An inexact subgradient algorithm for Equilibrium Problems

An inexact subgradient algorithm for Equilibrium Problems

Paulo SantosI, ^*; Susana Scheimberg^II

^IDM, UFPI, Teresina, Brazil.

^IIPESC/COPPE, IM, UFRJ, Rio de Janeiro, Brazil E-mails: [e-mail] / [e-mail]

ABSTRACT

We present an inexact subgradient projection type method for solving a nonsmooth Equilibrium Problem in a finite-dimensional space. The proposed algorithm has a low computational cost per iteration. Some numerical results are reported.

Mathematical subject classification: Primary: 90C33; Secondary: 49J52.

Key words: Equilibrium problem, Projection method, Subgradient method.

1 Introduction

Let C be a nonempty closed convex subset of and let be a bifunction such that f (x, x) = 0 for all x ∈ C and C x C is contained in the domain of f. We consider the following Equilibrium Problem:

The solution set of this problem (1) is denoted by S( f, C).

This formulation gives a unified framework for several problems in the sense that it includes, as particular cases, optimization problems, Nash equilibria problems, complementarity problems, fixed point problems, variational inequalities and vector minimization problems (see for instance [4]).

In this work we assume that the function f (x, •): —> (—∞, +∞] is convex and subdifferentiable at x, for all x ∈ C (see [8, 13, 15, 18]). In [8] the subdifferential of this function is called diagonal subdifferential. We define by diagonal subgradients the elements of this set.

The aim of this paper is to develop and to analyze an inexact projected diagonal subgradient method using a divergent series steplength rule. The algorithm is easy to implement and it has a low computational cost since only one inexact projection is done per iteration.

Recently, many algorithms have been developed for solving problem (1) combining diagonal subgradients with projections, see for instance, [6, 7, 15, 18, 19, 21] and references therein.

The paper is organized as follows: In Section 2 we recall useful basic notions. In Section 3 we define the algorithm and study its convergence. In Section 4, we report some computational experiments. In Section 5, we give some concluding remarks.

2 Preliminaries

In this section we present some basic concepts, properties, and notations that we will use in the sequel. Let Rⁿ be endowed with the Euclidean inner product {•, •) and the associated norm || • ||.

Definition 2.1. Let ξ > 0 and x ∈ . A point p_x ∈ C is called a ξ-projection of x onto C, if p_x is a ξ-solution of the problem

that is

where P_C(x) is the orthogonal projection of x onto C.

It is easy to show that the ξ -projection of x onto C is characterized by

Through this paper we will consider the following enlargement of the diagonal subdifferential.

Definition 2.2. The ∈-diagonal subdifferential ∂ ∈₂ f (x, x) of a bifunction f at x ∈ C, is given by

Let us note that the 0-diagonal subdifferential is the diagonal subdifferential ∂₂f (x, x), studied in [8].

The following well-known property will be useful in this paper.

Lemma 2.3.Let [vk} and {δ_k} be nonnegative sequences of real numbers satisfying Then the sequence (v_k} is convergent.

The next technical result will be used in the convergence analysis.

Lemma 2.4.Let θ, β and ξ be nonnegative real numbers satisfying θ² — β θ - ξ < 0, then,

Proof. Consider the quadratic function s( θ) = θ² — β θ — ξ, then s( θ) < 0 implies that

since θ > 0.

Multiplying the last inequality by β and using the property we obtain

The proof is complete.

In the convergence analysis we will assume that the solution set of (1) is contained in the solution set of its dual problem, which is given by

The solution set of this problem is denoted by S_d (f, C).

When f is a pseudomonotone bifunction on C (if x, y ∈ C and f (x, y) > 0, then f (y, x) < 0), it holds that S( f, C) ⊆ S_d(f, C). Moreover, this inclusion is also valid for monotone bifunctions (f (x, y) + f (y, x) < 0).

Now, we are in position to define our algorithm.

3 The algorithm and its convergence analysis

Take a positive parameter p and real sequences {p_k},{β_k }, { ∈_k} and {ξ_k} verifying the following conditions:

3.1 The Inexact Projected Subgradient Method (IPSM)

Step 0: Choose x⁰∈ C. Set k = 0.

Step 1: Let x^k ∈ C. Obtain g^k ∈ ∂ ∈^k ₂f (x^k, x^k). Define

Step 2: Compute x^k+¹∈ C such that:

Notice that the point x^k+¹ is a ξ_k-projection of (x^k — α_kg^k) onto C. In particular, if ξk = 0, then x^k+¹ = P_C (x^k — α_kg^k).

We also observe that the steplength rule (9) is similar with those given in [1] and [2]. In fact, in [1] is taking y_k = max{β_k, ||g^k||} with Σ β_k = +∞, while in [2] is considered p_k = 1 for all k ∈ N.

In the exact version of IPSM is considered ∈_k = ξ_k = 0 for all k ∈ N and the following stopping criteria are included: g^k = 0 (at step 1) and x^k = x^k+¹ (at step 2).

3.2 Convergence analysis

The first result concerns the exact version of the algorithm.

Proposition 3.1. If the exact version of Algorithm IPSM generates a finite sequence, then the last point is a solution of problem EP (f, C).

Proof. Since ∈_k = 0, we have that g^k ∈ ∂₂ f (x^k, x^k). If the algorithm stops at step 1 we have g^k = 0. So, our conclusion follows from (3).

Now, assume that the algorithm finishes at step 2, that is, x^k = x^k+¹. Suppose, for the sake of contradiction, that x^k ∉ S( f, C). Then, there exists x ∈ C such that f (x^k, x) < 0. Using again (3) we get

On the other hand, by replacing x^k+1 by x^k in (10) and taking in account that ξk = 0, it results

Therefore, from (11) and (12) we get a contradiction because α_k > 0. Hence, x^k ∈ S( f, C).

From now on, we assume that the algorithm IPSM generates an infinite sequence denoted by {x^k}.

We derive the following auxiliary property.

Lemma 3.2. For each k, the following inequalities hold

Proof. (i) From (9) it follows

(ii) By taking x = x^k in (10) it results

where the Cauchy-Schwarz inequality is used in the second inequality and the last follows from (13).

Therefore, the desired result is obtained from Lemma 2.4 with θ = ||x^k+¹ — x^k||, β= β_k and ξ = ξ_k, for each k ∈ N.

The next requirement will be used in the subsequent discussions.

A1. The solution set S( f, C) is nonempty;

Notice that this is a common assumption for EP (f, C) (see for example, [11, 13, 15, 18] and references therein). Regarding the existence of solutions for equilibrium problems we refer to [9, 12, 20] and references therein.

Proposition 3.3.Assume that A1 is verified. Then, for every x* ∈ S( f,C) and for each k, the folloing assertion holds

where δ_k = 2α_k∈_k+ 2β_k² + 4ξ.

Proof. By a simple algebraic manipulation we have that

By combining (16) and (10) with x = x * it follows

By applying the Cauchy-Schwarz inequality and Lemma 3.2 (i), it yields

In virtue of (18) and Lemma 3.2 (ii) it results

On the other hand, from the fact that g^k ∈∂₂^∈k f (x^k, x^k), we have that (g^k, x * — x^k) < f (x^k, x *) + ∈_k. Therefore, since α_k > 0 we obtain

The conclusion follows from (19) and (20).

The following requirement will be used to obtain the boundedness of the sequence {x^k} generated by IPSM.

We point out that this assumption is weaker than the pseudomonotonicity condition. In fact, consider the following example.

Example 3.4. Let EP (f, C) be defined by

Observe that S( f, C) = [0} and f (y, x*) = f (y, 0) = 0 for all y ∈ C. Hence, A2 holds. However, the bifunction fis not pseudomonotone on C. In fact, we have f (—0.5, 0.5) = f (0.5, —0.5) = 0.25 > 0.

Notice that f (x, •) is convex for all x ∈ C and is diagonal subdifferentiable with ∂2 f (x, x) = [2|x |x — x², 2|x|x + x²].

Furthermore, this example gives a negative answer to the conjecture given in [9], namely, if C is a nonempty, convex and closed set such that f(x, x) = 0, f (x, •) : C —> R is convex and lower semi-continuous, f (•, y) : C —> R is upper semi-continuous for all x ∈ C and the primal and dual equilibrium problems have the same nonempty solution set, then f is pseudomonotone.

We observe that in [12] an example which disproves the conjecture using a pseudoconvex function f (x, •) instead of a convex function is given.

Theorem 3.5. Assume that A1 and A2 are verified. Then,

Proof. (i) Let x* ∈ S( f, C) and k ∈ N. By A2 we have f (x^k, x*) < 0 which together with Proposition 3.3 implies

where δk = 2α_k ∈_k + 2β_k² + 4ξ_k.

Therefore, in virtue of (7), (8) and (9) we obtain

Hence, from (21), (22) and Lemma 2.3 it results that {||x^k — x*||²} is a convergent sequence.

(ii) The conclusion follows from part (i).

Now , we establish two different hypotheses on the data to obtain an asymptotic behavior of the sequence {x^k} .

A3. The ∈-diagonal subdifferential is bounded on bounded subsets of C.

A3'. The sequence {g^k} is bounded.

Let us note that, condition A3 has been considered in [14] in the setting of optimization problems. Also, a similar condition has been assumed in [10] for equilibrium problems (condition (A)). We observe that A3 and A3 hold under the conditions that there is a nonempty, open and convex set U containing C such that f is finite and continuous on U x U, f (x, x) = 0 and f (x, •) : C —> R is convex for all x ∈ C ([10], Proposition 4.3). Condition A3 has been required in [7] and [15] for equilibrium problems. This condition has also been assumed in [16] and [17] for saddle point problems.

Observe that Example 3.4 satisfies both assumptions.

Theorem 3.6. Suppose that A1 and A2 are verified. Then, under A3 or A3' it holds

Proof. Let x^* ∈ S( f, C). By Proposition 3.3 and 2 it results

Hence,

As m — +oo we have

which together ith (22) yields

On the other hand, by A3' or A3 we have that {||g^k||} is bounded. In fact, by Theorem 3.5 we get that {x^k} is bounded. Therefore, the assertion follows from A3. In consequence, using (6) and (9) we conclude that there exists L > p such that || g^k || < L for all k ∈ N. Therefore

Therefore

Consequently, by (26) and (27) e have

Then, the conclusion follows from (28) and (7).

In order to obtain the convergence of the whole sequence we introduce two additional assumptions.

A5. f (•, y) is upper semicontinuous for all y ∈ C.

Assumption A4 holds, for example, when the problem EP (f, C) corresponds to an optimization problem, or when it is a reformulation of the variational inequality problem with a paramonotone operator. Moreover, the requirement A4 can be considered as an extension of the cut property given in [5] from variational inequality problems to equilibrium problems. Assumption A4 can be recovered if we assume A2 and the following condition holds

which is considered, for instance, in [3].

We also note that Assumption A5 is a common requirement for EP (f, C) (see, for example, [11, 18] and references therein).

Example 3.7. We consider the equilibrium problem defined by C = (— ∞, 0] and f (x, y) = x²(|y| — |x|). Let us observe that A1-A5 hold. In fact, x* = 0 is the unique solution of EP(f, C), f (y, x*) = —|y|y² < 0 for all y ∈ C, f (x, y) is continuous and implies that , that is,

Theorem 3.8. Assume that A1, A2, A3 or A3', A4 and A5 are satisfied. Then, the whole sequence {x^k} converges to a solution of EP (f, C).

Proof. Let x* ∈ S( f, C). By Theorem 3.6, there exists a subsequence {x^kj} of {x^k} such that

In view of Theorem 3.5, we have that {x^kj} is bounded. So, there is x ∈ C and a subsequence of {x^kj}, without lost of generality, namely {x^kj}, such that

Combining assumption A5 together with Theorem 3.6 it follows

From assumption A2 we have f (x, x*) < 0, so, it results

Therefore, A4 implies that . Using again Theorem 3.5 we obtain that the sequence is convergent, which together with (30) it yields

Notice that Theorem 3.8 remains valid if we replace assumptions A2, A4 and A5 by the t-strongly pseudomonotone condition on f with respect to x* ∈ S( f, C), that is,

This condition is weaker than the strong monotonicity of f which has been assumed in [15] for solving equilibrium problems.

4 Numerical results

In this section we illustrate the algorithm IPSM. Some comparisons are also reported. In the two first examples we compare the iterates of IPSM with such one obtained by the Relaxation Algorithm given in [6], where a constrained optimization problem and a line search have been solved at each iteration. Example 4.3 shows the computational time of IPSM versus our implementation of the Extragradient method given in [21], where a constrained optimization problem, a line search and a projection, have been performed at each iteration. Also, we present a nonsmooth example verifying our assumptions. We take ξ_k = ∈ _k = 0, for all k ∈ N, in order to compare the performance of the algorithms.

The algorithm has been coded in MATLAB 7.8 on a 2GB RAM Pentium Dual Core.

Example 4.1. Consider the River Basin Pollution Problem given in [6] which consists of three players with payoff functions:

where u = (0.01, 0.05, 0.01) and v = (2.90, 2.88, 2.85), and the constraints are given by

We take y_k = max{3, ||g^k||},

Table 1 gives the results obtained by IPSM algorithm and by the Relaxation Algorithm (RA) used in [6].

Table 1

Example 4.2. Consider the Cournot oligopoly problem with shared constraints and nonlinear cost functions as described in [6]. The bifunction is defined by

where, n = 1.1, c = (10, 8,..., 2), K = (5, 5,..., 5), β = (1.2, 1.1,..., 0.8) and For this problem, we consider

In Table 2, we show the first three components of each iterate for sake of comparison of IPSM with the relaxation algorithm RA given in [6].

Again, despite the algorithms RA and IPSM obtain similar results at iteration 20, the computational effort is different.

Example 4.3. Consider two equilibrium problems given in [21], where

and the bifunction is of the form

The matrices P, Q and the vector q are defined by

where the i-th problem considers P = P_i, i = 1, 2.

For the first problem, we take Like in [21], we use tol = 10^-3 and x₀= (1, 3, 1, 1, 2).

In Table 3, we compare IPSM with two Extragradient Algorithms (EA) given in [21].

In the second problem, we use

Like in [21], we take tol = 10^—3 and x₀= (1, 3, 1, 1, 2).

In Table 4, we compare IPSM with algorithm EA given in [21].

Example 4.4. Consider the nonsmooth equilibrium problem defined by the bifunction f (x, y) = |y₁| — |x₁| + y²₂— and the constraint set C = {x ∈ R²₊ :x₁ + x₂ = 1}. The optimal point is x* = and the partial subdifferential ofthe equilibrium bifunction fis given by

We use yk = max{l, ||g^k||} and ||x — x*|| < 10^-4 as stop criteria. In Table 5, we show our results by considering different initial points.

5 Concluding remarks

In this paper we have presented a subgradient-type method, denoted by IPSM, for solving equilibrium problems and established its convergence under mild assumptions.

Numerical results were reported for test problems given in the literature of computational methods for solving nonsmooth equilibrium problems. The comparison with other two schemes has shown a satisfactory behavior of the algorithm IPSM in terms of the computational time.

Acknowledgements. We would like to thank two anonymous referees whose comments and suggestions greatly improved this work.

Received: 15/08/10.

Accepted: 05/01/11.

#CAM-317/10.

• [1] A. Auslender and M. Teboulle, Projected subgradient methods with non-Euclidean distances for non-differentiable convex minimization and variational inequalities Mathematical Programming, 120 (2009), 27-48.
• [2] J.Y. Bello Cruz and A.N. Iusem, Convergence of direct methods for paramonotone variational inequalities Computational Optimization and Applications, 46 (2010), 247-263.
• [3] M. Bianchi, G. Kassay and R. Pini, Existence of equilibria via Ekeland'sprinciple. Journal of Mathematical Analysis and Applications, 305 (2005), 502-512.
• [4] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems. Math. Student, 63 (1994), 123-145.
• [5] J.P. Crouzeix, P. Marcotte and D. Zhu, Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities Mathematical Programming, Ser. A 88 (2000), 521-539.
• [6] A. Heusinger and C. Kanzow, Relaxation Methods for Generalized Nash Equilibrium Problems with Inexact Line Search. Journal of Optimization Theory and Applications, 143 (2009), 159-183.
• [7] H. Iiduka and I. Yamada, A Subgradient-type method for the equilibrium problem over the fixed point set and its applications Optimization, 58 (2009), 251-261.
• [8] A.N. Iusem, On the Maximal Monotonicity of Diagonal Subdifferential Operators. Journal of Convex Analysis, 18 (2011), final page numbers not yet available.
• [9] A.N. Iusem and W. Sosa, New existence results for equilibrium problems. Nonlinear Analysis, 52 (2003), 621-635.
• [10] A.N. Iusem and . Sosa, Iterative algorithms for equilibrium problems. Optimization, 52 (2003), 301-316.
• [11] AN. Iusem and W. Sosa, On the proximal point method for equilibrium problems in Hilbert spaces. Optimization, 59 (2010), 1259-1274.
• [12] A.N. Iusem, G. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems. Mathematical Programming, 116 (2009), 621-635.
• [13] I.V. Konnov, Application of the proximal point method to nonmonotone equilibrium problems. Journal of Optimization Theory and Applications, 119 (2003), 317-333.
• [14] P.-E. Maingé, Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization. Set-Valued Analysis, 16 (2008), 899-912.
• [15] L.D. Muu and T.D. Quoc, Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model. Journal of Optimization Theory and Applications, 142 (2009), 185-204.
• [16] A. Nedic and A. Ozdaglar, Subgradient Methods for Saddle-Point Problems. Journal of Optimization Theory and Applications, 142 (2009), 205-228.
• [17] Y. Nesterov, Primal-dual Subgradient Methods for convex problems. Mathematical Programming, 120 (2009), 221-259.
• [18] T.T. Nguyen, J.J. Strodiot and V.H. Nguyen, The interior proximal extragradient method for solving equilibrium problems. Journal of Global Optimization, 44 (2009), 175-192.
• [19] T.T. Nguyen, J.J. Strodiot and V.H. Nguyen, A bundle method for solving equilibrium problems. Mathematical Programming, 116 (2009), 529-552.
• [20] S. Scheimberg and F.M. Jacinto, An extension of FKKM Lemma with an application to generalized equilibrium problems. Pacific Journal of Optimization, 6 (2010), 243-253.
• [21] .D.Q Tran, L.M.Dung and .V.H.Nguyen, Extragradient algorithms extended to equilibrium problems. Optimization, 57 (2008), 749-776.

*

The first author is supported by CNPq grant 150839/2009-0

Publication Dates

• Publication in this collection
  22 Mar 2011
• Date of issue
  2011

History

• Received
  15 Aug 2010
• Accepted
  05 Jan 2011