An interior approximal method for solving pseudomonotone equilibrium problems
© Anh et al.; licensee Springer. 2013
Received: 10 July 2012
Accepted: 10 March 2013
Published: 5 April 2013
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.
In this article, for solving Problem , we assume that the bifunction f and C satisfy the following conditions:
A1. , where A is a maximal matrix (), , and is nonempty.
A2. For each , the function is convex and subdifferentiable on C.
A4. f is continuous on .
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]).
the author showed that the sequence globally converges to a solution of Problem . However, the convergence depends on three positive parameters , , and β and in some cases, they are unknown or difficult to approximate.
where is some positive number. Recently, Tran et al.  extended these projection techniques to Problem 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.
with and . 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 . However, at each iteration k in the Armijo-type linesearch progress of the algorithm requires the computation of a subgradient of the bifunction , which is not easy in some cases. Moreover, most of current algorithms for solving Problem 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 , 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 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 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
where and .
for some positive constants β. It is easy to see that with , where , and , computing becomes Step 1 of the extragradient method proposed in . In Lemma 3.2(i), we will show that if then is a solution to Problem . Otherwise, a computationally inexpensive Armijo-type procedure is used to find a point such that the convex set and the hyperplane contain the solution set and strictly separates from the solution. Then we compute the next iterate by projecting onto the intersection of the feasible set C with and the half-space . The algorithm is described in more detail as follows.
Algorithm 2.1 Choose , and .
Increase k by 1, and return to Step 1.
3 Convergence results
In the next lemma, we show the existence of the nonnegative integer in Algorithm 2.1.
Lemma 3.1 For , , if then there exists the smallest nonnegative integer which satisfies (2.3).
Hence, it must be either or . The first case contradicts to , while the second one contradicts to the fact . The proof is completed. □
Let us discuss the global convergence of Algorithm 2.1.
If , then .
- (ii)Since , , for every and is convex on C, we have
- (iii)For . Then since f is pseudomonotone on C and , we have . So . To prove , we will use mathematical induction. Indeed, for we have . This holds. Suppose that
- (iv)Since is the projection of onto and (iii), by the definition of projection, we have
The cases remaining to consider are the following.
Case 1. .
This case must follow that . Since is bounded, there exists an accumulation point of . In other words, a subsequence converges to some such that , as . Then we see from Lemma 3.2(i) that .
Case 2. .
which implies , and hence . So, all cluster points of belong to the solution set .
The last inequality is due to . So, and the sequence has an unique cluster point . □
Now we consider the relation between the solution existence of Problem and the convergence of generated by Algorithm 2.1.
Lemma 3.4 (see )
Suppose that C is a compact convex subset of and f is continuous on C. Then the solution set of Problem is nonempty.
Consequently, the solution set of Problem is empty if and only if the sequence diverges to infinity.
The set has at least elements.
- (b), and for every .
- (c)is not a solution to Problem .
A similar discussion as above leads to the conclusion that the sequence converges to , which contradicts the emptiness of the solution set . The theorem is proved. □
4 Applications to Cournot-Nash equilibrium model
Then the problem of finding an equilibrium point of this model can be formulated as Problem . It follows from Lemma 3.2 (i) that is a solution of Problem if and only if . Thus, is an ϵ-solution to Problem , if . To illustrate our algorithm, we consider two academic numerical tests of the bifunction f in .
Example 4.1: Iterations of Algorithm 2.1 with
where , the components of d are chosen randomly in .
Example 4.2: The tolerance
CPU times (seconds)
CPU time (seconds)
The computations are performed by Matlab R2008a running on a PC Desktop Intel(R) Core(TM)i5 firstname.lastname@example.org GHz 3.33 GHz 4 Gb RAM.
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 , 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 , the auxiliary principle in , the inexact subgradient method in , and other methods in , 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.
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.
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