of PEP(𝐶,𝑓_𝜖).

Theorem 2.9. Suppose that the equilibrium bifunctions 𝑓,𝑔 are pseudomonotone, upper semicontinuous with respect to the first argument and lower semicontinuous, convex with respect to the second argument on 𝐶, then any cluster point of the sequence {𝑥_𝑘} with 𝑥_𝑘∈SOL(𝐶,𝑓_𝜖𝑘),𝜖_𝑘→0 is a solution to the original bilevel problem (1.1). In addition, if 𝑓 is strongly monotone and 𝑔 is coercive on 𝐶, then for each 𝜖_𝑘>0 the penalized problem PEP(𝐶,𝑓_𝜖𝑘) is solvable, and any sequence {𝑥_𝑘} with 𝑥_𝑘∈SOL(𝐶,𝑓_𝜖𝑘) converges to the unique solution of the bilevel problem (1.1) as 𝑘→∞.

Proof. Let {𝑥_𝑘} be any sequence with 𝑥_𝑘∈SOL(𝐶,𝑓_𝜖𝑘), and let 𝑥 be any of its cluster points. Without lost of generality, we may assume that 𝑥_𝑘→𝑥as𝑘→∞. Since 𝑥_𝑘∈SOL(𝐶,𝑓_𝜖𝑘), one has 𝑔𝑥_𝑘,𝑦+𝜖_𝑘𝑓𝑥_𝑘,𝑦≥0,∀𝑦∈𝐶.(2.13) For any 𝑧∈𝑆_𝑔, we have 𝑔(𝑧,𝑦)≥0,forall𝑦∈𝐶 and in particular, 𝑔(𝑧,𝑥_𝑘)≥0. Then, by the pseudomonotonicity of 𝑔, we have 𝑔(𝑥_𝑘,𝑧)≤0. Replacing 𝑦 by 𝑧 in (2.13), we obtain 𝑔𝑥_𝑘,𝑧+𝜖_𝑘𝑓𝑥_𝑘,𝑧≥0,(2.14) which implies that 𝜖_𝑘𝑓𝑥_𝑘,𝑧≥−𝑔𝑥_𝑘,𝑧≥0⟹𝑓𝑥_𝑘,𝑧≥0.(2.15) Let 𝑘→∞, by upper semicontinuity of 𝑓, we have 𝑓(𝑥,𝑧)≥0forall𝑧∈𝑆_𝑔.
To complete the proof, we need only to show that 𝑥∈𝑆_𝑔. Indeed, for any 𝑦∈𝐶, we have 𝑔𝑥_𝑘,𝑦+𝜖_𝑘𝑓𝑥_𝑘,𝑦≥0,∀𝑦∈𝐶.(2.16) Again, by upper semicontinuity of 𝑓 and 𝑔, we obtain in the limit, as 𝜖_𝑘→0, that 𝑔(𝑥,𝑦)≥0forall𝑦∈𝐶. Hence, 𝑥∈𝑆_𝑔.
Now suppose, in addition, that 𝑓 is strongly monotone on 𝐶. By Corollary 2.6, 𝑓_𝜖𝑘 is uniformly coercive on 𝐶. Thus, problem PEP(𝐶,𝑓_𝜖𝑘) is solvable and, for all 𝜖_𝑘>0, the solution sets of these problems are contained in a compact set 𝐵. So any infinite sequence {𝑥_𝑘} of the solutions has a cluster point, say, 𝑥. By the first part, 𝑥 is a solution of (1.1). Note that, from the assumption on 𝑔, the solution set 𝑆_𝑔 of the lower equilibrium (EP𝐶,𝑔) is a closed, convex, compact set. Since 𝑓 is lower semicontinuous and convex with respect to the second argument and is strongly monotone on 𝐶, the upper equilibrium problem EP(𝑆_𝑔,𝑓) has a unique solution. Using again the first part of the theorem, we can see that 𝑥_𝑘→𝑥as𝑘→∞

Remark 2.10. In a special case considered in [6], where both 𝑓 and 𝑔 are monotone, the penalized problem (PEP) is monotone too. In this case, (PEP) can be solved by some existing methods (see, e.g., [6, 11–14, 19]) and the references therein. However, when one of these two bifunctions is pseudomonotone, the penalized problem (PEP), in general, does not inherit any monotonicity property from 𝑓 and 𝑔. In this case, problem (PEP) cannot be solved by the above-mentioned existing methods.

3. Gap Function and Descent Direction

A well-known tool for solving equilibrium problem is the gap function. The regularized gap function has been introduced by Taji and Fukushima in [20] for variational inequalities, and extended by Mastroeni in [11] to equilibrium problems. In this section, we use the regularized gap function for the penalized equilibrium problem (PEP). As we have mentioned above, this problem, even when 𝑔 is pseudomonotone and 𝑓 is strongly monotone, is still difficult to solve.

Throughout this section, we suppose that both 𝑓 and 𝑔 are lower semicontinuous, convex on 𝐶 with respect to the second argument. First, we recall (see, e.g., [11]) the definition of a gap function for the equilibrium problem.

Definition 3.1. A function 𝜑∶𝐶→ℝ∪{+∞} is said to be a gap function for (PEP) if(i)𝜑(𝑥)≥0,forall𝑥∈𝐶, (ii)𝜑(𝑥)=0 if and only if 𝑥 is a solution for (PEP).

A gap function for (PEP) is 𝜑(𝑥)=−min_𝑦∈𝐶𝑓_𝜖(𝑥,𝑦). This gap function may not be finite and, in general, is not differentiable. To obtain a finite, differentiable gap function, we use the regularized gap function introduced in [20] and recently used by Mastroeni in [11] to equilibrium problems. From Proposition 2.2 and Theorem 2.1 in [11], the following proposition is immediate.

Proposition 3.2. Suppose that 𝑙∶𝐶×𝐶→ℝ is a nonnegative differentiable, strongly convex bifunction on 𝐶 with respect to the second argument and satisfies(a)𝑙(𝑥,𝑥)=0forall𝑥∈𝐶, (b)∇_𝑦𝑙(𝑥,𝑥)=0forall𝑥∈𝐶.
Then the function 𝜑_𝜖(𝑥)=−min_𝑦∈𝐶𝑔𝑥,𝑦+𝜖𝑓𝑥,𝑦+𝑙𝑥,𝑦(3.1) is a finite gap function for (PEP). In addition, if 𝑓 and 𝑔 are differentiable with respect to the first argument and ∇_𝑥𝑓(𝑥,𝑦),∇_𝑥𝑔(𝑥,𝑦) are continuous on 𝐶, then 𝜑_𝜖(𝑥) is continuously differentiable on 𝐶 and ∇𝜑_𝜖(𝑥)=−∇_𝑥𝑔𝑥,𝑦_𝜖(𝑥)−𝜖∇_𝑥𝑓𝑥,𝑦_𝜖(𝑥)+𝑙𝑥,𝑦_𝜖(𝑥)=−∇_𝑥𝑔_𝜖𝑥,𝑦_𝜖(𝑥),(3.2) where 𝑔_𝜖𝑓,𝑦𝑥,𝑦=𝑔𝑥,𝑦+𝜖𝑥,𝑦+𝑙𝑥,𝑦_𝜖(𝑥)=argmin_𝑦∈𝐶𝑔_𝜖𝑥,𝑦.(3.3)

Note that the function 𝑙(𝑥,𝑦)∶=(1/2)⟨𝑀(𝑦−𝑥),𝑦−𝑥⟩, where 𝑀 is a symmetric positive definite matrix of order 𝑛 that satisfies the assumptions on 𝑙.

We need some definitions on ∇-monotonicity.

Definition 3.3. A differentiable bifunction ℎ∶𝐶×𝐶→ℝ is called as follows:(a)strongly ∇-monotone on 𝐶 if there exists a constant 𝜏>0 such that, ∇_𝑥ℎ𝑥,𝑦+∇_𝑦ℎ𝑥,𝑦,𝑦−𝑥≥𝜏𝑦−𝑥²,∀𝑥,𝑦∈𝐶,(3.4)(b)strictly ∇-monotone on 𝐶 if ∇_𝑥ℎ𝑥,𝑦+∇_𝑦ℎx,𝑦,𝑦−𝑥>0,∀𝑥,𝑦∈𝐶,𝑥≠𝑦,(3.5)(c)∇-monotone on 𝐶 if ⟨∇_𝑥ℎ𝑥,𝑦+∇_𝑦ℎ𝑥,𝑦,𝑦−𝑥⟩≥0,∀𝑥,𝑦∈𝐶,(3.6)(d)strictly pseudo-∇-monotone on 𝐶 if ∇_𝑥ℎ𝑥,𝑦,𝑦−𝑥≤0⟹∇_𝑦ℎ𝑥,𝑦,𝑦−𝑥>0,∀𝑥,𝑦∈𝐶,𝑥≠𝑦,(3.7)(e)pseudo-∇-monotone on 𝐶 if ∇_𝑥ℎ𝑥,𝑦,𝑦−𝑥≤0⟹∇_𝑦ℎ𝑥,𝑦,𝑦−𝑥≥0,∀𝑥,𝑦∈𝐶.(3.8)

Remark 3.4. The definitions (a), (b), and (c) can be found, for example, in [8, 11]. The definitions (d) and (e), to our best knowledge, are not used before. From the definitions, we have (a)⟹(b)⟹(c)⟹(e),(a)⟹(b)⟹(d)⟹(e).(3.9) However, (c) may not imply (d) and vice versa as shown by the following simple examples.

Example 3.5. Consider the bifunction ℎ(𝑥,𝑦)=𝑒^𝑥2(𝑦²−𝑥²) defined on 𝐶×𝐶 with 𝐶=ℝ. This bifunction is not ∇-monotone on 𝐶, because ∇_𝑥ℎ𝑥,𝑦+∇_𝑦ℎ𝑥,𝑦,𝑦−𝑥=2𝑒^𝑥2𝑦−𝑥²𝑥²+𝑥𝑦+1(3.10) is negative for 𝑥=−1,𝑦=3. However, ℎ(𝑥,𝑦) is strictly pseudo- ∇-monotone. Indeed, we have ∇_𝑥ℎ𝑥,𝑦,𝑦−𝑥=2𝑥𝑒^𝑥2𝑦²−𝑥²−1𝑦−𝑥≤0⟺𝑥𝑦²−𝑥²∇−1𝑦−𝑥≤0,_𝑦ℎ𝑥,𝑦,𝑦−𝑥=2𝑦𝑒^𝑥2𝑦−𝑥>0⟺𝑦𝑦−𝑥>0.(3.11) It is not difficult to verify that 𝑥𝑦²−𝑥²−1𝑦−𝑥≤0⟹𝑦𝑦−𝑥>0,as𝑥≠𝑦.(3.12)
Hence this function is strictly pseudo- ∇-monotone but is not ∇-monotone.
Vice versa, considering the bifunction ℎ(𝑥,𝑦)=(𝑦−𝑥)^𝑇𝑀(𝑦−𝑥) defined on ℝ^𝑛×ℝ^𝑛, where 𝑀 is a matrix of order 𝑛×𝑛, we have the following:(i)ℎ is ∇-monotone, because ∇_𝑥ℎ𝑥,𝑦+∇_𝑦ℎ=−𝑥,𝑦,𝑦−𝑥𝑦−𝑥^𝑇𝑀+𝑀^𝑇+𝑦−𝑥^𝑇𝑀+𝑀^𝑇,𝑦−𝑥=0,∀𝑥,𝑦.(3.13) Clearly, ℎ is not strictly-∇-monotone,(ii)ℎ is strictly pseudo ∇-monotone if and only if ∇_𝑥ℎ𝑥,𝑦,𝑦−𝑥=−𝑦−𝑥^𝑇𝑀+𝑀^𝑇,𝑦−𝑥≤0(3.14) implies ∇_𝑦ℎ𝑥,𝑦,𝑦−𝑥=𝑦−𝑥^𝑇𝑀+𝑀^𝑇,𝑦−𝑥⟩>0,∀𝑥,𝑦,𝑥≠𝑦.(3.15) The latter inequality equivalent to 𝑀+𝑀^𝑇 is a positive definite matrix of order 𝑛×𝑛.

Remark 3.6. As shown in [8] when ℎ(𝑥,𝑦)=⟨𝑇(𝑥),𝑦−𝑥⟩ with 𝑇 a differentiable monotone operator on 𝐶, ℎ is monotone on 𝐶 if and only if 𝑇 is monotone on 𝐶, and in this case, monotonicity of ℎ on 𝐶 coincides with ∇-monotonicity of ℎ on 𝐶.

The following example shows that pseudomonotonicity may not imply pseudo-∇-monotonicity.

Example 3.7. Let ℎ(𝑥,𝑦)=−𝑎𝑥(𝑦−𝑥), defined on ℝ₊×ℝ₊, (𝑎>0). It is easy to see that ℎ𝑥,𝑦≥0⟹ℎ𝑦,𝑥≤0,∀𝑥,𝑦≥0.(3.16) Thus, ℎ is pseudomonotone on ℝ₊.
We have ∇_𝑥ℎ𝑥,𝑦,𝑦−𝑥=−𝑎𝑦−𝑥𝑦−2𝑥<0,∀𝑦>2𝑥>0.(3.17)
But ∇_𝑦ℎ𝑥,𝑦,𝑦−𝑥=−𝑎𝑥𝑦−𝑥<0,∀𝑦>2𝑥>0.(3.18)
So ℎ is not pseudo-∇-monotone on ℝ₊.

From the definition of the gap function 𝜑_𝜖, a global minimal point of this function over 𝐶 is a solution to problem (PEP). Since 𝜑_𝜖 is not convex, its global minimum is extremely difficult to compute. In [8], the authors have shown that under the strict ∇-monotonicity a stationary point is also a global minimum of gap function. By a counterexample, the authors in [8] also pointed out that the strict ∇-monotonicity assumption cannot be relaxed to ∇-monotonicity. The following theorem shows that the stationary property is still guaranteed under the strict pseudo-∇-monotonicity.

Theorem 3.8. Suppose that 𝑔_𝜖 is strictly pseudo- ∇-monotone on 𝐶. If 𝑥 is a stationary point of 𝜑_𝜖 over 𝐶, that is, ∇𝜑_𝜖(𝑥),𝑦−𝑥≥0,∀𝑦∈𝐶.(3.19) then 𝑥 solves (PEP).

Proof. Suppose that 𝑥 does not solve (PEP), then 𝑦_𝜖(𝑥)≠𝑥.
Since 𝑥 is a stationary point of 𝜑_𝜖 on 𝐶, from the definition of 𝜑_𝜖, we have ∇𝜑_𝜖(𝑥),𝑦−𝑥=−∇_𝑥𝑔_𝜖𝑥,𝑦_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥≥0.(3.20) By strict pseudo-∇-monotonicity of 𝑔_𝜖, it follows that ∇_𝑦𝑔_𝜖𝑥,𝑦_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥>0.(3.21) On the other hand, since 𝑦_𝜖(𝑥) minimizes 𝑔_𝜖(𝑥,⋅) over 𝐶, we have ∇_𝑦𝑔_𝜖𝑥,𝑦_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥≤0,(3.22) which is in contradiction with (3.21).

To compute a stationary point of a differentiable function over a closed-convex set, we can use the existing descent direction algorithms in mathematical programming (see, e.g., [8, 21]). The next proposition shows that if 𝑦(𝑥) is a solution of the problem min_𝑦∈𝐶𝑔_𝜖(𝑥,𝑦), then 𝑦(𝑥)−𝑥 is a descent direction on 𝐶 of 𝜑_𝜖 at 𝑥. Namely, we have the following proposition.

Proposition 3.9. Suppose that 𝑔_𝜖 is strictly pseudo-∇-monotone on 𝐶 and 𝑥 is not a solution to Problem (PEP), then ∇𝜑_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥<0.(3.23)

Proof. Let 𝑑_𝜖(𝑥)=𝑦_𝜖(𝑥)−𝑥. Since 𝑥 is not a solution to (PEP), then 𝑑_𝜖(𝑥)≠0. Suppose that, by contradiction, 𝑑_𝜖(𝑥) is not a descent direction on 𝐶 of 𝜑_𝜖 at 𝑥, then ∇𝜑_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥≥0⟺−∇_𝑥𝑔_𝜖𝑥,𝑦_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥≥0,(3.24) which, by strict pseudo-∇-monotonicity of 𝑔_𝜖, implies ∇_𝑦𝑔_𝜖𝑥,𝑦_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥>0.(3.25) On the other hand, since 𝑦_𝜖(𝑥) minimizes 𝑔_𝜖(𝑥,⋅) over 𝐶, by the well-known optimality condition, we have ∇_𝑦𝑔_𝜖𝑥,𝑦_𝜖(𝑥),𝑦_𝜖(𝑥)−𝑥≤0,(3.26) which contradicts (3.25).

Proposition 3.10. Suppose that 𝑔(𝑥,⋅) is strictly convex on 𝐶 for every 𝑥∈𝐶 and 𝑔 is strictly pseudo-∇-monotone on 𝐶. If 𝑥∈𝐶 is not a solution of (PEP), then there exists 𝜖>0 such that 𝑦_𝜖(𝑥)−𝑥 is a descent direction of 𝜑_𝜖 on 𝐶 at 𝑥 for all 0<𝜖≤𝜖.

Proof. By contradiction, suppose that the statement of the proposition does not hold, then there exist 𝜖_𝑘↘0 and 𝑥∈𝐶 such that ∇𝜑_𝜖𝑘(𝑥),𝑦_𝜖𝑘(𝑥)−𝑥≥0⟺−∇_𝑥𝑔_𝜖𝑘𝑥,𝑦_𝜖𝑘(𝑥),𝑦_𝜖𝑘(𝑥)−𝑥≥0.(3.27) Since 𝑔_𝜖(𝑥,⋅) is strictly convex differentiable on 𝐶, by Theorem 2.1 in [9], the function 𝜖↦𝑦_𝜖(𝑥) is continuous with respect to 𝜖, thus 𝑦_𝜖𝑘(𝑥) tends to 𝑦₀(𝑥) as 𝜖_𝑘→0, where 𝑦₀(𝑥)=argmin_𝑦∈𝐶𝑔(𝑥,𝑦). Since 𝑔_𝜖𝑘(𝑥,𝑦)=𝑔(𝑥,𝑦)+𝜖_𝑘𝑓(𝑥,𝑦) is continuously differentiable, letting 𝜖_𝑘→0 in (3.27), we obtain −∇_𝑥𝑔𝑥,𝑦₀(𝑥),𝑦₀(𝑥)−𝑥≥0.(3.28) By strict pseudo-∇-monotonicity of 𝑔, it follows that ∇_𝑦𝑔𝑥,𝑦₀(𝑥),𝑦₀(𝑥)−𝑥>0.(3.29) On the other hand, since 𝑦_𝜖𝑘(𝑥) minimizes 𝑔_𝜖𝑘(𝑥,⋅) over 𝐶, we have ∇_𝑦𝑔_𝜖𝑘𝑥,𝑦_𝜖𝑘(𝑥),𝑦_𝜖𝑘(𝑥)−𝑥≤0.(3.30) Taking the limit, we obtain ∇_𝑦𝑔𝑥,𝑦₀(𝑥),𝑦₀(𝑥)−𝑥≤0,(3.31) which contradicts (3.29).

To illustrate Theorem 3.8, let us consider the following examples.

Example 3.11. Consider the bifunctions 𝑔(𝑥,𝑦)=𝑒^𝑥2(𝑦²−𝑥²) and 𝑓(𝑥,𝑦)=10^𝑥2(𝑦²−𝑥²) defined on ℝ×ℝ. It is not hard to verify that,(i)𝑔(𝑥,𝑦),𝑓(𝑥,𝑦) are monotone, strictly pseudo-∇-monotone on ℝ,(ii)for all 𝜖>0 the bifunction 𝑔(𝑥,𝑦)+𝜖𝑓(𝑥,𝑦) is monotone and strictly pseudo-∇-monotone on ℝ and satisfying all of the assumptions of Theorem 3.8.

Example 3.12. Let 𝑓(𝑥,𝑦)=−𝑥²−𝑥𝑦+2𝑦² and 𝑔(𝑥,𝑦)=−3𝑥²𝑦+𝑥𝑦²+2𝑦³ defined on ℝ₊×ℝ₊ it is easy to see that,(i)𝑔,𝑓 are pseudomonotone, strictly ∇-monotone on ℝ₊,(ii)for all 𝜖>0 the bifunction 𝑔(𝑥,𝑦)+𝜖𝑓(𝑥,𝑦) is pseudomonotone and strictly ∇-monotosne on ℝ₊ and satisfying all of the assumptions of Theorem 3.8.

4. Application to the Tikhonov Regularization Method

The Tikhonov method [22] is commonly used for handling ill-posed problems. Recently, in [23] the Tikhonov method has been extended to the pseudomonotone equilibrium problem Find𝑥^∗∈𝐶suchthat𝑔𝑥^∗,𝑦≥0,∀𝑦∈𝐶,(EP𝐶,𝑔) where, as before, 𝐶 is a closed-convex set in ℝ^𝑛 and 𝑔∶𝐶→ℝ is a pseudomonotone bifunction satisfying 𝑔(𝑥,𝑥)=0 for every 𝑥∈𝐶.

In the Tikhonov regularization method considered in [23], problem (EP𝐶,𝑔) is regularized by the problems find𝑥^∗∈𝐶suchthat𝑔_𝜖𝑥^∗,𝑦∶=𝑔𝑥^∗,𝑦+𝜖𝑓𝑥^∗,𝑦≥0,∀𝑦∈𝐶,(EP𝐶,𝑔_𝜖) where 𝑓 is an equilibrium bifunction on 𝐶 and 𝜖>0 and play the role of the regularization bifunction and regularization parameter, respectively.

In [23], the following theorem has been proved.

Theorem 4.1. Suppose that 𝑓(⋅,𝑦),𝑔(⋅,𝑦) are upper semicontinuous and 𝑓(𝑥,⋅),𝑔(𝑥,⋅) are lower semicontinuous convex on 𝐶 for each 𝑥,𝑦∈𝐶 and that 𝑔 is pseudomonotone on 𝐶. Suppose further that 𝑓 is strongly monotone on 𝐶 satisfying the condition ∃𝛿>0:𝑓𝑥,𝑦≤𝛿𝑥−𝑥^𝑔𝑦−𝑥,∀𝑥,𝑦∈𝐶,(4.1) where 𝑥^𝑔∈𝐶 (plays the role of a guess solution) is given.
Then the following three statements are equivalent:(a)the solution set of (EP𝐶,𝑔_𝜖) is nonempty for each 𝜀>0 and lim_𝜀→0⁺𝑥(𝜀) exists, where 𝑥(𝜀) is arbitrarily chosen in the solution set of (EP𝐶,𝑔_𝜖),(b)the solution set of (EP𝐶,𝑔_𝜖) is nonempty for each 𝜀>0 and lim_𝜀→0⁺sup‖𝑥(𝜀)‖<∞, where 𝑥(𝜀) is arbitrarily chosen in the solution set of (EP𝐶,𝑔_𝜖),(c)the solution set of (EP𝐶,𝑔) is nonempty.
Moreover, if any one of these statements holds, then lim_𝜀→0⁺𝑥(𝜀) is equal to the unique solution of the strongly monotone equilibrium problem EP(𝑆_𝑔,𝑓), where 𝑆_𝑔 denotes the solution set of the original problem (EP𝐶,𝑔).

Note that, when 𝑔 is monotone on 𝐶, the regularized subproblems are strongly monotone and therefore, they can be solved by some existing methods. When 𝑔 is pseudomonotone, the subproblems, in general, are no longer strongly monotone, even not pseudomonotone. So solving them becomes a difficult task. However, the problem of finding the limit point of the sequences of iterates leads to the unique solution of problem EP(𝑆_𝑔,𝑓).

In order to apply the penalty and gap function methods described in the preceding sections, let us take, for instant, 𝑓𝑥,𝑦=𝑥−𝑥^𝑔,𝑦−𝑥.(4.2) Clearly, 𝑓 is both strongly monotone and strongly ∇-monotone with the same modulus 1. Moreover, 𝑓 satisfies the condition (4.1). Therefore, the problem of finding the limit point in the above Tikhonov regularization method can be formulated as the bilevel equilibrium problemfind𝑥∈𝑆_𝑔suchthat𝑓𝑥^∗,𝑦≥0,∀𝑦∈𝑆_𝑔,(4.3) which is of the form (1.1). Now, for each fixed 𝜖_𝑘>0, we consider the penalized equilibrium problem PEP(𝐶,𝑓_𝜖𝑘) defined asfind𝑥_𝑘∈𝐶suchthat𝑓_𝜖𝑘𝑥_𝑘,𝑦∶=𝑔𝑥_𝑘,𝑦+𝜖_𝑘𝑓𝑥_𝑘,𝑦≥0,∀𝑦∈𝐶.(4.4) As before, by SOL(𝐶,𝑓_𝜖𝑘), we denote the solution set of PEP(𝐶,𝑓_𝜖𝑘).

Applying Theorems 2.9 and 3.8, we obtain the following result.

Theorem 4.2. Suppose that the bifunction 𝑔 satisfies the following conditions:(i)𝑔(𝑥,⋅) is convex, lower semicontinuous for all 𝑥∈𝐶,(ii)𝑔 is pseudomonotone and coercive on 𝐶. Then for any 𝜖_𝑘>0, the penalized problem PEP(𝐶,𝑓_𝜖𝑘) is solvable, and any sequence {𝑥_𝑘} with 𝑥_𝑘∈SOL(𝐶,𝑓_𝜖𝑘) for all 𝑘 converges to the unique solution of the problem (4.3) as 𝑘→∞. (iii)In addition, if𝑔(𝑥,𝑦)+𝜖_𝑘𝑓(𝑥,𝑦) is strictly pseudo- ∇-monotone on𝐶 (in particular,𝑔(𝑥,𝑦) is∇-monotone), and𝑥_𝑘 is any stationary point of the mathematical programmin_𝑥∈𝐶𝜑_𝑘(𝑥) with 𝜑_𝑘(𝑥)∶=min_𝑦∈𝐶𝑔𝑥,𝑦+𝜖_𝑘𝑓𝑥,𝑦,(4.5) then{𝑥_𝑘} converges to the unique solution of the problem (4.3) as𝑘→∞.

5. Conclusion

We have considered a class of bilevel pseudomonotone equilibrium problems. The main difficulty of this problem is that its feasible domain is not given explicitly as in a standard mathematical programming problem. We have proposed a penalty function method to convert the bilevel problem into one-level ones. Then we have applied the regularized gap function method to solve the penalized equilibrium subproblems. We have generalized the pseudo-∇-monotonicity concept from ∇-monotonicity. Under the pseudo-∇-monotonicity property, we have proved that any stationary point of the gap function is a solution to the original bilevel problem. As an application, we have shown how to apply the proposed method to the Tikhonov regularization method for pseudomonotone equilibrium problems.

Acknowledgment

This work is supported by the National Foundation for Science Technology Development of Vietnam (NAFOSTED).