J. Adv. Math. Stud

J. Adv. Math. Stud. Vol. 8(2015), No. 1, 01-08 http://journal.fairpartners.ro

SENSITIVITY ANALYSIS FOR GENERAL VARIATIONAL INCLUSIONS INVOLVING DIFFERENCE OF OPERATORS MUHAMMAD ASLAM NOOR, RABIA KAMAL, KHALIDA INAYAT NOOR AND AWAIS GUL KHAN Abstract. It is known that the general variational inclusions involving difference of two (or more) monotone operators are equivalent to the resolvent equations. We use this equivalent form to develop the sensitivity analysis for general variational inclusions. This technique does not require the differentiability of the given data. Some special cases are discussed. Results obtained in this paper continue to hold for these problems.

1. INTRODUCTION Variational inequalities are being used as mathematical programming models to study a large number of equilibrium problems arising in finance, economics, transportation, operations research and engineering sciences. The behavior of such equilibrium problems as a result of changes in the problem data is always of concern. Such type of study is known as the sensitivity analysis. Recently much attention has been given to develop a general sensitivity analysis framework for variational inequalities and related problems. The techniques suggested so far vary with the problem being studied, see [4, 6, 7, 8, 9, 15, 16, 19, 27, 29, 30] and the references therein. Sensitivity analysis for variational inequalities has been studied by many authors including Dafermos [4], Kyparisis [6, 7], Liu [8], Moudafi and Noor [9], Noor and Noor [19], Qiu and Magnanti [27], Tobin [29] and Yen [30] using quite different techniques. Dafermos [4] used the fixed-point formulation of variational inequalities to study the sensitivity analysis. This technique has been modified and extended by many authors for studying the sensitivity analysis of various other classes of variational inequalities. This approach has strong geometrical flavor. It is well known that the variational inequalities are equivalent to the Wiener-Hopf equations, see Noor [14]. This fixed-point equivalence is obtained by a suitable and appropriate rearrangement of the Wiener-Hopf equations. The Wiener-Hopf equation approach is quite general, flexible unified and provides us with a new technique to study the sensitivity analysis of variational inequalities without assuming the differentiability of the given data. Noor [15, 18] and Noor and Noor [20] used the WienerHopf equations approach to study this problem. In this paper, we use the technique of Moudafi and Noor [9], Noor [15, 16, 18] and Noor and Noor [20] to develop the sensitivity framework for the general variational inclusions. It has been shown that the general variational inclusions involving difference of two monotone operators are equivalent to the general resolvent equations. This equivalent formulation is used to consider the sensitivity analysis. This is the main motivation of this paper. It is worth mentioning that this approach is easy to implement and provides an alternate Received: April 13, 2014. Revised: October 16, 2014. 2010 Mathematics Subject Classification: 49J40, 90C33. Key words and phrases: Variational inclusions, sensitivity analysis, resolvent equations. c 2015 Fair Partners Team for the Promotion of Science & Fair Partners Publishers

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Muhammad Aslam Noor, Rabia Kamal, Khalida Inayat Noor and Awais Gul Khan

approach to study the sensitivity analysis. The ideas and techniques of this paper may motivate further research work in this area. 2. FORMULATION AND BASIC RESULTS Let H be a real Hilbert space, whose norm and inner product are denoted by · and ·, ·, respectively. For given monotone operators T, A, g : H → H, consider a problem of finding u ∈ H such that 0 ∈ A (g (u)) − T u. (2.1) The problem of type (2.1) is called general variational inclusion involving difference of monotone operators. Note that the difference of two monotone operators is not a monotone operator as contrast to the sum of two monotone operators. Due to this fact, the problem of finding a zero of the difference of two monotone operators is very difficult as compared to finding the zeros of the sum of monotone operators, see [1, 10, 11, 25, 26]. We now discuss some special cases of problem (2.1). I. If g ≡ I, the identity operator, then problem (2.1) is equivalent to finding u ∈ H such that 0 ∈ A (u) − T u, (2.2) which is recently considered and studied by Moudafi [10, 11] and Noor et al. [25, 26], using essentially two different techniques. II. If A (u) ≡ ∂f (u) and T u ≡ ∂g (u), where f and g are two convex lower semicontinuous functions, then problem (2.2) is equivalent to finding u ∈ H such that 0 ∈ ∂f (u) − ∂g (u) ,

(2.3)

under some suitable conditions. It is well known that the necessary optimality condition for the problem of finding the minimum of f (u) − g (u) is equivalent to the problem (2.3). In fact, a wide class of problems arising in different branches of science such as multicommodity network, image restoration processing, discrete tomography, clustering. Particularly, these problems are suitable to model several nonconvex industrial problems such as portfolio optimization, fuel mixture, phylogenetic analysis. III. If A (g (u)) ≡ ∂ϕ (g (u)), the subdifferential of a proper, convex and lower-semicontinuous function ϕ : H → R ∪ {∞}, then problem (2.1) is equivalent to finding u ∈ H such that 0 ∈ ∂ϕ (g (u)) − T u, (2.4) a problem considered and studied by Adly and Oettli [1]. We note that problem (2.4) can be written as: find u ∈ H such that T u, g (v) − g (u) + ϕ (g (u)) − ϕ (g (v)) ≤ 0,

∀v ∈ H,

(2.5)

which is known as the general mixed variational inequality or the variational inequality of the second kind. For the applications, numerical methods and other aspects of these mixed variational inequalities, see [1-30] and the references therein. IV. If ϕ is an indicator function of a closed and convex set K in a real Hilbert space H, that is, 0, if u ∈ K, ϕ (u) = IK (u) = +∞, otherwise, then problem (2.5) is equivalent to finding u ∈ H : g(u) ∈ K such that T u, g (v) − g (u) ≤ 0,

for all v ∈ H : g (v) ∈ K,

(2.6)

Sensitivity analysis for general variational inclusions involving difference of operators

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which is known as the general variational inequality, introduced and studied by Noor [13] in 1988. The general variational inequalities have been studied extensively in recent years. For the formulation, numerical methods, applications, and other aspects of the general variational inequalities (2.6), see [17-24] and the references therein. V. If g ≡ I, the identity operator, then problem (2.6) reduces to: find u ∈ K such that T u, v − u ≤ 0,

for all v ∈ K,

(2.7)

which is known as the classical variational inequalities, introduced and studied by Stampacchia [28] in 1964. For appropriate and suitable choice of the operators and the space, one can obtain several new and known classes of variational inclusions, variational inequalities and complementarity problems. See [1-30] and the references therein for recent developments. We also need the following well-known fundamental results and concepts. Definition 2.1. [2] If A is a maximal monotone operator on H, then for a constant ρ > 0, the resolvent operator associated with A is defined by JA [u] = (I + ρA)

−1

[u] ,

for all u ∈ H,

where I is the identity operator. It is known that a monotone operator is maximal, if and only if, its resolvent operator is defined everywhere. Furthermore, the resolvent operator is a single valued and nonexpansive, that is, JA [u] − JA [v] ≤ u − v , for all u, v ∈ H Remark 2.1. Since ∂ϕ be a subdifferential of a proper, convex, and lower semicontinuous function ϕ : H → R ∪ {+∞} is a maximal monotone operator, we define −1

Jϕ = (I + ρ∂ϕ)

,

the resolvent operator associated with ∂ϕ and ρ > 0 is a constant. We now establish the equivalence between the problem (2.1) and the fixed point problem using the resolvent operator technique. This alternative formulation has been used to discuss the existence of a solution of the problem (2.1) and to suggest and analyze an iterative method for solving the variational inclusions of type (2.1), see [24]. Lemma 2.1. [24] Let A be a maximal monotone operator. Then function u ∈ H, is a solution of the problem (2.1) , if and only if, u ∈ H satisfies the relation g (u) = JA [g (u) + ρT u] , −1

where JA = (I + ρA)

(2.8)

is the resolvent operator and ρ > 0 is a constant.

Lemma 2.1 implies that the general variational inclusion (2.1) is equivalent to the fixed point problem (2.8). This alternative equivalent formulation is very useful from the numerical and theoretical point of view. Related to the problem (2.1), we now consider the problem of solving the resolvent equations. Let RA = I − JA , where JA is the resolvent operator and I is the identity operator. For given nonlinear operators T, A, g, consider the problem of finding z ∈ H such that T g −1 JA [z] − ρ−1 RA [z] = 0. (2.9)

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Equations of the type (2.9) are called general resolvent equations associated with problem (2.1). These were introduced and studied by Noor et al [24]. We now consider the parametric version of problems (2.1) and (2.9). To formulate the problem, let M be an open subset of H in which the parameter λ takes values. Let T (u, λ) be given operator defined on H × M and take value in H. From now onward, we denote Tλ (·) ≡ T (·, λ), unless otherwise specified. The parametric general variational inclusion involving difference of monotone operators problem is to find (u, λ) ∈ H × M such that 0 ∈ A (g (u)) − Tλ u.

(2.10)

We also assume that the problem (2.9) has a unique solution u for some λ ∈ M . Related to the problem (2.9), we consider the parametric general resolvent equations. We consider the problem of finding (z, λ), (u, λ) ∈ H × M , such that Tλ g −1 JA [z] − ρ−1 RA [z] = 0,

(2.11)

where ρ > 0 is a constant. Using the technique of Noor [15, 16, 18] and Noor and Noor [20], one can easily establish the equivalence between problems (2.10) and (2.11). Lemma 2.2. The problem (2.10) has a solution (u, λ) ∈ H × M, if and only if, the problem (2.11) have a solution (z, λ) , (u, λ) ∈ H × M , provided g (u) = JA [z] ,

(2.12)

z = g (u) + ρTλ u,

(2.13)

where ρ > 0 is a constant. From Lemma 2.2, we see that the problem (2.10) and (2.11) are equivalent. We use this equivalence to study the sensitivity analysis of problem (2.1). We assume that for some λ ∈ M , problem (2.11) has a solution z and X is a closure of a ball in H centered at z. We want to investigate those conditions under which, for each λ in a neighborhood of λ, problem (2.11) has a unique solution z (λ) near z and the function z (λ) is (Lipschitz) continuous and differentiable. Definition 2.2. A nonlinear operator Tλ on X × M is said to be strongly monotone, if there exists a constant α > 0 such that 2

Tλ u − Tλ v, u − v ≥ α u − v ,

for all λ ∈ M , u, v ∈ X.

Definition 2.3. A nonlinear operator Tλ on X × M is said to be Lipschitz continuous, if there exists a constant β > 0 such that Tλ u − Tλ v ≤ β u − v ,


Definition 2.4. A nonlinear operator Tλ on X × M is said to be strongly antimonotone, if there exists a constant γ > 0 such that 2

Tλ u − Tλ v, u − v ≤ −γ u − v ,


3. MAIN RESULTS We consider the case, when the solutions of the parametric resolvent equations (2.13) lie in the interior of X. Following the ideas of Dafermos [4] and Noor [15, 17], we consider the map Fλ (z) = g (u) + ρTλ u, for all (z, λ) ∈ X × M, (3.1)


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and g (u) = JA [z] . (3.2) We have to show that the map Fλ (z) has a fixed point, which is a solution of the resolvent equations (2.11). First of all, we prove that the map Fλ (z), defined by (3.1), is a contraction map with respect to z uniformly λ ∈ M . Lemma 3.1. Let Tλ (·) be a locally strongly antimonotone with constant α > 0 and locally Lipschitz continuous with constant β > 0. If the operator g is strongly monotone with constant δ > 0 and Lipschitz continuous with constant σ > 0, respectively, then, for all z1 , z2 ∈ X and λ ∈ M , we have Fλ (z1 ) − Fλ (z2 ) ≤ θ z1 − z2 , where

for

k + 1 − 2ρα + ρ2 β 2 2 θ= , k 1− 2 2 2 ρ − α < α − β k (2 − k) , β2 β2

where

α>β

(3.3)

k (2 − k),

k < 1,

k = 2 1 − 2δ + σ 2 .

(3.4)

(3.5)

Proof. For all z1 , z2 ∈ X and λ ∈ M , we have, form (3.1), Fλ (z1 ) − Fλ (z2 ) = g (u1 ) + ρTλ u1 − g (u2 ) − ρTλ u2 ≤ u1 − u2 + ρ (Tλ u1 − Tλ u2 ) + u1 − u2 − (g (u1 ) − g (u2 )) .

(3.6)

Using locally strongly antimonotonicity and locally Lipschitz continuity of Tλ with constants α > 0 and β > 0, respectively, we have 2 u1 − u2 + ρ (Tλ u1 − Tλ u2 ) = u1 − u2 2 + 2ρ Tλ u1 − Tλ u2 , u1 − u2 + ρ2 Tλ u1 − Tλ u2 2 2

2

2

≤ u1 − u2 − 2ρα u1 − u2 + ρ2 β 2 u1 − u2 2 = 1 − 2ρα + ρ2 β 2 u1 − u2 .

(3.7)

Similarly, using strongly monotonicity with constants δ > 0 and Lipschitz continuity with constant σ > 0, of the operator g, we have 2 2 u1 − u2 − (g (u1 ) − g (u2 )) ≤ 1 − 2δ + σ 2 u1 − u2 . (3.8) From (3.6) − (3.8), we have Fλ (z1 ) − Fλ (z2 ) ≤

k 2 2 + 1 − 2ρα + ρ β u1 − u2 . 2

From (3.2), using (3.8), we have u1 − u2 ≤ u1 − u2 − (g (u1 ) − g (u2 )) + JA [z1 ] − JA [z2 ] k ≤ u1 − u2 + z1 − z2 , 2

(3.9)

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which implies u1 − u2 ≤

1 1−

k 2

z1 − z2 .

(3.10)

Combining (3.9) and (3.10), we have k + 1 − 2ρα + ρ2 β 2 Fλ (z1 ) − Fλ (z2 ) ≤ 2 z1 − z2 = θ z1 − z2 , k 1− 2 where

k + 1 − 2ρα + ρ2 β 2 2 θ= . k 1− 2 From (3.3) − (3.5), it follows that θ < 1 and consequently the map Fλ (z), defined by (3.1) is a contraction map and has a fixed point z (λ), which is a solution of the problem (2.11). Remark 3.1. From Lemma 3.1, we see that the map Fλ (z) defined by (3.1) has a unique fixed point z (λ), that is, z (λ) = Fλ (z). Also, by assumption, the function z, for λ = λ is a solution of the parametric resolvent equations (2.11). Again using Lemma 3.1, we see that z, for λ = λ is a fixed point of Fλ (z) and it is also a fixed point of Fλ (z). Consequently, we conclude that z λ = z = Fλ z λ . Using Lemma 3.1, we can prove the continuity of the solution z (λ) of the parametric resolvent equations (2.11) using the technique of Noor [15, 17]. However, for the sake of completeness and to convey an idea of the techniques involved, we give its proof. Lemma 3.2. If the operator Tλ (·) is locally Lipschitz continuous with respect to the parameter λ, with constant μ > 0 and the mapping λ → JA [z] is continuous (or Lipschitz continuous), then the function z (λ) satisfying (2.8) is continuous (or Lipschitz continuous) at λ = λ. Proof. For all λ ∈ M , using Lemma 3.1, we have z (λ) − z λ = Fλ (z (λ)) − F z λ λ ≤ Fλ (z (λ)) − Fλ z λ + Fλ z λ − Fλ z λ ≤ θ z (λ) − z λ + Fλ z λ − Fλ z λ . (3.11) From (3.1), using the fact that the operator Tλ (·) is a Lipschitz continuous with respect to λ the parameter, we have Fλ z λ − F z λ = g u λ + ρTλ u λ − g u λ − ρT u λ λ λ = ρ Tλ u λ − Tλ u λ (3.12) ≤ ρμ λ − λ . Combining (3.11) and (3.12), we have z (λ) − z λ ≤ ρμ λ − λ , 1−θ This is the desired result.

for all λ, λ ∈ M.


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We now state and prove the main result of this paper and this is the motivation our next result. Theorem 3.1. Let u be the solution of the problem (2.10) and z be the solution of the problem (2.11) for λ = λ. Let Tλ (u) be the locally strongly antimonotone and Lipschitz continuous operator on X. Let the operator g is strongly monotone and Lipschitz continuous. If the map λ → JA [z] is continuous (or Lipschitz continuous) at λ = λ, then there exists a neighborhood N ⊂ M of λ such that for λ ∈ N , then theparametric resolvent equations (2.11) have a unique solution z (λ) in the interior of X, z λ = z and z (λ) is continuous (or Lipschitz continuous) at λ = λ. Proof. Its proof follows from Lemmas 3.1, 3.2 and Remark 3.1.

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[20] M. A. Noor and K. I. Noor: Sensitivity analysis of some quasi variational inequalities, J. Adv. Math. Stud., 6(2013), 43-52. [21] M. A. Noor, K. I. Noor and A. G. Khan: Some iterative schemes for solving extended general quasi variational inequalities, Appl. Math. Inf. Sci., 7(2013), 917-925. [22] M. A. Noor, K. I. Noor and A. G. Khan: Parallel schemes for solving a system of extended general quasi variational inequalities, Appl. Math. Comput. 245(2014), 566-574. [23] M. A. Noor, K. I. Noor and A. G. Khan: Three step algorithms for solving extended general variational inequalities, J. Adv. Math. Stud. 7(2014), 38-49. [24] M. A. Noor, K. I. Noor, K. I. and R. Kamal: General variational inclusions involving difference of operators, J. Inequal. Appl., 98(2014), 16 pages. [25] M. A. Noor, K. I. Noor, A. Hamdi and E. H. El-Shemas: On difference of two monotone operators, Optim. Lett., 3(2009), 329-335. [26] M. A. Noor, K. I. Noor, E. H. El-Shemas and A. Hamdi: Resolvent iterative methods for difference of two monotone operators, Int. J. Optim. Theory Methods Appl., 1(2009), 15-25. [27] Y. Qiu and T. L. Magnanti: Sensitivity analysis for variational inequalities defined on polyhedral sets, Math. Oper. Res., 14(1989), 410-432. [28] G. Stampacchia: Formes bilineaires coercivites sur les ensembles convexes, C. R. Math. Acad. Sci. Paris, 258(1964), 4413-4416. [29] R. L. Tobin: Sensitivity analysis for variational inequalities, J. Optim. Theory Appl., 48(1986), 191204. [30] N. D. Yen and G. M. Lee: Solution sensitivity of a class of variational inequalities, J. Math. Anal. Appl., 215(1997), 48-55. COMSATS Institute of Information Technology Department of Mathematics Park Road, Islamabad, Pakistan E-mail address: [email protected] (Corresponding Author) COMSATS Institute of Information Technology Department of Mathematics Park Road, Islamabad, Pakistan E-mail address: [email protected] COMSATS Institute of Information Technology Department of Mathematics Park Road, Islamabad, Pakistan E-mail address: [email protected] COMSATS Institute of Information Technology Department of Mathematics Park Road, Islamabad, Pakistan E-mail address: [email protected]