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A cutting hyperplane method for solving pseudomonotone non-Lipschitzian equilibrium problems
Journal of Inequalities and Applications volume 2012, Article number: 288 (2012)
We present a new method for solving equilibrium problems, where the underlying function is continuous and satisfies a pseudomonotone assumption. First, we construct an appropriate hyperplane which separates the current iterative point from the solution set. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the half-space containing the solution set. We also analyze the global convergence of the method under minimal assumptions.
MSC: 65K10, 90C25.
The typical form of equilibrium problems is formulated by the Ky Fan inequality as follows (see ):
where C is a nonempty closed convex subset of and is a bifunction such that for all , shortly . In this paper, we suppose that is convex on C for all , f is continuous on and the solution set S of is nonempty.
Although has a simple formulation, it includes many important problems in applied mathematics such as variational inequalities, complementarity problems, (vector) optimization problems, fixed point problems and saddle point problems (see [2–4]). In recent years, equilibrium problems have become an attractive field for many researchers in both theory and applications (see [5–11]). There is a myriad of literature related to equilibrium problems and their applications in electricity market, transportation, economics and network [12, 13].
Theory of equilibrium problems has been studied extensively and intensively in terms of the existence of solutions and generalizations to many abstract ways. However, methods for solving are still limited and have not satisfied the need of applications. There are popular approaches to solving to our knowledge. The first approach is based on the gap function (see ), the second way is to use the proximal point method  and the third one is the auxiliary subproblem principle . Recently, basing on the fixed point property that is a solution to if and only if it is the unique solution of the problem
where , and Armijo linesearch techniques, Tran et al. in  introduced extragradient algorithms for solving equilibrium problems and obtained the convergence under the assumption that the bifunction f is pseudomonotone as the following:
By replacing a quadratic term in the subproblem (1.1) by the Bregman distance function, Nguyen et al. in  proposed the interior proximal extragradient method for solving , where f is pseudomonotone and C only is a polyhedron convex set. The method has also been extensively studied to solve and variational inequalities (see [15–17]).
A special case of Problem is the variational inequality problem, shortly , which is to find a point such that
where C is a nonempty closed convex subset of and . A typical method to solve Problem is the projection method, which is based on the property that x is a solution to Problem if and only if it coincides with zeros of the projected residual function , where is the metric projection on C. Solodov and Svaiter in  proposed a projection method which starts with a point and generates a sequence defined, for all , , , by
Under pseudomonotone and continuous assumptions of F, the authors showed that the sequences globally converge to a solution of the variational inequality problem . Note that if , then is a solution to the problem.
In this paper, by combining the extragradient methods in  and Armijo-type linesearch techniques in (1.3), we propose a new method for solving Problem , which is called to be cutting hyperplane method. First, we construct an appropriate hyperplane which separates the current iterative point from the solution set. Next, we combine this technique with Armijo-type linesearch techniques to obtain a convergent iteration scheme for pseudomonotone equilibrium problems. Then, the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the half-space containing the solution set. Compared with the extragradient method in  and the current methods, our iteration method is quite simple. The fundamental difference here is that the global convergence of the method only requires the continuity and pseudomonotonicity of the bifunction f. Moreover, we also show that the cluster point of the sequence in our scheme is the limit of the projection of the iteration point onto the solution set of Problem .
The rest of the paper is organized as follows. In Section 2, we give formal definitions of our target and the pseudomonotonicity of f. We then propose the cutting hyperplane method. Section 3 is devoted to the proof of its global convergence to a solution of . In the last section, we apply the method for oligopolistic equilibrium market models with concave cost functions and a generalized form of the bifunction defined by the Cournot-Nash equilibrium model considered in [13, 19–21].
2 Proposed method
Definition 2.1 A bifunction f is said to be
(a) strongly monotone on C if there exists a constant such that
(b) monotone on C if
(c) pseudomonotone on C if
It is observed that (a) ⇒ (b) ⇒ (c).
If f is a mapping defined by
where is a multivalued mapping such that for all , then can be formulated as the multivalued variational inequality (shortly, MVI):
Find , such that
In this case, it is known that solutions coincide with zeros of the following projected residual function:
where and . In other words, with , , the point is a solution of (MVI) if and only if , where (see ). Applying this idea to the equilibrium problems , we obtain the following solution scheme.
Let be a current approximation to the solution of . First, we compute
for some positive constant β (as Step 1 of Algorithm 2 in ). Set . It is easy to see that if , then is a solution to Problem . Otherwise, we search the line segment between and for a point such that the hyperplane
strictly separates from the solution set S of . To find such , we may use a computationally inexpensive Armijo-type procedure in . We find the smallest nonnegative number such that
We set and choose . Then we compute the next iterate by projecting onto the intersection of the feasible set C with the half-space
This means that
Instead of (2.2), Tran et al. in  used a linesearch technique as follows:
where , and is a strongly convex (with modulus ) and continuously differentiable function. It is clear to see that (2.2) is different and simpler than the technique (3.1). Both of them are Armijo-type linesearch techniques, so a small part of the proof of the following lemma is close to the proof of Lemma 4.2 in .
Lemma 3.1 If for , then there exists the smallest nonnegative integer such that
Proof For and , we suppose to obtain a contradiction that for every nonnegative integer m, we have
Passing to the limit in the above inequality, as , by continuity of , we obtain
On the other hand, since is a solution to the convex optimization problem
With , the last inequality implies
Combining (3.2) with (3.3), we obtain
Hence, it must be either or . The first case contradicts , while the second one contradicts the fact that . □
Lemma 3.2 (see )
Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Suppose that, for all , the sequence satisfies
Then the sequence converges strongly to some .
Let us discuss the global convergence of Scheme (2.1)-(2.3).
Lemma 3.3 Let be the sequence generated by Scheme (2.1)-(2.3). Then the following hold:
(i) If , then .
(ii) If , then .
(iii) If , then .
(iv) , where .
(ii) From , we have
Combining this inequality with (2.2) and , we obtain
This implies that . It means that .
(iii) Suppose . Then for all , and since f is pseudomonotone on C, we get
From , we have
From this inequality and (3.4), it follows that
(iv) We know that
Since and , for every , there exists such that
where . In particular, for , we easily deduce that the corresponding and thus that . Therefore, for every , we have
because . Also, we have
Since , using the Pythagorean theorem, we can reduce that
From (3.5) and (3.6), we have
Using Lemma 3.3, we can prove the global convergence of Scheme (2.1)-(2.3) under moderate assumptions.
Theorem 3.4 Let f be pseudomonotone and be bounded on C. Then
where , and the sequence generated by Scheme (2.1)-(2.3) converges to a solution of .
Proof We first show that the sequence is bounded. Since , we have
Substituting , we have
which implies that
Hence, we have
From , it follows that
Replacing y by and combining with assumptions and , we have
Combining this inequality with (2.2) and the assumption , we obtain
Hence, (3.8) reduces to
Combining (3.7) with (3.9), we obtain
This implies that the sequence is nonincreasing and hence convergent. So, there exists a subsequence which converges to . We consider the function , where is the indicator function on C. Then g is the strongly convex function on C and hence ∂g is strongly monotone with a constant . By the definition of a strongly monotone mapping, we have
Since and , we choose and . So,
By the assumption that is upper semicontinuous on C and converges to , the sequence is bounded. Combining this and (3.11), the sequence is also bounded. Therefore, the sequences and are bounded. We suppose that
This together with (3.10) implies
Since is convergent, it is easy to see that
The cases remaining to consider are the following.
Case 1. . This case must follow that . In other words, the subsequence converges to and converges to . Then we have . Then we see from Lemma 3.3 that , and besides we can take , in particular in (3.12). Thus is a convergent sequence. Since is an accumulation point of , the sequence converges to zero, i.e., converges to .
Case 2. . Since is the smallest nonnegative integer, does not satisfy (2.2). Hence, we have
Passing to the limit in (3.13) as , and using the continuity of f, , , we have
where . From Scheme (2.1)-(2.3), we have
Since f is continuous, passing to the limit as , we obtain
Combining this with (3.14), we have
which implies , and hence . Letting and repeating the previous arguments, we conclude that the whole sequence converges to . This completes the proof. □
Corollary 3.5 Under assumptions of Theorem 3.4, the sequence converges to , where
Proof It is well known that f is pseudomonotone, so S is convex. By Theorem 3.4, the sequence converges to a solution . Set . By the definition of , we have
It follows from Theorem 3.4 that
Then, by Lemma 3.2, we have
Passing the limit in (3.15) and combining this with (3.16), we have
This means that and
4 Illustrative examples and numerical results
As an example for equilibrium problems , we consider the Cournot-Nash oligopolistic market equilibrium model (see [21, 25, 26]). In this model, it is assumed that there are n-firms producing a common homogenous commodity and that the price of the firm i depends on the total quantity of the commodity. Let denote the cost of the firm i when its production level is . Suppose that the profit of the firm i is given by
where is the cost function of the firm i that is assumed to be dependent only on its production level.
Let () denote the strategy set of the firm i. Each firm seeks to maximize its own profit by choosing the corresponding production level under the presumption that the production of other firms is parametric input. In this context, a Nash equilibrium is a production pattern 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 is said to be a Nash equilibrium point if
When is affine, this market problem can be formulated as a special Nash equilibrium problem in the n-person noncooperative game theory.
Then it has been proved in  that the problem of finding an equilibrium point of this model can be formulated as the following equilibrium problem in the sense of Blum and Oettli:
In classical Cournot-Nash models (see ), the price and cost functions for each firm are assumed to be affine of the following forms:
Combining this with (4.1), (4.2), (4.3) and (4.4), we obtain that
It follows from Definition 2.1 that the following result holds.
Proposition 4.1 If the parametric μ satisfies for all , then the function f defined by (4.5) is pseudomonotone on C and it can be not monotone on C.
Now we consider a generalized form of the bifunction defined by the above Cournot-Nash equilibrium model. Let C be a polyhedral convex set given by
where , . The equilibrium bifunction is of the form
where , is convex for each fixed and continuous on C. The function defined by (4.6) also is a generalized form of the bifunction defined by the Cournot-Nash equilibrium model considered in . By using Definition 2.1, it is easy to have the following property of f.
Proposition 4.2 If there exists a bifunction which satisfies for all , then the function f defined by (4.6) is pseudomonotone and it can be not monotone on .
To illustrate our scheme, we consider two academic numerical tests of the function .
Case 1. , , where
In this case, the bifunction f defined in (4.6) is pseudomonotone, continuous and differentiable on C. The subproblem needed to solve at Step 1 is of the strongly convex quadratic programming
In Step 2, is defined by the form
Thus, and the sequence is uniform bounded. Note that is the unique solution to
Subproblems (4.7) and (4.8) can then be solved efficiently, for example, by the Matlab optimization toolbox. Lemma 3.3 shows that if , then is a solution to problems . So, we can say that is an ϵ-solution to problems if we have with . Taking , , and , we obtained the iterates in Table 1.
The approximate solution obtained after 14 iterations is
In Table 2, we compare Scheme (2.1)-(2.3) with Algorithm 2a in . Now, there have been some changes in this case: , and the first component of vector q are chosen randomly in , and the first component of vector d is chosen randomly in . In both cases, we use Algorithm 2a with the same equilibrium bifunction, the quadratic regularization function and parameters , , .
Case 2. We use Scheme (2.1)-(2.3) with the same equilibrium problems and dates as in Case 1. Unless the bifunction , where D is defined by the components of the , is , is chosen by . This example is given by Bnouhachem (see ). Under these assumptions, it can be proved that f is continuous and pseudomonotone on C.
We also see that in Step 1 the solution can be written by
and in Step 2,
Thus, the sequence is uniform bounded. There have been some changes in this case: , where , but d is defined, the components are chosen randomly in . Choosing , , and and comparing Scheme (2.1)-(2.3) and Algorithm 2.1 in , we obtained the computation presented in Table 3.
We perform Scheme (2.1)-(2.3) and Algorithm 2a  in Matlab R2008a running on a PC Desktop Intel(R) Core(TM)2 Duo CPU T5750@ 2.00GHz 1.32 GB, 2Gb RAM.
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We are very grateful to the anonymous referees for their really helpful and constructive comments on improving the paper. This work was completed while the first author was studying at the Kyungnam University for the NRF Postdoctoral Fellowship for Foreign Researchers. And the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).
The authors declare that they have no competing interests.
The main idea of this paper was proposed by PNA. PNA, JKK and NDH prepared the manuscript initially and performed all the steps of the proof in this research. All authors read and approved the final manuscript.
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Anh, P.N., Kim, J.K. & Hien, N.D. A cutting hyperplane method for solving pseudomonotone non-Lipschitzian equilibrium problems. J Inequal Appl 2012, 288 (2012). https://doi.org/10.1186/1029-242X-2012-288
- pseudomonotone equilibrium problems
- cutting hyperplane method