Article
Kyungpook Mathematical Journal 2022; 62(1): 89-106
Published online March 31, 2022
Copyright © Kyungpook Mathematical Journal.
An Iterative Method for Equilibrium and Constrained Convex Minimization Problems
Maryam Yazdi, Mohammad Mehdi Shabani, Saeed Hashemi Sababe*
Young Researchers and Elite Club, Malard Branch, Islamic Azad University, Malard, Iran
e-mail : msh_yazdi@yahoo.com
Faculty of sciences, Imam Ali University, Tehran, Iran
e-mail : mohammadmehdishabani@yahoo.com
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada and Young Researchers and Elite Club, Malard Branch, Islamic Azad University, Malard, Iran
e-mail : s.hashemi@ualberta.ca">s.hashemi@ualberta.ca, hashemi_1365@yahoo.com
Received: February 1, 2021; Accepted: November 8, 2021
Abstract
We are concerned with finding a common solution to an equilibrium problem associated with a bifunction, and a constrained convex minimization problem. We propose an iterative fixed point algorithm and prove that the algorithm generates a sequence strongly convergent to a common solution. The common solution is identified as the unique solution of a certain variational inequality.
Keywords: Nonexpansive mapping, equilibrium problem, fixed point, convex minimization, averaged mapping, iterative method, variational inequality
1. Introduction
Consider two problems in a Hilbert space: A constrained convex optimization (CCO) and an equilibrium problem (EP) associated with a bifunction satisfying certain properties. It is known that CCO can be solved by the gradient-projection algorithm (GPA). It is also known that EP is equivalent to a fixed point problem. Therefore, both problems can be solved by fixed point algorithms.
Let
Recall that the equilibrium problem (EP) associated to a bifunction
The set of solutions of EP (1.1) is denoted by
If ϕ is of the form
Consequently, EP (1.1) includes, as special cases, numerous problems from various areas such as in physics, optimization and economics, and has been received a lot of attention; see, e.g., [7, 8, 5, 9, 12, 13, 14, 16, 17, 18, 19, 22, 28] and the references therein.
In 2010, Tian [20] considered the following iterative method:
where
Consider the constrained convex minimization problem:
where
where the initial guess
is a contraction for
However, if the gradient
In 2012, Tian and Liu [21] studied a composite iterative scheme by the viscosity approximation method [15, 26] for finding a common solution of an equilibrium problem and a constrained convex minimization problem:
where
In this paper, motivated by the above results, we will propose an iterative fixed point algorithm to find a point which is a common solution to an equilibrium problem associated with a bifunction and a constrained convex minimization problem. We will prove strong convergence of the algorithm under certain conditions and identify the limit of the sequence generated by the algorithm as the unique solution of some variational inequality.
2. Preliminaries
Let
for all
Note that
The following inequality is convenient in applications. Recall that for all
Next, given a Hilbert space
Lemma 2.1. ([10]) Let
Let
(
(
(
(
Under these conditions, it is easy to see that the solution set
Following Combettes and Hirstoaga [6], we can define, for each fixed
for
Lemma 2.2. ([6]) The mapping
(i)
(ii)
Property (ii) shows an equivalence of EP (1.1) to the fixed point problem of
Definition 2.3. ([21]) A mapping
Note that firmly nonexpansive mappings (e.g., projections) are
Proposition 2.4. ([4, 23])} The composite of finitely many averaged mappings is averaged. That is, if each of the mappings
Definition 2.5. A nonlinear operator
(i) β-strongly monotone if there exists
(ii) ν-inverse strongly monotone (for short, ν-ism) if there exists
It can be easily seen that any projection
Inverse strongly monotone (also referred to as co-coercive) operators have extensively been used in practical problems from various areas, for instance, traffic assignment problems; see, for example, [3, 11] and references therein.
Proposition 2.6. ([4, 23]) Let
-
(i)
T is nonexpansive if and only if the complementI-T is-ism. -
(ii) If
T is ν-ism, then for, is -ism. -
(iii)
T is averaged if and only if the complementI-T is ν-ism for some.
Indeed, for
Lemma 2.7. ([1, 25]) Assume
where
Then
3. An Iterative Scheme
In this section we introduce an iterative fixed point algorithm for finding a point in
In the rest of this paper, we always assume that
for each
Lemma 3.1. Suppose a continuously differentiable, convex function
and let
-
(i) The mapping
is α-av, where ; namely, one can write , where and is nonexpansive. -
(ii) The function
is uniformly Lipschitz continuous in over bounded x ∈ C .
Lemma 3.2. ([27, Lemma 3.1]) Suppose
is a
The following is the main result of this paper.
Theorem 3.3. Assume the bifunction
(
(
(
(
Let
where the initial point
Suppose
(
Then, the sequences
To prove Theorem 3.3 we first establish some lemmas.
Lemma 3.4. Let
Then, for
Since
it turns out that (noting
This finishes the proof.
Lemma 3.5. For
and
In particular,
Similarly, we have
Adding up the last two relations and using the monotonicity
This can be rewritten as
and (3.4) follows immediately.
Next using the nonexpansivity of
The proof of the lemma is complete.
Step 1. The sequences
Now since
As a result, we get, by induction,
for all
Step 2. Asymptotic regularity of
Since
where
To estimate
Observe by Lemmas 3.4 and 3.5
Now since
Substituting (3.10)) into (3.7) yields
where
By condition
Step 3. We claim that
(3a)
(3b)
(3c)
Indeed, since
and (3a) follows. To show (3b), we first observe that the firm nonexpansivity of
For the sake of brevity, we shall use the
It turns out that
This proves (3b). To verify (3c), it suffices to verify that
Step 4. We have the following asymptotic variational inequality:
where
To prove (3.12), take a subsequence
By virtue of VI (3.13), it suffices to show
We may assume
The demiclosedness principle of nonexpansive mappings then ensures that
This results in
Hence,
Step 5. Strong convergence of
Here
where
Since
It turns out from (3a) of Step 3 and (3.12) that
It is now immediately clear that
Remark 3.6. The choices of the parameters
where
Remark 3.7. Theorem 3.3 is a generalization of [21, Theorem 3.2]. In addition, we used the condition
4. Numerical Test
In this section, we give a numerical example to illustrate the scheme (3.2) given in Theorem 3.3.
Example 4.1. Let
Set
Then
So its discriminant is
Thus, by Lemma 2.2, we get
for
converge to
Table 1 indicates the values of the sequences
-
Table 1 . The values of the sequences
{xn} and{un} .Numerical results for x1=12 andx1 =-18 n xn un n xn un 1 12 2 1 -18 -3 2 2.944 0.49067 2 -4.416 -0.736 3 0.42394 0.07065 3 -0.6359 -0.10598 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 15 2.2373e-15 3.7289e-16 15 -3.356e-15 -5.593e-16 16 1.088e-16 1.8133e-17 16 -1.632e-16 -2.72e-17 17 5.1872e-18 8.6453e-19 17 -7.7808e-18 -1.2968e-18 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 28 6.1677e-33 1.028e-33 28 -9.251e-33 -1.5419e-33 29 2.5625e-34 4.2709e-35 29 -3.8438e-34 -6.4064e-35 30 1.0574e-35 0 30 -1.5862e-35 0
Figure 1 presents the behavior of the sequences
-
Figure 1. The convergence of
{xn} and{un} with different initial valuesx1
Acknowledgements.
A part of this research was carried out while the third author was visiting the university of Alberta. The authors are grateful to Professor Hong-Kun Xu for his comments.
References
- K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda,
Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space , Nonlinear Anal.,67 (2007), 2350-2360. - E. Blum and W. Oettli,
From optimization and variatinal inequalities to equilibrium problems , Math. Student.,63 (1994), 123-145. - D. P. Bretarkas and E. M. Gafin,
Projection methods for variational inequalities with applications to the traffic assignment problem , Math. Programming Stud.,17 (1982), 139-159. - C. Byrne,
A unified treatment of some iterative algorithms in signal processing and image reconstruction , Inverse Problems,20 (2004), 103-120. - V. Colao, G. Marino and H. K. Xu,
An iterative method for finding common solutions of equilibrium and fixed point problems , J. Math. Anal. Appl.,344 (2008), 340-352. - P. L. Combettes and S. A. Hirstoaga,
Equilibrium programming in Hilbert spaces , J. Nonlinear Convex Anal.,6 (2005), 117-136. - L. C. Ceng, A. Petrusel, J. C. Yao, Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities and fixed Points of nonexpansive mappings,
zero points of accretive operators in Banach spaces , Fixed Point Theory,19 (2018), 487-502. - L. C. Ceng, A. Petrusel, J. C. Yao and Y. Yao,
Systems of variational inequalities with hierarchical variational inequality constraints for Lipschitzian pseudocontractions , Fixed Point Theory,20 (2019), 113-133. - V. Colao, G. Marino and H. K. Xu,
An explicit scheme of equilibrium for a finite family of nonexpansive mappings , J. Math. Anal. Appl.,344 (2008), 340-352. - K. Geobel, W. A. Kirk and Topics in Metric Fixed Point Theory. Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press; 1999.
- D. Han and H. K. Lo,
Solving non additive traffic assignment problems: a descent method for cocoercive variational inequalities , European J. Oper. Res.,159 (2004), 529-544. - S. Husain and N. Singh,
A hybrid iterative algorithm for a split mixed equilibrium problem and a hierarchical fixed point problem , Appl. Set-Valued Anal. Optim.,1 (2019), 149-169. - J. S. Jung,
A general composite iterative method for equilibrium problems and fixed point problems , J. Comput. Anal. Appl.,12 (2010), 124-140. - G. Marino and H. K. Xu,
A general iterative method for nonexpansive mappings in Hilbert spaces , J. Math. Anal. Appl.,318 (2006), 43-52. - A. Moudafi,
Viscosity approximation methods for fixed point problems , J. Math. Anal. Appl.,241 (2000), 46-55. - J. W. Peng, J. C. Yao and A viscosity approximation scheme for system of equilibrium problems,
nonexpansive mappings and monotone mappings , Nonlinear Anal.,12 (2009), 6001-6010. - S. Plubtieng and R. Punpaeng,
A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces , J. Math. Anal. Appl.,336 (2007), 455-469. - A. Razani and M. Yazdi,
Viscosity approximation method for equilibrium and fixed point problems , Fixed Point Theory,14(2) (2013), 455-472. - A. Razani and M. Yazdi,
A New iterative method for generalized equilibrium and fixed point problems of nonexpansive mappings , Bull. Malays. Math. Sci. Soc.,35(4) (2021), 1049-1061. - M. Tian,
A general iterative alghoritm for nonexpansive mappings in Hilbert spaces , Nonlinear Anal.,73 (2010), 689-694. - M. Tian and L. Liu,
Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem , Fixed Point Theory Appl.,201(2012) (2012), 1-17. - S. Wang, C. Hu and G. Chia,
Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems , Appl. Math. Comput.,215 (2010), 3891-3898. - H. K. Xu,
Averaged mappings and the gradient-projection algorithm , J. Optim. Theory Appl.,150 (2011), 360-378. - H. K. Xu,
An iterative approach to quadratic optimization , J. Optim. Theory Appl.,116 (2003), 659-678. - H. K. Xu,
Iterative algorithms for nonlinear operators , J. London Math. Soc.,66 (2002), 240-256. - H. K. Xu,
Viscosity approximation methods for nonexpansive mappings , J. Math. Anal. Appl.,298 (2004), 279-291. - I. Yamada. The hybrid steepest descent for variational inequality problems over the intersection of fixed points sets of nonexpansive mappings. Stud. Comput. Math. North-Holland, Amsterdam; 2001.
- M. Yazdi and S. Hashemi Sababe A new extragradient method for equilibrium,
split feasibility and fixed point problems , J. Nonlinear Convex Anal.,22(4) (2021), 759-773.