# An interior approximal method for solving pseudomonotone equilibrium problems

- Pham N Anh
^{1}Email author, - Pham M Tuan
^{2}and - Le B Long
^{1}

**2013**:156

https://doi.org/10.1186/1029-242X-2013-156

© Anh et al.; licensee Springer. 2013

**Received: **10 July 2012

**Accepted: **10 March 2013

**Published: **5 April 2013

## Abstract

In this paper, we present an interior approximal method for solving equilibrium problems for pseudomonotone bifunctions without Lipschitz-type continuity on polyhedra. The method can be viewed as combining a special interior proximal function, which replaces the usual quadratic function, Armijo-type linesearch techniques and the cutting hyperplane methods. Convergence properties of the method are established, among them the global convergences are proved under few assumptions. Finally, we present some preliminary computational results to Cournot-Nash oligopolistic market equilibrium models.

**MSC:**65K10, 90C25.

## Keywords

## 1 Introduction

*C*be a nonempty closed convex subset of ${\mathcal{R}}^{n}$ and a bifunction $f:C\times C\to \mathcal{R}$ satisfying $f(x,x)=0$ for all $x\in C$. We consider equilibrium problems in the sense of Blum and Oettli [1] (shortly $EP(f,C)$), which are to find ${x}^{\ast}\in C$ such that

In this article, for solving Problem $EP(f,C)$, we assume that the bifunction *f* and *C* satisfy the following conditions:

A_{1}. $C=\{x\in {R}^{n}:Ax\le b\}$, where *A* is a $p\times n$ maximal matrix ($rankA=n$), $b\in {\mathcal{R}}^{p}$, and $intC=\{x:Ax<b\}$ is nonempty.

A_{2}. For each $x\in C$, the function $f(x,\cdot )$ is convex and subdifferentiable on *C*.

_{3}.

*f*is pseudomonotone on $C\times C$,

*i.e.*, for each $x,y\in C$, it holds

A_{4}. *f* is continuous on $C\times C$.

A_{5}. $Sol(f,C)\ne \mathrm{\varnothing}$.

Equilibrium problems appear in many practical problems arising, for instance, physics, engineering, game theory, transportation, economics, and network (see [2–5]). In recent years, both theory and applications became attractive for many researchers (see [1, 6–14]).

*C*. Under the assumptions that

*f*is strongly monotone with constant $\beta >0$ on $C\times C$,

*i.e.*,

*f*is Lipschitz-type continuous with constants ${c}_{1}>0$, ${c}_{2}>0$,

*i.e.*,

the author showed that the sequence $\{{x}^{k}\}$ globally converges to a solution of Problem $EP(f,C)$. However, the convergence depends on three positive parameters ${c}_{1}$, ${c}_{2}$, and *β* and in some cases, they are unknown or difficult to approximate.

*C*, or onto some related set, in order to iteratively reach a solution. In particular, Korpelevich [16] proposed an algorithm for solving the variational inequalities. In each iteration of the algorithm, in order to get the next iterate ${x}^{k+1}$, two orthogonal projections onto

*C*are calculated, according to the following iterative step. Given the current iterate ${x}^{k}$, calculate

where $\lambda >0$ is some positive number. Recently, Tran *et al.* [17] extended these projection techniques to Problem $EP(f,C)$ involving monotone equilibrium bifunctions but it must satisfy a certain Lipschitz-type continuous condition. To avoid this requirement, they proposed linesearch procedures commonly used in variational inequalities to obtain projection-type algorithms for solving equilibrium problems.

*d*replaces the function

*g*in (1.1):

with $\mu \in (0,1)$ and $y\in {\mathcal{R}}_{+}^{n}=\{{({x}_{1},\dots ,{x}_{n})}^{T}\in {\mathcal{R}}^{n}:{x}_{i}>0\phantom{\rule{0.25em}{0ex}}\mathrm{\forall}i=1,\dots ,n\}$. Then the interior proximal linesearch extragradient methods can be viewed as combining the function *d* and Armijo-type linesearch techniques. Convergence of the iterative sequence is established under the weaker assumptions that *f* is pseudomonotone on $C\times C$. However, at each iteration *k* in the Armijo-type linesearch progress of the algorithm requires the computation of a subgradient of the bifunction $\partial f({x}^{k},\cdot )({y}^{k})$, which is not easy in some cases. Moreover, most of current algorithms for solving Problem $EP(f,C)$ are based on Lipschitz-type continuous assumptions or the computation of subgradients of the bifunction *f* (see [21–25]).

Our main purpose of this paper is to give an iterative algorithm for solving a pseudomonotone equilibrium problem without Lipschitz-type continuity of the bifunction and the computation of subgradients. To summarize our approach, first we use an interior proximal function *d* as in [22], which replaces the usual quadratic function in auxiliary problems. Next, we construct an appropriate hyperplane and a convex set, which separate the current iterative point from the solution set and we also combine this technique with the Armijo-type linesearch technique. Then the next iteration is obtained as the projection of the current iterate onto the intersection of the feasible set with the convex set and the half-space containing the solution set.

The paper is organized as follows. In Section 2, we recall the auxiliary problem principle of Problem $EP(f,C)$ and propose a new iterative algorithm. Section 3 is devoted to the proof of its global convergence and also show the relation between the solution set of $EP(f,C)$ and the cluster point of the iterative sequences in the algorithm. In Section 4, we apply our algorithm for solving generalized variational inequalities. Applications to the Nash-Cournot oligopolistic market equilibrium model and the numerical results are reported in the last section.

## 2 Proposed algorithm

*d*is defined by (1.2). Then the gradient ${\mathrm{\nabla}}_{1}D(x,y)$ of $D(\cdot ,y)$ at

*x*for every $y\in C$ is defined by

where ${X}_{y}=diag({l}_{1}(y),\dots ,{l}_{p}(y))$ and $log\frac{l(x)}{l(y)}=(log\frac{{l}_{1}(x)}{{l}_{1}(y)},\dots ,log\frac{{l}_{p}(x)}{{l}_{p}(y)})$.

*D*:

for some positive constants *β*. It is easy to see that with $f(x,y)=\u3008F(x),y-x\u3009$, where $F:C\to {\mathcal{R}}^{n}$, and $D(x,y)=\frac{1}{2}{\parallel x-y\parallel}^{2}$, computing ${y}^{k}$ becomes Step 1 of the extragradient method proposed in [16]. In Lemma 3.2(i), we will show that if $\parallel {y}^{k}-{x}^{k}\parallel =0$ then ${x}^{k}$ is a solution to Problem $EP(f,C)$. Otherwise, a computationally inexpensive Armijo-type procedure is used to find a point ${z}^{k}$ such that the convex set ${C}_{k}:=\{x\in {\mathcal{R}}^{n}:f({z}^{k},x)\le 0\}$ and the hyperplane ${H}_{k}:=\{x\in {\mathcal{R}}^{n}:\u3008x-{x}^{k},{x}^{0}-{x}^{k}\u3009\le 0\}$ contain the solution set $Sol(f,C)$ and strictly separates ${x}^{k}$ from the solution. Then we compute the next iterate ${x}^{k+1}$ by projecting ${x}^{0}$ onto the intersection of the feasible set *C* with ${C}_{k}$ and the half-space ${H}_{k}$. The algorithm is described in more detail as follows.

**Algorithm 2.1** Choose ${x}^{0}\in C$, $0<\sigma <\frac{\beta}{2{\parallel {\overline{A}}^{-1}\parallel}^{2}}$ and $\gamma \in (0,1)$.

Step 1.

Increase *k* by 1, and return to Step 1.

## 3 Convergence results

In the next lemma, we show the existence of the nonnegative integer ${m}_{k}$ in Algorithm 2.1.

**Lemma 3.1** *For* $\gamma \in (0,1)$, $0<\sigma <\frac{\beta}{2{\parallel {\overline{A}}^{-1}\parallel}^{2}}$, *if* $\parallel r({x}^{k})\parallel >0$ *then there exists the smallest nonnegative integer* ${m}_{k}$ *which satisfies* (2.3).

*Proof*Assume on the contrary, (2.3) is not satisfied for any nonnegative integer

*i*,

*i.e.*,

*f*, we have

Hence, it must be either $r({x}^{k})=0$ or $\sigma \ge \frac{\beta}{2{\parallel {\overline{A}}^{-1}\parallel}^{2}}$. The first case contradicts to $r({x}^{k})\ne 0$, while the second one contradicts to the fact $\sigma <\frac{\beta}{2{\parallel {\overline{A}}^{-1}\parallel}^{2}}$. The proof is completed. □

Let us discuss the global convergence of Algorithm 2.1.

**Lemma 3.2**

*Let*$\{{x}^{k}\}$

*be the sequence generated by Algorithm*2.1

*and*$Sol(f,C)\ne \mathrm{\varnothing}$.

*Then the following hold*.

- (i)
*If*$r({x}^{k})=0$,*then*${x}^{k}\in Sol(f,C)$. - (ii).${x}^{k}\notin {C}_{k}$
- (iii).$Sol(f,C)\subseteq C\cap {C}_{k}\cap {H}_{k}$
- (iv).${lim}_{k\to \mathrm{\infty}}\parallel {x}^{k+1}-{x}^{k}\parallel =0$

*Proof*(i) Since ${y}^{k}$ is the solution to problem (2.2) and an optimization result in convex programming, we have

- (ii)Since ${z}^{k}={x}^{k}-{\gamma}^{{m}_{k}}r({x}^{k})$, $r({x}^{k})>0$, $f(x,x)=0$ for every $x\in C$ and $f(x,\cdot )$ is convex on
*C*, we have$\begin{array}{rcl}0& =& f({z}^{k},{z}^{k})\\ =& f({z}^{k},(1-{\gamma}^{{m}_{k}}){x}^{k}+{\gamma}^{{m}_{k}}{y}^{k})\\ \le & (1-{\gamma}^{{m}_{k}})f({z}^{k},{x}^{k})+{\gamma}^{{m}_{k}}f({z}^{k},{y}^{k})\\ \le & (1-{\gamma}^{{m}_{k}})f({z}^{k},{x}^{k})-{\gamma}^{{m}_{k}}\sigma {\parallel r\left({x}^{k}\right)\parallel}^{2}\\ <& (1-{\gamma}^{{m}_{k}})f({z}^{k},{x}^{k}).\end{array}$

- (iii)For ${x}^{\ast}\in Sol(f,C)$. Then since
*f*is pseudomonotone on*C*and $f({x}^{\ast},{z}^{k})\ge 0$, we have $f({z}^{k},{x}^{\ast})\le 0$. So ${x}^{\ast}\in {C}_{k}$. To prove $Sol(f,C)\subseteq {H}_{k}$, we will use mathematical induction. Indeed, for $k=0$ we have ${H}_{0}={\mathcal{R}}^{n}$. This holds. Suppose that$Sol(f,C)\subseteq {H}_{m}\phantom{\rule{1em}{0ex}}\text{for}k=m\ge 0.$

- (iv)Since ${x}^{k}$ is the projection of ${x}^{0}$ onto $C\cap {C}_{k-1}\cap {H}_{k-1}$ and (iii), by the definition of projection, we have$\parallel {x}^{k}-{x}^{0}\parallel \le \parallel {x}^{\ast}-{x}^{0}\parallel \phantom{\rule{1em}{0ex}}\mathrm{\forall}{x}^{\ast}\in Sol(f,C).$

□

**Theorem 3.3**

*Suppose that assumptions*A

_{1}

*to*A

_{5}

*hold*, $\partial f(x,\cdot )(x)$

*is upper semicontinuous on*

*C*,

*and the sequence*$\{{x}^{k}\}$

*is generated by Algorithm*2.1.

*Then*$\{{x}^{k}\}$

*globally converges to a solution*${x}^{\ast}$

*of Problem*$EP(f,C)$,

*where*

*Proof*For each ${\overline{w}}^{k}\in \partial f({z}^{k},\cdot )({z}^{k})$, set

*C*and $\{{x}^{k}\}$ is bounded, there exists $M>0$ such that

The cases remaining to consider are the following.

Case 1. ${lim\hspace{0.17em}sup}_{k\to \mathrm{\infty}}{\gamma}^{{m}_{k}}>0$.

This case must follow that ${lim\hspace{0.17em}inf}_{k\to \mathrm{\infty}}\parallel r({x}^{k})\parallel =0$. Since $\{{x}^{k}\}$ is bounded, there exists an accumulation point $\overline{x}$ of $\{{x}^{k}\}$. In other words, a subsequence $\{{x}^{{k}_{i}}\}$ converges to some $\overline{x}$ such that $r(\overline{x})=0$, as $i\to \mathrm{\infty}$. Then we see from Lemma 3.2(i) that $\overline{x}\in Sol(f,C)$.

Case 2. ${lim}_{k\to \mathrm{\infty}}{\gamma}^{{m}_{k}}=0$.

*f*and

*f*, we have

*f*is continuous, passing onto the limit, as $j\to \mathrm{\infty}$, we obtain

which implies $r(\overline{x})=0$, and hence $\overline{x}\in Sol(f,C)$. So, all cluster points of $\{{x}^{k}\}$ belong to the solution set $Sol(f,C)$.

The last inequality is due to $\stackrel{\u02c6}{x}={Pr}_{Sol(f,C)}({x}^{0})$. So, ${x}^{\ast}=\stackrel{\u02c6}{x}$ and the sequence $\{{x}^{k}\}$ has an unique cluster point ${Pr}_{Sol(f,C)}({x}^{0})$. □

Now we consider the relation between the solution existence of Problem $EP(f,C)$ and the convergence of $\{{x}^{k}\}$ generated by Algorithm 2.1.

**Lemma 3.4** (see [4])

*Suppose that* *C* *is a compact convex subset of* ${\mathcal{R}}^{n}$ *and* *f* *is continuous on* *C*. *Then the solution set of Problem* $EP(f,C)$ *is nonempty*.

**Theorem 3.5**

*Suppose that assumptions*A

_{1}

*to*A

_{4}

*hold*,

*f*

*is continuous*, $\partial f(x,\cdot )(x)$

*is upper semicontinuous on*

*C*,

*the sequence*$\{{x}^{k}\}$

*is generated by Algorithm*2.1,

*and*$Sol(f,C)=\mathrm{\varnothing}$.

*Then we have*

*Consequently*, *the solution set of Problem* $EP(f,C)$ *is empty if and only if the sequence* $\{{x}^{k}\}$ *diverges to infinity*.

*Proof*The first, we show that $C\cap {C}_{k}\cap {H}_{k}\ne \mathrm{\varnothing}$ for every $k\ge 0$. On the contrary, suppose that there exists ${k}_{0}\ge 1$ such that

*M*such that

- (a)
The set $\{{\overline{x}}^{k}\}$ has at least ${k}_{0}+1$ elements.

- (b), ${C}_{k}={\overline{C}}_{k}$ and ${H}_{k}={\overline{H}}_{k}$ for every $k=0,1,\dots ,{k}_{0}$.${x}^{k}={\overline{x}}^{k}$
- (c)is not a solution to Problem $EP(f,\overline{C})$.${x}^{{k}_{0}}$

A similar discussion as above leads to the conclusion that the sequence $\{{x}^{k}\}$ converges to ${Pr}_{Sol(f,C)}({x}^{0})$, which contradicts the emptiness of the solution set $Sol(f,C)$. The theorem is proved. □

## 4 Applications to Cournot-Nash equilibrium model

*n*-firms producing a common homogenous commodity and that the price ${p}_{i}$ of firm

*i*depends on the total quantity ${\sigma}_{x}={\sum}_{i=1}^{n}{x}_{i}$ of the commodity. Let ${h}_{i}({x}_{i})$ denote the cost of the firm

*i*when its production level is ${x}_{i}$. Suppose that the profit of firm

*i*is given by

*i*that is assumed to be dependent only on its production level. There is a common strategy space $C\subseteq {\mathcal{R}}^{n}$ for all firms. Each firm seeks to maximize its own profit by choosing the corresponding production level under the presumption that the production of the other firms are parametric input. In this context, a Nash equilibrium is a production pattern in which in which no firm can increase its profit by changing its controlled variables. Thus, under this equilibrium concept, each firm determines its best response given other firms’ actions. Mathematically, a point ${x}^{\ast}={({x}_{1}^{\ast},\dots ,{x}_{n}^{\ast})}^{T}\in C$ is said to be a Nash equilibrium point if ${x}^{\ast}$ is a solution of the problem:

Then the problem of finding an equilibrium point of this model can be formulated as Problem $EP(f,C)$. It follows from Lemma 3.2 (i) that ${x}^{k}$ is a solution of Problem $EP(f,C)$ if and only if $r({x}^{k})=0$. Thus, ${x}^{k}$ is an *ϵ*-solution to Problem $EP(f,C)$, if $\parallel r({x}^{k})\parallel \le \u03f5$. To illustrate our algorithm, we consider two academic numerical tests of the bifunction *f* in ${\mathcal{R}}^{5}$.

**Example 4.1**We consider an application of Cournot-Nash oligopolistic market equilibrium model taken from [17]. The equilibrium bifunction is defined by

*f*is pseudomonotone on

*C*and the interior approximal function (2.1) is defined through

**Example 4.1: Iterations of Algorithm 2.1 with**
$\mathbf{\parallel}\mathit{r}\mathbf{(}{\mathit{x}}^{\mathit{k}}\mathbf{)}\mathbf{\parallel}\mathbf{\le}\mathbf{0.0001}$

Iteration (k) | ${\mathit{x}}_{\mathbf{1}}^{\mathit{k}}$ | ${\mathit{x}}_{\mathbf{2}}^{\mathit{k}}$ | ${\mathit{x}}_{\mathbf{3}}^{\mathit{k}}$ | ${\mathit{x}}_{\mathbf{4}}^{\mathit{k}}$ | ${\mathit{x}}_{\mathbf{5}}^{\mathit{k}}$ |
---|---|---|---|---|---|

0 | 1 | 3 | 1 | 1 | 1 |

10 | 4.3842 | 0.0001 | 3.4633 | 1.7683 | 1.3842 |

20 | 4.4657 | 0.0013 | 3.1371 | 1.9315 | 1.4657 |

50 | 4.4901 | 0.0017 | 3.0404 | 1.9796 | 1.4898 |

100 | 4.4976 | 0.0082 | 3.0117 | 1.9938 | 1.4969 |

150 | 4.5001 | 0.0099 | 3.0031 | 1.9979 | 1.4990 |

180 | 4.5012 | 0.0107 | 3.0005 | 1.9989 | 1.4995 |

200 | 4.5106 | 0.0142 | 2.9716 | 2.0071 | 1.4965 |

250 | 4.5309 | 0.0412 | 2.9176 | 2.0206 | 1.4897 |

300 | 4.5370 | 0.0494 | 2.9013 | 2.0247 | 1.4877 |

350 | 4.5389 | 0.0519 | 2.8963 | 2.0259 | 1.4870 |

355 | 4.5395 | 0.0526 | 2.8948 | 2.0263 | 1.4868 |

358 | 4.5396 | 0.0528 | 2.8943 | 2.0264 | 1.4868 |

360 | 4.5397 | 0.0529 | 2.8942 | 2.0264 | 1.4868 |

361 | 4.5397 | 0.0529 | 2.8942 | 2.0265 | 1.4868 |

**Example 4.2**

where $arctan(x-y)={(arctan({x}_{1}-{y}_{1}),\dots ,arctan({x}_{5}-{y}_{5}))}^{T}$, the components of *d* are chosen randomly in $(0,10)$.

*f*satisfies convergent assumptions of Theorem 3.3 in this paper and Theorem 3.1 in [21]. We choose the parameters in Algorithm 2.1: $\parallel {\overline{A}}^{-1}\parallel =1$, $\beta =5$, $\sigma =1.2$, $\gamma =0.5$, $\mu =0.2$. In the algorithm (shortly (

*IPLE*)) proposed by Nguyen

*et al.*[21], the parameters are chosen as follows: $\theta =0.5$, $\tau =0.7$, $\alpha =0.4$, $\mu =2$, ${c}_{k}=0.5+\frac{1}{k+1}$ for all $k\ge 1$. We compare Algorithm 2.1 with (

*IPLE*). The iteration numbers and the computational time for 5 problems are given in Table 2.

**Example 4.2: The tolerance**
$\mathbf{\parallel}\mathit{r}\mathbf{(}{\mathit{x}}^{\mathit{k}}\mathbf{)}\mathbf{\parallel}\mathbf{\le}\mathbf{0.19}$

Algorithm 2.1 | Algorithm (IPLE) | ||||
---|---|---|---|---|---|

Problem | No. iterations | CPU times (seconds) | Problem | No. iterations | CPU time (seconds) |

No. 1 | 547 | 16.5140 | No. 1 | 421 | 15.1388 |

No. 2 | 379 | 8.0249 | No. 2 | 372 | 14.4729 |

No. 3 | 1,045 | 21.2740 | No. 3 | 1,026 | 23.9581 |

No. 4 | 781 | 15.7751 | No. 4 | 1,216 | 32.3385 |

No. 5 | 625 | 14.1925 | No. 5 | 297 | 9.4139 |

The computations are performed by Matlab R2008a running on a PC Desktop Intel(R) Core(TM)i5 650@3.2 GHz 3.33 GHz 4 Gb RAM.

## 5 Conclusion

This paper presented an iterative algorithm for solving pseudomonotone equilibrium problems without Lipschitz-type continuity of the bifunctions. Combining the interior proximal extragradient method in [22], the Armijo-type linesearch and cutting hyperplane techniques, the global convergence properties of the algorithm are established under few assumptions. Compared with the current methods such as the interior proximal extragradient method, the dual extragradient algorithm in [14], the auxiliary principle in [15], the inexact subgradient method in [29], and other methods in [4], the fundamental difference here is that our algorithm does not require the computation of subgradient of a convex function. We show that the cluster point of the sequence in our algorithm is the projection of the starting point onto the solution set of the equilibrium problems. Moreover, we also give the relation between the existence of solutions of equilibrium problems and the convergence of the iteration sequence.

## Declarations

### Acknowledgements

We are very grateful to the anonymous referees for their really helpful and constructive comments in improving the paper. The work was supported by National Foundation for Science and Technology Development of Vietnam (NAFOSTED), code 101.02-2011.07.

## Authors’ Affiliations

## References

- Blum E, Oettli W: From optimization and variational inequality to equilibrium problems.
*Math. Stud.*1994, 63: 127–149.MathSciNetGoogle Scholar - Bigi G, Castellani M, Pappalardo M: A new solution method for equilibrium problems.
*Optim. Methods Softw.*2009, 24: 895–911. 10.1080/10556780902855620MathSciNetView ArticleGoogle Scholar - Daniele P, Giannessi F, Maugeri A:
*Equilibrium Problems and Variational Models*. Kluwer Academic, Dordrecht; 2003.View ArticleGoogle Scholar - Konnov IV:
*Combined Relaxation Methods for Variational Inequalities*. Springer, Berlin; 2000.Google Scholar - Moudafi A: Proximal point algorithm extended to equilibrium problem.
*J. Nat. Geom.*1999, 15: 91–100.MathSciNetGoogle Scholar - Anh PN: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities.
*J. Optim. Theory Appl.*2012. doi:10.1007/s10957–012–0005-xGoogle Scholar - Anh PN: A hybrid extragradient method extended to fixed point problems and equilibrium problems.
*Optimization*2012. doi:10.1080/02331934.2011.607497Google Scholar - Anh PN, Kim JK: Outer approximation algorithms for pseudomonotone equilibrium problems.
*Comput. Math. Appl.*2011, 61: 2588–2595. 10.1016/j.camwa.2011.02.052MathSciNetView ArticleGoogle Scholar - Anh PN, Muu LD, Nguyen VH, Strodiot JJ: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities.
*J. Optim. Theory Appl.*2005, 124: 285–306. 10.1007/s10957-004-0926-0MathSciNetView ArticleGoogle Scholar - Iusem AN, Sosa W: On the proximal point method for equilibrium problems in Hilbert spaces.
*Optimization*2010, 59: 1259–1274. 10.1080/02331931003603133MathSciNetView ArticleGoogle Scholar - Zeng LC, Yao JC: Modified combined relaxation method for general monotone equilibrium problems in Hilbert spaces.
*J. Optim. Theory Appl.*2006, 131: 469–483. 10.1007/s10957-006-9162-0MathSciNetView ArticleGoogle Scholar - Heusinger A, Kanzow C: Relaxation methods for generalized Nash equilibrium problems with inexact line search.
*J. Optim. Theory Appl.*2009, 143: 159–183. 10.1007/s10957-009-9553-0MathSciNetView ArticleGoogle Scholar - Konnov IV: Combined relaxation methods for monotone equilibrium problems.
*J. Optim. Theory Appl.*2001, 111: 327–340. 10.1023/A:1011930301552MathSciNetView ArticleGoogle Scholar - Quoc TD, Anh PN, Muu LD: Dual extragradient algorithms to equilibrium problems.
*J. Glob. Optim.*2012, 52: 139–159. 10.1007/s10898-011-9693-2MathSciNetView ArticleGoogle Scholar - Mastroeni G: On auxiliary principle for equilibrium problems. Nonconvex Optimization and Its Applications 68. In
*Equilibrium Problems and Variational Models*. Edited by: Daniele P, Giannessi F, Maugeri A. Kluwer Academic, Dordrecht; 2003:289–298.View ArticleGoogle Scholar - Korpelevich GM: The extragradient method for finding saddle points and other problems.
*Matecon*1976, 12: 747–756.Google Scholar - Tran DQ, Dung ML, Nguyen VH: Extragradient algorithms extended to equilibrium problems.
*Optimization*2008, 57: 749–776. 10.1080/02331930601122876MathSciNetView ArticleGoogle Scholar - Anh PN: A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems.
*Acta Math. Vietnam.*2009, 34: 183–200.MathSciNetGoogle Scholar - Bnouhachem A: An LQP method for pseudomonotone variational inequalities.
*J. Glob. Optim.*2006, 36: 351–356. 10.1007/s10898-006-9013-4MathSciNetView ArticleGoogle Scholar - Forsgren A, Gill PE, Wright MH: Interior methods for nonlinear optimization.
*SIAM Rev.*2002, 44: 525–597. 10.1137/S0036144502414942MathSciNetView ArticleGoogle Scholar - Nguyen TTV, Strodiot JJ, Nguyen VH: The interior proximal extragradient method for solving equilibrium problems.
*J. Glob. Optim.*2009, 44: 175–192. 10.1007/s10898-008-9311-0MathSciNetView ArticleGoogle Scholar - Anh PN: An LQP regularization method for equilibrium problems on polyhedral.
*Vietnam J. Math.*2008, 36: 209–228.MathSciNetGoogle Scholar - Auslender A, Teboulle M, Bentiba S: A logarithmic-quadratic proximal method for variational inequalities.
*Comput. Optim. Appl.*1999, 12: 31–40. 10.1023/A:1008607511915MathSciNetView ArticleGoogle Scholar - Auslender A, Teboulle M, Bentiba S: Iterior proximal and multiplier methods based on second order homogeneous kernels.
*Math. Oper. Res.*1999, 24: 646–668.MathSciNetView ArticleGoogle Scholar - Bigi G, Passacantando M: Gap functions and penalization for solving equilibrium problems with nonlinear constraints.
*Comput. Optim. Appl.*2012, 53: 323–346. 10.1007/s10589-012-9481-zMathSciNetView ArticleGoogle Scholar - Marcotte P: Algorithms for the network oligopoly problem.
*J. Oper. Res. Soc.*1987, 38: 1051–1065.View ArticleGoogle Scholar - Mordukhovich BS, Outrata JV, Cervinka M: Equilibrium problems with complementarity constraints: case study with applications to oligopolistic markets.
*Optimization*2007, 56: 479–494. 10.1080/02331930701421079MathSciNetView ArticleGoogle Scholar - Murphy FH, Sherali HD, Soyster AL: A mathematical programming approach for determining oligopolistic market equilibrium.
*Math. Program.*1982, 24: 92–106. 10.1007/BF01585096MathSciNetView ArticleGoogle Scholar - Santos P, Scheimberg S: An inexact subgradient algorithm for equilibrium problems.
*Comput. Appl. Math.*2011, 30: 91–107.MathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.