The Dual Gap Function for Variational Inequalities - CiteSeerX

Appl Math Optim 48:129–148 (2003) DOI: 10.1007/s00245-003-0771-9

© 2003 Springer-Verlag New York Inc.

The Dual Gap Function for Variational Inequalities∗ Jianzhong Zhang,1 Changyu Wan,2 and Naihua Xiu3 1 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong [email protected] 2 Institute

of Operations Research, Qufu Normal University, Qufu 273165, People’s Republic of China 3 Department of Applied Mathematics, Northern Jiaotong University, Beijing 100044, People’s Republic of China [email protected]

Communicated by J. Stoer

Abstract. In this paper we further study the dual gap function G, which was introduced by Marcotte and Zhu [14], for the variational inequality problem (VIP). We characterize the directional derivative and subdifferential of G. Based on these, we get a better understanding of the concepts of a global error bound, weak sharpness, and minimum principle sufficiency property for the pseudo-monotone VIP. Key Words. Variational inequalities, Dual gap function, Directional derivative, Subdifferential, Error bound, Weak sharpness. AMS Classification.

90C33, 49M99.

∗ This research was partly supported by the CityU Strategic Research Grant #7001427, and the National Natural Science Foundation of China (10171055, 10271002).

130

1.

Jianzhong Zhang, Changyu Wan, and Naihua Xiu

Introduction

We consider the following variational inequality problem (VIP for short): find a vector x ∗ ∈ X such that F(x ∗ ), x − x ∗  ≥ 0,

∀x ∈ X,

(1.1)

where F(x) is a mapping from R n into itself, X is a nonempty closed convex set in R n , and ·, · and || · || denote the usual Euclidean inner product and norm in R n , respectively. We denote the solution set of problem (1.1) by X ∗ . A popular approach for solving VIP is to reformulate it as an equivalent minimization problem. In particular, Auslender [2] suggested the equivalent problem of minimizing the (nondifferentiable, nonconvex, and possibly extended-valued) gap function g(x) := sup{F(x), x − y | y ∈ X },

x ∈ Rn ,

(1.2)

over the constraint set X . To overcome the nondifferentiability of g, Fukushima [8] and Auchmuty [1] independently proposed the regularized gap function. To overcome the constraint restriction, Mangasarian and Solodov [12] proposed in the case of nonn linear complementarity problems (NCP) (i.e., X = R+ ) a reformulation involving the unconstrained minimization of an implicit Lagrangian. This was subsequently extended by Peng [15] and then Taji et al. [18] and Yamashita et al. [19] to the D-gap function for the VIP. For the details, see survey papers by Fukushima [9] and Larsson and Patriksson [10]. More recently, Marcotte and Zhu [14] introduced a new gap function for problem (1.1): G(x) := sup{F(y), x − y | y ∈ X },

x ∈ Rn ,

(1.3)

which is called the dual gap function while g(x) is called the primal gap function. Since function G is the pointwise supremum of affine functions, it is closed and convex on R n . Moreover, G(x) ≥ 0 for any x ∈ X ; and when F is continuous and pseudo-monotone on X , x ∗ ∈ X ∗ if and only if G(x ∗ ) = 0. Thus, any solution of the VIP is a global minimizer for the convex optimization problem min{G(x) | x ∈ X }

(1.4)

with zero optimal value. By using some characterizations of G, Marcotte and Zhu studied the relationship among a global error bound, weak sharpness of the solution set, minimum principle sufficiency (MPS) property, and finite termination of descent algorithms for the solutions of problem (1.1). However, most of their main results require the assumptions that F is pseudo-monotone+ on X (see Section 2) and X is compact. These two conditions are quite strong and not easy to satisfy. In order to relax these conditions, we find that the topic of the dual gap function G in [14] should be further investigated. The aim of this paper is to find more properties of the dual gap function, and to establish some applications of these properties.

The Dual Gap Function

131

We begin this piece of work with a review of some necessary results of convex analysis in Section 2. We then show some new characterizations of the directional derivative and subdifferential of the dual gap function G in Section 3. As a result, we finally get a better understanding of the concepts of global error bound, weak sharpness of the solution set, and MPS property for the pseudo-monotone VIP in Section 4. These results may be regarded as extensions of the ones by Burke and Ferris [3], in which they introduced and characterized the notion of weak sharp minima to mathematical programming (MP). This paper improves the conclusions of [14], because our results do not request the assumptions that F is pseudo-monotone+ and X is compact.

2.

Preliminaries

We first recall the related concepts and conclusions in convex analysis (see, e.g., [16]), which are the main tools for our theoretical analysis. Let A and B be two subsets in R n . Define A\B = {x ∈ R n | x ∈ A, x ∈ / B}, A ± α B = {x ± αy | x ∈ A, y ∈ B} ψ ∗ (x|A) = sup{x, y | y ∈ A},

(α ∈ R), x ∈ Rn ,

where ψ ∗ (x|A) is called the support function for the set A. ¯ the convex hull of A by Co A, For a given set A, we denote the closure of A by A, the affine hull of A by aff A, the interior of A by int A, and the relative interior of A by ri A. The polar Ao of A is defined as Ao = {y ∈ R n | y, x ≤ 0, ∀x ∈ A}. If A is a nonempty convex set in R n , its normal cone at x ∈ R n is
if x ∈ A, {y ∈ R n | y, z − x ≤ 0, ∀z ∈ A}, N A (x) = ∅, otherwise, and its tangent cone at x ∈ R n is T A (x) = [N A (x)]o . Let f be a convex function from R n into R¯ := R ∪ {−∞} ∪ {+∞}. Then f is said to be proper if the effective domain K ( f ) := {x ∈ R n | f (x) < +∞} = ∅ and the set {x ∈ R n | f (x) = −∞} = ∅. Moreover, if f is proper convex, then K ( f ) is convex and ri K ( f ) = ∅. A proper convex function f is said to be closed at x ∈ R n if it is lower semi-continuous at this point, and to be closed in R n if it is lower semi-continuous at each point of R n .

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Jianzhong Zhang, Changyu Wan, and Naihua Xiu

Setting x ∈ K ( f ), the cone of all feasible directions at this point is defined as ¯ D K ( f ) (x) = {d ∈ R n | ∃λ¯ > 0, x + λd ∈ K ( f ), ∀λ ∈ (0, λ)}, and D K ( f ) (x) ⊆ TK ( f ) (x). The directional derivative of the function f at x in the direction d ∈ R n is defined as f  (x; d) = lim λ↓0

f (x + λd) − f (x) . λ

If f is proper convex and x ∈ K ( f ), then f  (x; d) exists for any d ∈ R n , where the derivative value is either finite or infinite. We use ∂ f (x) to express the subdifferential of f at x ∈ K ( f ) which is the set of all subgradients of f at x. ∂ f (x) is a closed convex set. If f is proper convex, then f is subdifferentiable at each point of ri K ( f ). We next review the concepts of monotonicity and generalized monotonicity. The mapping F is said to be monotone on X if for any x, y ∈ X , F(y) − F(x), y − x ≥ 0. The mapping F is said to be pseudo-monotone on X , if for any x, y ∈ X , F(x), y − x ≥ 0

⇒

F(y), y − x ≥ 0.

The mapping F is said to be quasi-monotone on X , if for any x, y ∈ X , F(x), y − x > 0

⇒

F(y), y − x ≥ 0.

The mapping F is said to be monotone+ on X , if it is monotone on X and for any x, y ∈ X , F(y) − F(x), y − x = 0

⇒

F(y) = F(x).

The mapping F is said to be pseudo-monotone+ on X , if it is pseudo-monotone on X and for any x, y ∈ X , F(x), y − x ≥ 0,

F(y), y − x = 0

⇒

F(y) = F(x).

Several results about these concepts can be found in [13], [17], [21], and [22]. Finally, the projection of a point x ∈ R n onto the set A is defined as (see, e.g., [20]) PA [x] = arg min{||x − y|| | ∀y ∈ A}. From the definition of the normal and tangent cones, we know that a vector x ∗ ∈ X ∗ if and only if −F(x ∗ ) ∈ N X (x ∗ ) or equivalently PTX (x ∗ ) [−F(x ∗ )] = 0.

3.

Properties of the Dual Gap Function

In this section we first show some characteristic expressions of the directional derivative of the dual gap function G defined in (1.3). We then establish several structural expressions for the subdifferential of G. Finally, we give some sufficient conditions for ensuring the existence of the directional derivative and the subdifferential.

The Dual Gap Function

3.1.

133

The Directional Derivative

Let x¯ ∈ R n . Define the level set of the function F(y), x¯ − y over X as θ (x, ¯ δ) = {y ∈ X | F(y), x¯ − y ≥ δ},

δ ∈ R.

This set is closely related to the value G(x). ¯ If G(x) ¯ is finite, then the directional derivative of G at point x¯ in the direction d ∈ R n possesses the following property. Lemma 3.1. Let G be as defined in (1.3). If x¯ ∈ K (G), then G  (x; ¯ d) exists for any d ∈ R n , and G  (x; ¯ d) ≥ lim sup{F(y), d | y ∈ θ(x, ¯ δ)}. δ↑G(x) ¯

(3.1)

Proof. As x¯ ∈ K (G), G is proper convex in R n , and hence the directional derivative G  (x; ¯ d) exists for any d ∈ R n . From the definition (1.3), we know that for any λ > 0 and any δ < G(x), ¯ G(x¯ + λd) = sup{F(y), x¯ + λd − y | y ∈ X } ≥ sup{F(y), x¯ − y + λF(y), d | y ∈ θ(x, ¯ δ)} ≥ δ + λ sup{F(y), d | y ∈ θ(x, ¯ δ)}. This implies that G(x¯ + λd) − δ ≥ sup{F(y), d | y ∈ θ(x, ¯ δ)}. λ Letting δ ↑ G(x) ¯ and then λ ↓ 0, we derive (3.1). The proof is completed. In the lemma above, the direction d ∈ R n is arbitrary. If d is taken in the cone of feasible directions of K (G) at x, ¯ then the function F(y), d is locally bounded in y in some neighborhood. Lemma 3.2. Let G be as defined in (1.3). If x¯ ∈ K (G), d ∈ D K (G) (x), ¯ and δ0 < G(x), ¯ then there are numbers M > 0 and λ∗ > 0 such that for any λ ∈ (0, λ∗ ] and δ0 ≤ δ < G(x), ¯ sup{F(y), d | y ∈ θ (x¯ + λd, δ)} ≤ M.

(3.2)

Proof. Since K (G) is convex and d ∈ D K (G) (x), ¯ there is a number λ¯ > 0 such that for ¯ any λ ∈ (0, λ], x¯ + λd ∈ K (G). By the lower semi-continuity of G, there is a number λ > 0 such that for any λ ∈ (0, λ ] and any δ ∈ [δ0 , G(x)), ¯ θ (x¯ + λd, δ) = ∅.

(3.3)

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Jianzhong Zhang, Changyu Wan, and Naihua Xiu

Taking λ∗∗ = min{λ¯ , λ }, from x¯ + λ∗∗ d ∈ K (G) we obtain G(x¯ + λ∗∗ d) < +∞.

(3.4)

Letting λ∗ = 12 λ∗∗ , from (3.3) we have for any λ ∈ (0, λ∗ ] and any δ ∈ [δ0 , G(x)), ¯ G(x¯ + λ∗∗ d) = sup{F(y), x¯ + λ∗∗ d − y | y ∈ X } ≥ sup{F(y), x¯ + λ∗∗ d − y | y ∈ θ(x¯ + λd, δ)} = sup{F(y), x¯ + λd − y + (λ∗∗ − λ)F(y), d | y ∈ θ(x¯ + λd, δ)} ≥ δ0 + (λ∗∗ − λ) sup{F(y), d | y ∈ θ(x¯ + λd, δ)}, from which and (3.4) we obtain sup{F(y), d | y ∈ θ (x¯ + λd, δ)} ≤

G(x¯ + λ∗∗ d) − δ0 λ∗∗ − λ

≤ (2/λ∗∗ )|G(x¯ + λ∗∗ d) − δ0 | := M. The proof is completed. In view of Lemmas 3.1 and 3.2, we can prove the main result in this subsection, which gives a structural expression of the directional derivative. Theorem 3.3. Let G be as defined in (1.3). If x¯ ∈ K (G) and d ∈ D K (G) (x), ¯ then G  (x; ¯ d) = lim sup{F(y), d | y ∈ θ(x, ¯ δ)}. δ↑G(x) ¯

(3.5)

Proof. Due to the given assumptions and as δ0 < G(x), ¯ there are two numbers M > 0 and λ∗ > 0 such that for any λ ∈ (0, λ∗ ) and any δ ∈ [δ0 , G(x)), ¯ (3.2) and (3.3) hold. So, G(x¯ + λd) = sup{F(y), x¯ + λd − y | y ∈ X } = sup{F(y), x¯ + λd − y | y ∈ θ(x¯ + λd, δ)} ≤ sup{F(y), x¯ + λd − y | y ∈ θ(x, ¯ δ) ∪ [θ(x¯ + λd, δ)\θ(x, ¯ δ)]} = max{sup{F(y), x¯ − y + λF(y), d | y ∈ θ(x, ¯ δ)}, sup{F(y), x¯ − y + λF(y), d | y ∈ [θ(x¯ +λd, δ)\θ(x, ¯ δ)]}} ≤ max{G(x) ¯ + λ sup{F(y), d | y ∈ θ(x, ¯ δ)}, δ + λ sup{F(y), d | y ∈ [θ(x¯ + λd, δ)\θ(x, ¯ δ)]}} ≤ max{G(x) ¯ + λ sup{F(y), d | y ∈ θ(x, ¯ δ)}, δ + λ sup{F(y), d | y ∈ θ(x¯ + λd, δ)}} ≤ max{G(x) ¯ + λ sup{F(y), d | y ∈ θ(x, ¯ δ)}, δ + λM}.

(3.6)

The Dual Gap Function

135

Since θ (x, ¯ δ) = ∅ by δ0 ≤ δ < G(x), ¯ we have sup{F(y), d | y ∈ θ (x, ¯ δ)} > −∞. From (3.6) and δ0 ≤ δ < G(x) ¯ we know that for sufficiently small λ ∈ (0, λ∗ ], G(x¯ + d) ≤ G(x) ¯ + λ sup{F(y), d | y ∈ θ(x, ¯ δ)}, which leads to G  (x; ¯ d) ≤ sup{F(y), d | y ∈ θ(x, ¯ δ)}.

(3.7)

Combining (3.1) and (3.7) yields (3.5). Now, we define the set (x) associated with the dual gap function G as (x) = arg max{F(y), x − y | y ∈ X }. For x ∈ R n , the set (x) is possibly empty. However, if it is nonempty at some point, then the directional derivative of G at this point can be expressed as follows. Lemma 3.4. Let G be as defined in (1.3). If x¯ ∈ R n satisfies (x) ¯ = ∅, θ(x, ¯ δ) is an upper semi-continuous set function at point δ = G(x), ¯ and if there exists a scalar β > 0 such that F is uniformly continuous on (x) ¯ + βB where B denotes the unit ball in R n , then for any d ∈ D K (G) (x), ¯ G  (x; ¯ d) = sup{F(y), d | y ∈ (x)}. ¯

(3.8)

Proof. Given ε > 0 arbitrarily. From the assumptions, there exists a scalar α ∈ (0, β) such that for any y ∈ (x) ¯ + αB, ||F(y) − F(z)|| < ε,

(3.9)

where z = P(x) ¯ δ) at point δ = G(x), ¯ there ¯ [y]. By the upper semi-continuity of θ( x, ¯ G(x)), exists a scalar δ¯ < G(x) ¯ such that for any δ ∈ (δ, ¯ θ (x, ¯ δ) ⊆ (x) ¯ + αB.

(3.10)

By using (3.9) and (3.10), we have that for any y ∈ θ(x, ¯ δ), z = P(x) ¯ [y], and d ∈ D K (G) (x), ¯ F(y), d ≤ F(z), d + ε||d||. This, plus the fact (x) ¯ ⊆ θ (x, ¯ δ), yield that sup{F(y), d | y ∈ (x)} ¯ ≤ sup{F(y), d | y ∈ θ(x, ¯ δ)} ≤ sup{F(y), d | y ∈ (x)} ¯ + ε||d||,

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Jianzhong Zhang, Changyu Wan, and Naihua Xiu

which infers from Theorem 3.3 that sup{F(y), d | y ∈ (x)} ¯ ≤ lim sup{F(y), d | y ∈ θ(x, ¯ δ)} δ↑G(x) ¯

= G  (x; ¯ d) ≤ sup{F(y), d | y ∈ (x)} ¯ + ε||d||. The arbitrariness of ε proves (3.8). Theorem 3.5. Assume that F is continuous on X and G is as defined in (1.3). If for ¯ is bounded, then for any x¯ ∈ K (G), there is a scalar δ¯ < G(x) ¯ such that θ(x, ¯ δ) ¯ d ∈ D K (G) (x), G  (x; ¯ d) = max{F(y), d | y ∈ (x)}. ¯

(3.11)

¯ is compact and the function F(y), x¯ − y is continuous, the set Proof. Since θ (x, ¯ δ) (x) ¯ is nonempty and compact. By the continuity of F, there is a scalar β > 0 such that F is uniformly continuous on (x) ¯ + βB. So, it suffices to prove by Lemma 3.4 that θ (x, ¯ δ) is upper semi-continuous at point δ = G(x). ¯ We argue by contradiction. Suppose that there are a scalar α ∈ (0, β) and two sequences {δk } ↑ G(x) ¯ and {yk } with yk ∈ θ(x, ¯ δk ), such that yk ∈ / (x) ¯ + αB,

k = 1, 2, . . . .

(3.12)

¯ for sufficiently large k, the sequence {yk } is bounded and Since yk ∈ θ (x, ¯ δk ) ⊆ θ (x, ¯ δ) hence has at least one limit point, say lim j→∞ yk j = y¯ . From the continuity of F and the inequality F(yk j ), x¯ − yk j  ≥ δk j ,

j = 1, 2, . . . ,

we obtain F( y¯ ), x¯ − y¯  ≥ G(x). ¯ This shows that y¯ ∈ (x) ¯ ⊆ (x) ¯ + αB. Thus, yk j ∈ (x) ¯ + αB for sufficiently large j, a contradiction to (3.12). Hence the proof is completed. When X is a compact set in R n , the continuity of F implies that K (G) = R n and ¯ is bounded for any x¯ ∈ R n and any δ¯ < G(x). θ (x, ¯ δ) ¯ Thus, from Theorem 3.5 we immediately obtain the following result, which was first proven by Danskin [4] (also see [5]). Corollary 3.6. Assume that F is continuous on X and X is compact. If G is as defined in (1.3), then for x¯ ∈ R n and d ∈ R n , G  (x; ¯ d) = max{F(y), d | y ∈ (x)}. ¯

(3.13)

The Dual Gap Function

3.2.

137

The Subdifferential

Lemma 3.7. Let G be as defined in (1.3). If x¯ ∈ K (G), then  [Co F(θ (x, ¯ δ)) + N K (G) (x)] ¯ ⊆ ∂G(x). ¯ δ 0 such that x ∈ int K (G),

∀x ∈ x¯ + εB.

From Theorem 3.8(ii) we derive  Co F(θ (x, δ)) = ∂G(x). δ