On Gap Functions for Quasi-Variational Inequalities

Abstract

For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D-gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization-based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D-gap functions for a quasi-variational inequality, which is a generalization of the variational inequality and is used to formulate generalized equilibrium problems. These extensions are shown to formulate equivalent optimization problems for quasi-variational inequalities and are shown to be continuous and directionally differentiable.

1. Introduction

In this paper, we consider merit functions for quasi-variational inequalities, denoted by QVIP, to find a vector 𝑥^∗∈𝑆(𝑥^∗) such that ⟨𝐹(𝑥^∗),𝑥−𝑥^∗⟩≥0∀𝑥∈𝑆(𝑥^∗), (1.1) where 𝐹∶ℝ^𝑛→ℝ^𝑛 is a mapping, the symbol ⟨⋅,⋅⟩ denotes the inner product in ℝ^𝑛, and 𝑆∶ℝ^𝑛→2^ℝ𝑛 is a set-valued mapping of which 𝑆(𝑥) is a closed convex set in ℝ^𝑛 for each 𝑥. When the mapping 𝑆 is a constant closed convex set for all 𝑥∈ℝ^𝑛, QVIP reduces to a well-known variational inequality [1, 2]: ⟨𝐹(𝑥^∗),𝑥−𝑥^∗⟩≥0∀𝑥∈𝑆. (1.2)

QVIP is used to study and formulate generalized equilibrium problems, such as the generalized Nash equilibrium problem in which a strategy set of each player varies according to the other players' strategies [3, 4]. For variational inequalities, various merit functions, such as the gap function, the regularized gap function [5], the D-gap functions [6, 7], and so on, have been proposed. These functions are used to make an equivalent optimization formulation for a variational inequality, and based on these formulations, several optimization-based methods are proposed for solving variational inequalities [8, 9].

Fukushima [10] has proposed gap functions for QVIP, which lead to equivalent optimization formulation for QVIP. In this paper, we extend the results of Fukushima in two directions. One is to show the directional differentiability under more general setting (Theorem 2.5) and to give one sufficient condition for stationary point to be a solution (Proposition 2.7). The other is to propose a so-called D-gap function for QVIP, which leads to an unconstrained minimization optimization formulation for QVIP, and to show its directional differentiability (Section 3).

Notations used in this paper are summarized as follows. The superscript 𝑇 denotes a transpose of vector or matrix, and ‖⋅‖ denotes the Euclidean norm in ℝ^𝑛 defined by ‖𝑥‖=⟨𝑥,𝑥⟩.

2. Regularized Gap Function for Qvip

In this section, we first generalize the regularized gap function for a variational inequality to a quasi-variational inequality and show its properties.

The (generalized) regularized gap functions 𝑓_𝛼∶ℝ^𝑛→ℝ for QVIP (1.1) are defined as 𝑓_𝛼(𝑥)=max_𝑦{−⟨𝐹(𝑥),𝑦−𝑥⟩−𝛼𝜙(𝑥,𝑦)∣𝑦∈𝑆(𝑥)}, (2.1) where an 𝛼 is a positive constant and a function 𝜙∶ℝ^𝑛×ℝ^𝑛→ℝ satisfies the following conditions.

(C1)𝜙 is continuously differentiable on ℝ^𝑛×ℝ^𝑛.(C2)𝜙 is nonnegative on ℝ^𝑛×ℝ^𝑛 and 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.(C3)𝜙 is strongly convex uniformly in 𝑥, that is, there is a 𝜇>0 such that

𝜙(𝑥,𝑦₁)−𝜙(𝑥,𝑦₂)≥⟨∇_𝑦𝜙(𝑥,𝑦₂),𝑦₁−𝑦₂⟩+𝜇‖𝑦₁−𝑦₂‖²∀𝑦₁,𝑦₂∈ℝ^𝑛. (2.2)

Remark 2.1. It is easy to verify that a function 𝜙(𝑥,𝑦)=⟨𝑥−𝑦,𝐺(𝑥−𝑦)⟩ for a positive definite symmetric matrix 𝐺 satisfies the above conditions (C1)∼(C3). In this case, the function (2.1) reduces to 𝑓(𝑥)=max_𝑦1{−⟨𝐹(𝑥),𝑦−𝑥⟩−2⟨𝑦−𝑥,𝐺(𝑦−𝑥)⟩∣𝑦∈𝑆(𝑥)} (2.3) with 𝛼=1/2. This is just a regularized gap function [5] originally proposed for variational inequalities (1.2).
By the strong convexity of 𝜙 and the closed convexity of 𝑆(𝑥), the maximum in (2.1) is uniquely attained and is given by the unique solution 𝑦_𝛼(𝑥) of the following mathematical programming problem: min_𝑦⟨𝐹(𝑥),𝑦−𝑥⟩+𝛼𝜙(𝑥,𝑦)s.t.𝑦∈𝑆(𝑥), (2.4) and the function (2.1) is written as 𝑓_𝛼(𝑥)=−⟨𝐹(𝑥),𝑦_𝛼(𝑥)−𝑥⟩−𝛼𝜙(𝑥,𝑦_𝛼(𝑥)). (2.5)

Lemma 2.2. A point 𝑥^∗ is a solution to QVIP (1.1) if and only if 𝑦_𝛼(𝑥^∗)=𝑥^∗.

Proof. Since the optimization problem (2.4) is convex with respect to 𝑦, the point 𝑦_𝛼(𝑥^∗) is a solution to (2.4) if and only if ⟨𝐹(𝑥^∗)+𝛼∇_𝑦𝜙(𝑥^∗,𝑦),𝑦−𝑦_𝛼(𝑥^∗)⟩≥0∀𝑦∈𝑆(𝑥^∗), (2.6) which is equivalent to ⟨𝛼∇_𝑦𝜙(𝑥^∗,𝑦),𝑦−𝑦_𝛼(𝑥^∗)⟩≥⟨𝐹(𝑥^∗),𝑦_𝛼(𝑥^∗)−𝑦⟩. (2.7) Then by substituting a solution 𝑥^∗ of QVIP (1.1) to 𝑦, we have ⟨∇_𝑦𝜙(𝑥^∗,𝑦),𝑥^∗−𝑦_𝛼(𝑥^∗)⟩≥0. (2.8) On the other hand, from the condition (C3) we have ⟨∇_𝑦𝜙(𝑥^∗,𝑦_𝛼(𝑥^∗)),𝑥^∗−𝑦_𝛼(𝑥^∗)⟩+𝜇‖𝑥^∗−𝑦_𝛼(𝑥^∗)‖²≤𝜙(𝑥^∗,𝑥^∗)−𝜙(𝑥^∗,𝑦_𝛼(𝑥^∗))≤0, (2.9) where the last inequality follows from the condition (C2). The above two inequalities lead to 𝑦_𝛼(𝑥^∗)=𝑥^∗.
Conversely, suppose that 𝑦_𝛼(𝑥^∗)=𝑥^∗. Then the inequality (2.6) reduces to ⟨𝐹(𝑥^∗),𝑦−𝑥^∗⟩≥0∀𝑦∈𝑆(𝑥^∗), (2.10) which shows that 𝑥^∗ is a solution to (1.1).

The next theorem shows that the function (2.1) or (2.5) leads to an equivalent optimization problem for quasi-variational inequalities. The theorem is inherently equivalent to [10, Theorem 2], but for completeness, we provide its proof. We note that our proof is more elementary and simpler than that of [10, Theorem 2].

Theorem 2.3. Let 𝑓_𝛼 be the function defined by (2.1) or (2.5). Then 𝑓_𝛼(𝑥)≥0 for all 𝑥∈𝑆(𝑥). Furthermore, 𝑓_𝛼(𝑥^∗)=0 and 𝑥^∗∈𝑆(𝑥^∗) if and only if 𝑥^∗ is a solution to QVIP (1.1). Hence, problem (1.1) is equivalent to finding a global optimal solution to the problem: minimize𝑓_𝛼(𝑥)subject𝑡𝑜𝑥∈𝑆(𝑥). (2.11)

Proof. The first assertion is obvious from the definition (2.1) and (C2). To prove the last assertion, suppose that 𝑥^∗ is a solution to QVIP. Then, we have ⟨𝐹(𝑥^∗),𝑥^∗−𝑥⟩−𝛼𝜙(𝑥^∗,𝑦)≤0∀𝑦∈𝑆(𝑥^∗). (2.12) Therefore, from the definition (2.1), we have 𝑓_𝛼(𝑥^∗)=0.
For the “only if” part, we consider the regularized gap function for fixed 𝑥^∗, 𝑓_𝛼(𝑥)=max_𝑦{−⟨𝐹(𝑥),𝑦−𝑥⟩−𝛼𝜙(𝑥,𝑦)∣𝑦∈𝑆(𝑥^∗)}. (2.13) Then, it follows from 𝑓_𝛼(𝑥^∗)=0 and 𝑥^∗∈𝑆(𝑥^∗) that 𝑓_𝛼(𝑥^∗)=0, which implies that 𝑥^∗ is a solution to the variational inequality [11] ⟨𝐹(𝑥^∗),𝑥−𝑥^∗⟩≥0∀𝑥∈𝑆(𝑥^∗). (2.14) This means that 𝑥^∗ is a solution to QVIP.

The next theorem gives a sufficient condition for the continuity of the function (2.1).

Theorem 2.4. Let 𝑓_𝛼 be the function defined by (2.1) or (2.5). If the set-valued mapping 𝑆 is continuous with respect to 𝑥 in the sense of set-valued mapping [12], then 𝑓_𝛼 is also continuous in 𝑥.

Proof. In a similar way to [13], 𝑦_𝛼 is shown to be continuous in 𝑥. Therefore, the function 𝑓_𝛼 is also continuous.

When the set-valued mapping 𝑆 is expressed as a finite number of convex inequalities, such that 𝑆(𝑥)={𝑦∈ℝ^𝑛∣𝑔_𝑖(𝑥,𝑦)≤0,𝑖=1,…,𝑚}, (2.15) where the functions 𝑔_𝑖(𝑥,𝑦)∶ℝ^𝑛×ℝ^𝑛→ℝ are continuous with respect to 𝑥 and 𝑦, and 𝑔_𝑖(𝑥,⋅) are convex for each 𝑥, then one sufficient condition for the continuity of the set-valued mapping 𝑆 is that Slater's constraint qualification holds, that is, for each 𝑥, there exists a vector 𝑦 (possibly depending on 𝑥) such that 𝑔_𝑖(𝑥,𝑦)<0∀𝑖=1,…,𝑚. (2.16) In this case, 𝑓_𝛼 is also continuous and 𝑦_𝛼(𝑥) satisfies the KKT condition: 𝐹(𝑥)+𝛼∇_𝑦𝜙(𝑥,𝑦_𝛼(𝑥))+^𝑚_𝑖=1𝜆_𝑖∇_𝑦𝑔_𝑖(𝑥,𝑦_𝛼𝑔(𝑥))=0_𝑖(𝑥,𝑦_𝛼(𝑥))≤0,𝜆_𝑖≥0,𝜆_𝑖𝑔_𝑖(𝑥,𝑦_𝛼(𝑥))=0,𝑖=1,…,𝑚. (2.17)

Unfortunately, the function 𝑓_𝛼 defined by (2.1) or (2.5) is not necessarily differentiable. However, the next theorem gives one sufficient condition of the directional differentiability of the function 𝑓_𝛼 with the set 𝑆 given by (2.15).

Theorem 2.5. Let the mapping 𝐹∶ℝ^𝑛→ℝ^𝑛 be continuously differentiable. Let also the set-valued mapping 𝑆 be defined as (2.15), where the functions 𝑔_𝑖(𝑥,𝑦)∶ℝ^𝑛×ℝ^𝑛→ℝ are continuous with respect to 𝑥 and 𝑦, and 𝑔_𝑖(𝑥,⋅) is convex for each 𝑥. If Slater's constraint qualification (2.16) holds, then the function 𝑓_𝛼 defined by (2.1) or (2.5) is directionally differentiable in any direction 𝑑∈ℝ^𝑛, and its directional derivative 𝑓^′_𝛼(𝑥;𝑑) is given by 𝑓^′_𝛼(𝑥;𝑑)=min_{𝜆∈𝑀(𝑥)}(𝐹(𝑥)+∇𝐹(𝑥)(𝑥−𝑦_𝛼(𝑥))−𝛼∇_𝑥𝜙(𝑥,𝑦_𝛼(𝑥))−^𝑚_𝑖=1𝜆_𝑖∇_𝑥𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)))^𝑇𝑑, (2.18) where 𝑀(𝑥) is defined by 𝑀(𝑥)={𝜆∈ℝ^𝑚|𝐹(𝑥)+𝛼∇_𝑦𝜙(𝑥,𝑦_𝛼(𝑥))+∑^𝑚_𝑖=1𝜆_𝑖∇_𝑦𝑔_𝑖(𝑥,𝑦_𝛼𝜆(𝑥))=0_𝑖≥0,𝜆_𝑖𝑔_𝑖(𝑥,𝑦_𝛼(𝑥))=0,𝑖=1,…,𝑚}. (2.19)

Proof. This directly follows from [14, Theorem 2].

Remark 2.6. Fukushima [10, Theorem 3] has also proven the directional differentiability of 𝑓_𝛼 for the function (2.3) with 𝑆(𝑥) polyhedral convex. This situation is a special case of this theorem.

The next proposition gives a sufficient condition for a stationary point of the equivalent optimization problem (2.11) to be a solution of QVIP (1.1) with the set 𝑆 given by (2.15).

Proposition 2.7. Suppose that 𝑥∈𝑆(𝑥). Suppose also that ∇𝐹(𝑥) is positive definite and 𝜆_𝑖⟨∇_𝑥𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)),𝑦_𝛼(𝑥)−𝑥⟩≥0 for all 𝑖 and 𝜆∈𝑀(𝑥). If the function 𝜙 used in the regularized gap function (2.1) or (2.5) is 𝜙(𝑥,𝑦)=𝜙^{0𝑥002𝑑𝑐}(𝑥−𝑦), where ^0𝑥0𝜙^02𝑑𝑐∶ℝ^𝑛→ℝ is differentiable strongly convex function, then a stationary point 𝑥 of the problem (2.11) is a solution to QVIP (1.1).

Proof. It suffices to show that 𝑥 is a solution to QVIP if the following inequality holds: 𝑓^′(𝑥;𝑦_𝛼(𝑥)−𝑥)≥0. (2.20)
It is easy to see that 𝜙(𝑥,𝑦)=𝜙^{0𝑥002𝑑𝑐}(𝑥−𝑦) satisfies the conditions (C1)∼(C3) and that ∇_𝑥𝜙(𝑥−𝑦)=−∇_𝑦𝜙(𝑥−𝑦). Then from the definition of directional derivative (2.18) and the KKT condition (2.17) for 𝑦_𝛼(𝑥), we have 𝑓^′(𝑥;𝑦_𝛼(𝑥)−𝑥)=min_{𝜆∈𝑀(𝑥)}(𝐹(𝑥)+∇𝐹(𝑥)(𝑥−𝑦_𝛼(𝑥))−𝛼∇_𝑥𝜙(𝑥,𝑦_𝛼(𝑥))−^𝑚_𝑖=1𝜆_𝑖∇_𝑥𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)))^𝑇(𝑦_𝛼(𝑥)−𝑥)=−(𝑦_𝛼(𝑥)−𝑥)^𝑇∇𝐹(𝑥)(𝑦_𝛼(𝑥)−𝑥)−(^𝑚_𝑖=1𝜆_𝑖∇_𝑦𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)))^𝑇(𝑦_𝛼(𝑥)−𝑥)−max_{𝜆∈𝑀(𝑥)}(^𝑚_𝑖=1𝜆_𝑖∇_𝑥𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)))^𝑇(𝑦_𝛼(𝑥)−𝑥). (2.21) Since functions 𝑔_𝑖 are convex with respect to 𝑦, we have 𝑔_𝑖(𝑥,𝑥)−𝑔_𝑖(𝑥,𝑦_𝛼(𝑥))≥∇_𝑦𝑔_𝑖(𝑥,𝑦_𝛼(𝑥))^𝑇(𝑥−𝑦_𝛼(𝑥)). (2.22) Then we have from the fact 𝑥∈𝑆 and (2.17) that −(^𝑚_𝑖=1𝜆_𝑖∇_𝑦𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)))^𝑇(𝑦_𝛼(𝑥)−𝑥)≤^𝑚_𝑖=1𝜆_𝑖(𝑔_𝑖(𝑥,𝑥)−𝑔_𝑖(𝑥,𝑦_𝛼(𝑥))≤−^𝑚_𝑖=1𝜆_𝑖𝑔_𝑖(𝑥,𝑦_𝛼(𝑥))≤0. (2.23) Therefore, it follows from (2.21) and the assumption 𝜆_𝑖⟨∇_𝑥𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)),𝑦_𝛼(𝑥)−𝑥⟩≥0 that 0≤𝑓^′(𝑥;𝑦_𝛼(𝑥)−𝑥)≤−(𝑦_𝛼(𝑥)−𝑥)^𝑇∇𝐹(𝑥)(𝑦_𝛼(𝑥)−𝑥)−max_{𝜆∈𝑀(𝑥)}(^𝑚_𝑖=1𝜆_𝑖∇_𝑥𝑔_𝑖(𝑥,𝑦_𝛼(𝑥)))^𝑇(𝑦_𝛼(𝑥)−𝑥)≤−(𝑦_𝛼(𝑥)−𝑥)^𝑇∇𝐹(𝑥)(𝑦_𝛼(𝑥)−𝑥), (2.24) which leads to 𝑦_𝛼(𝑥)=𝑥 from the positive definiteness of ∇𝐹. This shows from Lemma 2.2 that 𝑥 is a solution to QVIP.

Remark 2.8. When the functions 𝑔_𝑖 are all defined as linearized approximation of convex functions 𝑐_𝑖∶ℝ^𝑛→ℝ at 𝑥, that is, 𝑔_𝑖(𝑥,𝑦)=𝑐_𝑖(𝑥)+∇𝑐_𝑖(𝑥)^𝑇(𝑦−𝑥), then we have ∇_𝑥𝑔_𝑖(𝑥,𝑦_𝛼(𝑥))=∇𝑐_𝑖(𝑥)+∇²𝑐_𝑖(𝑥)(𝑦_𝛼(𝑥)−𝑥)−∇𝑐_𝑖(𝑥)=∇²𝑐_𝑖(𝑥)(𝑦_𝛼(𝑥)−𝑥). (2.25) Since the Hesse matrix ∇²𝑐_𝑖(𝑥) is positive semidefinite from the convexity of 𝑐_𝑖, the assumption of Proposition 2.7 is satisfied. This result has been already obtained by Taji and Fukushima [13] for this setting, and the above proposition is considered as a generalization in some sense.

3. D-Gap Function for Qvip

For 0<𝛼<𝛽, we consider the function 𝑔_𝛼𝛽 defined by Φ_𝛼𝛽(𝑥)=𝑓_𝛼(𝑥)−𝑓_𝛽(𝑥). (3.1) This is a so-called D-gap function and is originally introduced for the variational inequality (1.2) by Peng [6]. D-gap functions are shown to construct a differentiable equivalent unconstrained optimization formulation for VIPs.

We have the next proposition.

Proposition 3.1. For 0<𝛼<𝛽, for the function Φ_𝛼𝛽 defined by (3.1), (𝛽−𝛼)𝜙(𝑥,𝑦_𝛽(𝑥))≤Φ_𝛼𝛽(𝑥)≤(𝛽−𝛼)𝜙(𝑥,𝑦_𝛼(𝑥)). (3.2)

Proof. From the definition (3.1) and the fact that 𝑦_𝛼(𝑥),𝑦_𝛽∈𝑆(𝑥), we have Φ_𝛼𝛽(𝑥)=max_{𝑦∈𝑆(𝑥)}{−⟨𝐹(𝑥),𝑦−𝑥⟩−𝛼𝜙(𝑥,𝑦)}−max_{𝑦∈𝑆(𝑥)}{−⟨𝐹(𝑥),𝑦−𝑥⟩−𝛽𝜙(𝑥,𝑦)}≥⟨𝐹(𝑥),𝑥−𝑦_𝛽(𝑥)⟩−𝛼𝜙(𝑥,𝑦_𝛽(𝑥))−⟨𝐹(𝑥),𝑥−𝑦_𝛽(𝑥)⟩+𝛽𝜙(𝑥,𝑦_𝛽(𝑥))=(𝛽−𝛼)𝜙(𝑥,𝑦_𝛽(𝑥)). (3.3) This shows the left-side hand of the inequality. The right-hand side is shown in a similar way.

This proposition establishes the equivalence between a QVIP and the unconstrained minimization of a D-gap function Φ_𝛼𝛽.

Theorem 3.2. Let the function Φ_𝛼𝛽 be defined as (3.1). Then, for 0<𝛼<𝛽, Φ_𝛼𝛽(𝑥)≥0 for all 𝑥∈ℝ^𝑛. Moreover, Φ_𝛼𝛽(𝑥)=0 if and only if 𝑥 is a solution to QVIP. Hence, the problem (1.1) is equivalent to finding a global optimal solution to the unconstrained minimization problem: minimizeΦ_𝛼𝛽(𝑥). (3.4)

Proof. The first half follows directly from Proposition 3.1. The last half also follows from Lemma 2.2 and Proposition 3.1.

The continuity and the directional differentiability of the D-gap function (3.1) directly follow from those of the regularized gap function (2.1).

Theorem 3.3. Suppose that the set-valued mapping 𝑆 is continuous in 𝑥, then the D-gap function Φ_𝛼𝛽 defined by (3.1) is continuous in 𝑥. Moreover, if the set-valued mapping 𝑆 is defined by (2.15) and if Slater's constraint qualification (2.16) holds, then the D-gap function Φ_𝛼𝛽 is directionally differentiable in all direction 𝑑.

Proof. These results directly follow from the definition of the D-gap function Φ_𝛼𝛽 and Theorems 2.4 and 2.5.

Acknowledgment

The author would like to thank an anonymous referee for introducing the paper [10].