Auxiliary Principle Technique for Solving ... - Semantic Scholar
J Optim Theory Appl (2011) 149: 441–445 DOI 10.1007/s10957-010-9785-z
Auxiliary Principle Technique for Solving Bifunction Variational Inequalities M.A. Noor · K.I. Noor · E. Al-Said
Published online: 6 January 2011 © Springer Science+Business Media, LLC 2011
Abstract In this paper, we use the auxiliary principle technique to suggest and analyze an implicit iterative method for solving bifunction variational inequalities. We also study the convergence criteria of this new method under pseudomonotonicity condition. Keywords Variational inequalities · Auxiliary principle · Convergence · Iterative methods 1 Introduction Much attention has been given to study the bifunction variational inequality, which can be viewed as a useful and significant extension of the variational inequalities. Crespi et al. [1–4], Fang and Hu [5], Lalitha and Mehra [6] and Noor [7] have studied various aspects of the bifunction variational inequalities. There is a substantial number of numerical methods including projection technique and its variant forms, Wiener-Hopf equations, auxiliary principle and resolvent equations methods for solving variational inequalities. However, it is known that projection, Wiener-Hopf equations, proximal and resolvent equations techniques cannot be extended and generalized to suggest and analyze similar iterative methods for solving bifunction variational inequalities. This fact has motivated to use the auxiliary principle technique, Communicated by F. Giannessi. This research is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia, under grant No. KSU.VPP.108. M.A. Noor () · K.I. Noor Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan e-mail: [email protected] M.A. Noor · E. Al-Said Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia
442
J Optim Theory Appl (2011) 149: 441–445
which is mainly due to Glowinski, Lions and Tremolieres [8], to suggest and analyze an implicit iterative method for solving the bifunction variational inequalities. We also study the convergence of this new method under the pseudomonotonicity, which, of course, is a weaker condition than monotonicity. Our method of proof is very simple.
2 Preliminaries Let H be a real Hilbert space, whose inner product and norm are denoted by ., . and ., respectively. Let K be a nonempty, closed and convex set in H . For a given continuous bifunction T (.,.) : K × K −→ H, consider the problem of finding u ∈ K, such that T (u, v − u) ≥ 0,
∀v ∈ K,
(1)
which is called a bifunction variational inequality. A number of problems arising in various branches of pure and applied sciences can be studied via bifunction variational inequalities. It has been shown [7] that the minimum of a Gateaux differentiable convex function on a convex set can be characterized by the bifunction variational inequality (1). In a similar way, one can show that the minimum of a Lipschitz continuous nonconvex is a solution of (1). For the formulation, well-posedness, existence results for bifunction variational inequalities, see [1–7] and the references therein. If T (u, v − u) = G(u), v − u, where G is a nonlinear operator, then (1) is equivalent to the problem of finding u ∈ K, such that G(u), v − u ≥ 0,
∀v ∈ K,
(2)
which is the simplest form of the classical variational inequality introduced by Stampacchia [9]. For the recent applications, numerical methods and formulations of variational inequalities, and equilibrium problems, see [1–14] and the references therein. Definition 2.1 The bifunction function T (., .) : K × K −→ H is said to be pseudomonotone, iff T (u, v − u) ≥ 0
⇒
−T (v, u − v) ≥ 0,
∀u, v ∈ K.
3 Main Results In this section, we describe an iterative method for solving (1) by using the technique of the auxiliary principle technique, which is due to Glowinski et al. [8]. For a given u ∈ K satisfying (1), we consider the auxiliary bifunction variational inequality problem, which consists in finding w ∈ K, such that ρT (w, v − w) + E (w) − E (u), v − w ≥ 0,
∀v ∈ K,
(3)
J Optim Theory Appl (2011) 149: 441–445
443
where ρ > 0 is a constant and E (u) is the differential of a strong convex function E at u ∈ K. From the strongly convexity of the differentiable function E(u), it follows that problem (3) has a unique solution. Clearly, if w = u, then w is a solution of (1). This observation enables one to suggest and analyze the following iterative method for solving (1). Algorithm 3.1 For a given u0 ∈ H, calculate the approximate solution un+1 by the iterative scheme ρT (un+1 , v − un+1 ) + E (un+1 ) − E (un ), v − un+1 ≥ 0,
∀v ∈ K,
(4)
where ρ > 0 is a constant. Note that, if T (u, v − u) = G(u), v − u, then Algorithm 3.1 reduces to the following iterative scheme for solving (2). Algorithm 3.2 For a given u0 ∈ H, find the approximate solution un+1 by the iterative scheme ρG(un+1 ) + E (un+1 ) − E (un ), v − un+1 ≥ 0,
∀v ∈ K,
where ρ > 0 is a constant. For appropriate and suitable choice of the bifunction and the spaces, one can obtain a number of iterative methods for solving the bifunction variational inequalities and related optimization problems. We now study the convergence criteria of Algorithm 3.1 and this is the main motivation of next result. Theorem 3.1 Let the function T (., .) be pseudomonotone and let E(u) be a strong convex function with modulus β > 0. Then the approximate solution un+1 , obtained from Algorithm 3.1, converges to a solution u ∈ K of (1). Proof Let u ∈ K be a solution of (1). Then, using the pseudomonotonicity of T (., .), we have −T (v, u − v) ≥ 0,
∀v ∈ K.
(5)
Taking v = un+1 in (5) and v = u in (4), we have −T (un+1 , u − un+1 ) ≥ 0,
(6)
ρT (un+1 , u − un+1 ) + E (un+1 ) − E (un ), u − un+1 ≥ 0.
(7)
Now we consider the generalized Bregman function as B(u, z) = E(u) − E(z) − E (z), u − z ≥ βu − z2 , where we have used the fact that the function E(u) is strong convex.
(8)
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J Optim Theory Appl (2011) 149: 441–445
Combining (8), (6) and (7), we have B(u, un ) − B(u, un+1 ) = E(un+1 ) − E(un ) − E (un ), u − un + E (un+1 ), u − un+1 = E(un+1 ) − E(un ) − E (un ) − E (un+1 ), u − un+1 − E (un ), un+1 − un ≥ βun+1 − un 2 + E (un+1 ) − E (un ), u − un+1 ≥ βun+1 − un 2 − ρT (un+1 , u − un+1 ) ≥ βun+1 − un 2 . If un+1 = un , then clearly un is a solution of (1). Otherwise, for β > 0, the sequence B(u, un ) − B(u, un+1 ) is non-negative and we must have lim (un+1 − un ) = 0.
n→∞
It follows that the sequence {un } is bounded. Let u¯ be a cluster point of the subse¯ Now, using the quence {uni }, and let {uni } be a subsequence converging toward u. technique of Zhu and Marcotte [13], it can be shown that the entire sequence {un } converges to the cluster point u¯ satisfying (1).
Conclusion In this paper, we have suggested an implicit iterative method for solving bifunction variational inequalities using the auxiliary principle technique. We have also studied the convergence of the proposed method under suitable conditions. Comparison of this method with other methods is an open problem, which requires further efforts.
References 1. Crespi, G.P.J., Ginchev, J., Rocca, M.: Minty variational inequalities, increase along rays property and optimization. J. Optim. Theory Appl. 123, 479–496 (2004) 2. Crespi, G.P.J., Ginchev, J., Rocca, M.: Existence of solutions and star-shapedness in Minty variational inequalities. J. Glob. Optim. 32, 485–494 (2005) 3. Crespi, G.P.J., Ginchev, J., Rocca, M.: Increasing along rays property for vector functions. J. Nonconvex Anal. 7, 39–50 (2006) 4. Crespi, G.P.J., Ginchev, J., Rocca, M.: Some remarks on the Minty vector variational principle. J. Math. Anal. Appl. 345, 165–175 (2008) 5. Fang, Y.P., Hu, R.: Parametric well-posedness for variational inequalities defined by bifunction. Comput. Math. Appl. 53, 1306–1316 (2007) 6. Lalitha, C.S., Mehra, M.: Vector variational inequalities with cone-pseudomonotone bifunction. Optimization 54, 327–338 (2005) 7. Noor, M.A.: Some new classes of nonconvex functions. Nonlinear Funct. Anal. Appl. 11, 165–171 (2006) 8. Gloawinski, R., Lions, L.J., Tremolieres, R.: Numerical Analysis of Variational Inequalities. NorthHolland, Amsterdam (1981)
J Optim Theory Appl (2011) 149: 441–445
445
9. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964) 10. Noor, M.A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000) 11. Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004) 12. Noor, M.A., Noor, K.I., Rassias, T.M.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993) 13. Zhu, D.L., Marcotte, P.: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6, 714–726 (1996) 14. Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199– 277 (2004)
Auxiliary Principle Technique for Solving Bifunction Variational Inequalities M.A. Noor · K.I. Noor · E. Al-Said
Published online: 6 January 2011 © Springer Science+Business Media, LLC 2011
Abstract In this paper, we use the auxiliary principle technique to suggest and analyze an implicit iterative method for solving bifunction variational inequalities. We also study the convergence criteria of this new method under pseudomonotonicity condition. Keywords Variational inequalities · Auxiliary principle · Convergence · Iterative methods 1 Introduction Much attention has been given to study the bifunction variational inequality, which can be viewed as a useful and significant extension of the variational inequalities. Crespi et al. [1–4], Fang and Hu [5], Lalitha and Mehra [6] and Noor [7] have studied various aspects of the bifunction variational inequalities. There is a substantial number of numerical methods including projection technique and its variant forms, Wiener-Hopf equations, auxiliary principle and resolvent equations methods for solving variational inequalities. However, it is known that projection, Wiener-Hopf equations, proximal and resolvent equations techniques cannot be extended and generalized to suggest and analyze similar iterative methods for solving bifunction variational inequalities. This fact has motivated to use the auxiliary principle technique, Communicated by F. Giannessi. This research is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia, under grant No. KSU.VPP.108. M.A. Noor () · K.I. Noor Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan e-mail: [email protected] M.A. Noor · E. Al-Said Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia
442
J Optim Theory Appl (2011) 149: 441–445
which is mainly due to Glowinski, Lions and Tremolieres [8], to suggest and analyze an implicit iterative method for solving the bifunction variational inequalities. We also study the convergence of this new method under the pseudomonotonicity, which, of course, is a weaker condition than monotonicity. Our method of proof is very simple.
2 Preliminaries Let H be a real Hilbert space, whose inner product and norm are denoted by ., . and ., respectively. Let K be a nonempty, closed and convex set in H . For a given continuous bifunction T (.,.) : K × K −→ H, consider the problem of finding u ∈ K, such that T (u, v − u) ≥ 0,
∀v ∈ K,
(1)
which is called a bifunction variational inequality. A number of problems arising in various branches of pure and applied sciences can be studied via bifunction variational inequalities. It has been shown [7] that the minimum of a Gateaux differentiable convex function on a convex set can be characterized by the bifunction variational inequality (1). In a similar way, one can show that the minimum of a Lipschitz continuous nonconvex is a solution of (1). For the formulation, well-posedness, existence results for bifunction variational inequalities, see [1–7] and the references therein. If T (u, v − u) = G(u), v − u, where G is a nonlinear operator, then (1) is equivalent to the problem of finding u ∈ K, such that G(u), v − u ≥ 0,
∀v ∈ K,
(2)
which is the simplest form of the classical variational inequality introduced by Stampacchia [9]. For the recent applications, numerical methods and formulations of variational inequalities, and equilibrium problems, see [1–14] and the references therein. Definition 2.1 The bifunction function T (., .) : K × K −→ H is said to be pseudomonotone, iff T (u, v − u) ≥ 0
⇒
−T (v, u − v) ≥ 0,
∀u, v ∈ K.
3 Main Results In this section, we describe an iterative method for solving (1) by using the technique of the auxiliary principle technique, which is due to Glowinski et al. [8]. For a given u ∈ K satisfying (1), we consider the auxiliary bifunction variational inequality problem, which consists in finding w ∈ K, such that ρT (w, v − w) + E (w) − E (u), v − w ≥ 0,
∀v ∈ K,
(3)
J Optim Theory Appl (2011) 149: 441–445
443
where ρ > 0 is a constant and E (u) is the differential of a strong convex function E at u ∈ K. From the strongly convexity of the differentiable function E(u), it follows that problem (3) has a unique solution. Clearly, if w = u, then w is a solution of (1). This observation enables one to suggest and analyze the following iterative method for solving (1). Algorithm 3.1 For a given u0 ∈ H, calculate the approximate solution un+1 by the iterative scheme ρT (un+1 , v − un+1 ) + E (un+1 ) − E (un ), v − un+1 ≥ 0,
∀v ∈ K,
(4)
where ρ > 0 is a constant. Note that, if T (u, v − u) = G(u), v − u, then Algorithm 3.1 reduces to the following iterative scheme for solving (2). Algorithm 3.2 For a given u0 ∈ H, find the approximate solution un+1 by the iterative scheme ρG(un+1 ) + E (un+1 ) − E (un ), v − un+1 ≥ 0,
∀v ∈ K,
where ρ > 0 is a constant. For appropriate and suitable choice of the bifunction and the spaces, one can obtain a number of iterative methods for solving the bifunction variational inequalities and related optimization problems. We now study the convergence criteria of Algorithm 3.1 and this is the main motivation of next result. Theorem 3.1 Let the function T (., .) be pseudomonotone and let E(u) be a strong convex function with modulus β > 0. Then the approximate solution un+1 , obtained from Algorithm 3.1, converges to a solution u ∈ K of (1). Proof Let u ∈ K be a solution of (1). Then, using the pseudomonotonicity of T (., .), we have −T (v, u − v) ≥ 0,
∀v ∈ K.
(5)
Taking v = un+1 in (5) and v = u in (4), we have −T (un+1 , u − un+1 ) ≥ 0,
(6)
ρT (un+1 , u − un+1 ) + E (un+1 ) − E (un ), u − un+1 ≥ 0.
(7)
Now we consider the generalized Bregman function as B(u, z) = E(u) − E(z) − E (z), u − z ≥ βu − z2 , where we have used the fact that the function E(u) is strong convex.
(8)
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J Optim Theory Appl (2011) 149: 441–445
Combining (8), (6) and (7), we have B(u, un ) − B(u, un+1 ) = E(un+1 ) − E(un ) − E (un ), u − un + E (un+1 ), u − un+1 = E(un+1 ) − E(un ) − E (un ) − E (un+1 ), u − un+1 − E (un ), un+1 − un ≥ βun+1 − un 2 + E (un+1 ) − E (un ), u − un+1 ≥ βun+1 − un 2 − ρT (un+1 , u − un+1 ) ≥ βun+1 − un 2 . If un+1 = un , then clearly un is a solution of (1). Otherwise, for β > 0, the sequence B(u, un ) − B(u, un+1 ) is non-negative and we must have lim (un+1 − un ) = 0.
n→∞
It follows that the sequence {un } is bounded. Let u¯ be a cluster point of the subse¯ Now, using the quence {uni }, and let {uni } be a subsequence converging toward u. technique of Zhu and Marcotte [13], it can be shown that the entire sequence {un } converges to the cluster point u¯ satisfying (1).
Conclusion In this paper, we have suggested an implicit iterative method for solving bifunction variational inequalities using the auxiliary principle technique. We have also studied the convergence of the proposed method under suitable conditions. Comparison of this method with other methods is an open problem, which requires further efforts.
References 1. Crespi, G.P.J., Ginchev, J., Rocca, M.: Minty variational inequalities, increase along rays property and optimization. J. Optim. Theory Appl. 123, 479–496 (2004) 2. Crespi, G.P.J., Ginchev, J., Rocca, M.: Existence of solutions and star-shapedness in Minty variational inequalities. J. Glob. Optim. 32, 485–494 (2005) 3. Crespi, G.P.J., Ginchev, J., Rocca, M.: Increasing along rays property for vector functions. J. Nonconvex Anal. 7, 39–50 (2006) 4. Crespi, G.P.J., Ginchev, J., Rocca, M.: Some remarks on the Minty vector variational principle. J. Math. Anal. Appl. 345, 165–175 (2008) 5. Fang, Y.P., Hu, R.: Parametric well-posedness for variational inequalities defined by bifunction. Comput. Math. Appl. 53, 1306–1316 (2007) 6. Lalitha, C.S., Mehra, M.: Vector variational inequalities with cone-pseudomonotone bifunction. Optimization 54, 327–338 (2005) 7. Noor, M.A.: Some new classes of nonconvex functions. Nonlinear Funct. Anal. Appl. 11, 165–171 (2006) 8. Gloawinski, R., Lions, L.J., Tremolieres, R.: Numerical Analysis of Variational Inequalities. NorthHolland, Amsterdam (1981)
J Optim Theory Appl (2011) 149: 441–445
445
9. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964) 10. Noor, M.A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000) 11. Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004) 12. Noor, M.A., Noor, K.I., Rassias, T.M.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993) 13. Zhu, D.L., Marcotte, P.: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6, 714–726 (1996) 14. Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199– 277 (2004)
