A class of uncertain variational inequality problems
- Qiqiong Chen^{1} and
- Yuanguo Zhu^{1}Email author
https://doi.org/10.1186/s13660-015-0750-0
© Chen and Zhu 2015
Received: 14 April 2015
Accepted: 7 July 2015
Published: 25 July 2015
Abstract
In this paper, we propose a new class of variational inequality problems, say, uncertain variational inequality problems based on uncertainty theory in finite Euclidean spaces \(R^{n}\). It can be viewed as another extension of classical variational inequality problems besides stochastic variational inequality problems. Note that both stochastic variational inequality problems and uncertain variational inequality problems involve uncertainty in the real world, thus they have no conceptual solutions. Hence, in order to solve uncertain variational inequality problems, we introduce the expected value of uncertain variables (vector). Then we convert it into a classical deterministic variational inequality problem, which can be solved by many algorithms that are developed on the basis of gap functions. Thus the core of this paper is to discuss under what conditions we can convert the expected value model of uncertain variational inequality problems into deterministic variational inequality problems. Finally, as an application, we present an example in a noncooperation game from economics.
Keywords
variational inequality uncertainty theory optimization descent method line search1 Introduction
The variational inequality problem (VIP for short) is an important discipline of mathematics. Over the past several decades, VIP has developed into a very fruitful discipline in the field of mathematical programming. The developments include a rich mathematical theory, a host of effective solution algorithms, a multitude of interesting connections to numerous disciplines such as optimization theory, economics, engineering, game theory and networks (see [1–8]) and so forth. Although there are many derivatives of VIP in infinite spaces such as Hilbert spaces, Banach spaces etc. from the classical VIP of \(R^{n}\), we just pay our attention to the finite dimensional Euclidean space \(R^{n}\).
SVIP (4) was investigated in references such as [9–15], etc. And its applications can be found in inventory or pricing competition among several firms that provide substitutable goods or services (see [16] and [17]), a supply chain network model (see [18]), stochastic user equilibrium traffic network (see [19]) and wireless network (see [20]).
On the other hand, in the real world, there are some circumstances when no samples are available in the determinacy phenomena, for example, data of a new stock, data of devastating military experiments, etc., in which it is impossible for us to estimate a probability distribution. It seems that to invite some relevant experts to evaluate their belief degree that each event will happen makes sense. Here the belief degree represents the degree with which we believe the indeterminacy quantity falls into the left side of a given point about the indeterminacy phenomena. There also exist the cases that people’s viewpoints may vary from person to person for the same event; even for the same person, his/her view to the event may vary from time to time. Hence, in order to deal with those uncertainty phenomena, Prof. Liu, who came from Tsinghua University of China, founded a new theory - uncertainty theory - in 2007 (see [21]), which is quite different from probability theory. While SVIP is based on probability theory, we present in this paper its counterpart UVIP on the basis of uncertainty theory. There is no doubt that both of them are natural extensions of deterministic VIP.
For solving the deterministic VIP, we can reform it as a system of (unconstraint) equations. We can also cast it as a minimization problem by introducing gap functions [1]. Hence, on the basis of gap functions, we can design a host of effective solution algorithms for solving VIP.
In 1992, Fukushima [22] designed a regularized gap function as \(g(x)=\max\{F(x)^{T}(x-y)-\frac{1}{2}\|x-y\|^{2}\}\). Then, on the basis of it, he casted a VIP as a constrained differentiable optimization problem in the sense that any global solution of the reformulated optimization problem is a solution to VIP, and vice versa. It is presented as the following theorem.
(Theorem 3.3 in [22])
His research shows that the objective function is continuously differentiable whenever the mapping involved in the VIP is continuously differentiable (see Theorem 3.2 in [22]). Moreover, he discussed descent methods for solving the equivalent problem (see Lemma 5 in Section 2).
We highlight that the main idea for this paper is to discuss how to convert a UVIP into a classical VIP. Then one can solve it by many mature algorithms, which can be documented in [1, 22].
This paper is organized as follows. In Section 2, we review some definitions and lemmas which are useful in the sequel; in Section 3, we present a class of uncertain variational inequalities, discuss its expected value model and convert it into a class of classical deterministic variational inequalities in detail; as an application, we investigate in Section 4 an example developed from economics; we conclude the paper with Section 5.
2 Preliminaries
Before starting our discussion, in this section, let us review a few necessary definitions and lemmas.
We first state the axiomatic system of uncertainty theory as follows.
Let Γ be a nonempty set (universal set), and let \(\mathcal{L}\) be a σ-algebra over Γ. Then \((\Gamma, \mathcal{L})\) is called a measurable space and each element Λ in \(\mathcal{L}\) is called a measurable set, which is also called an event in uncertainty theory.
Definition 1
(Liu [21])
A set function \(\mathcal{M}\) defined on the σ-algebra \(\mathcal{L}\) over Γ is called an uncertain measure if it satisfies the following three axioms.
Axiom 1
(Normality axiom)
\(\mathcal{M}(\Gamma)=1\) for the universal set Γ.
Axiom 2
(Duality axiom)
\(\mathcal{M}\{\Lambda\} +\mathcal{M}\{\Lambda^{c}\}=1\) for any event Λ.
Axiom 3
(Subadditivity axiom)
Remark 1
Uncertain measure is interpreted as the personal belief degree (not frequency) of an uncertain event that may occur. It depends on the personal knowledge concerning the event. The uncertain measure will change if the state of knowledge changes.
Similar to stochastic variable defined on a probability space, an uncertain variable is a real-valued function defined on an uncertainty space. A formal definition is given as follows.
Definition 2
(Liu [21])
An uncertain variable is a function ξ from an uncertainty space \((\Gamma, \mathcal{L}, \mathcal{M})\) to the set of real numbers such that \(\{\xi\in\mathcal{B}\}\) is an event for any Borel set B. A k-dimensional uncertain vector is a function ξ from an uncertainty space \((\Gamma,\mathcal{L},\mathcal{M})\) to the set of k-dimensional real vectors such that \(\boldsymbol{\xi}\in B\) is an event for any Borel set B of k-dimensional real vector.
Note that the vector \(\boldsymbol{\xi}=(\xi_{1},\xi_{2},\ldots,\xi_{k})\) is an uncertain vector if and only if \(\xi_{1},\xi_{2},\ldots,\xi_{k}\) are uncertain variables.
In order to describe uncertain variables, Liu [21] introduced a new concept: uncertainty distribution. Note that it is a carrier of incomplete information of uncertain variable.
Definition 3
(Liu [21])
Besides the uncertainty distribution, we present the regular uncertainty distribution as follows, which is used more often.
Definition 4
(Liu [23])
Remark 2
It is clear that a regular uncertainty distribution \(\Phi(x)\) has an inverse function on the range of x with \(0<\Phi(x)<1\), and the inverse function \(\Phi^{-1}(\alpha)\) exists on the open interval \((0,1)\). Hence we have the following definition of inverse uncertain distribution.
Definition 5
(Liu [23])
Let ξ be an uncertain variable with regular uncertainty distribution \(\Phi(x)\). Then the inverse function \(\Phi^{-1}(\alpha )\) is called the inverse uncertainty distribution of ξ.
Since the expected value is the average value of the uncertain variable in the sense of uncertain measure and it represents the size of uncertain variable, it is natural to present the expected value of an uncertain variable as follows.
Definition 6
(Liu [21])
It is easy to know that if an uncertain variable \(\xi\sim\mathcal {L}(a,b)\), i.e., it is subject to a linear uncertainty distribution, then it has an expected value \(E[\xi]=\frac{a+b}{2}\).
Since the independence between uncertain variables is very important while describing many results, we state it formally.
Definition 7
(Liu [24])
Definition 8
(Liu [21])
Next, we present some lemmas which will be used in the discussion that follows.
Lemma 1
(Liu [21])
Lemma 2
(Liu [23])
Lemma 3
(Liu and Ha [25])
Note that a constant x is also an uncertain variable with which its inverse distribution is x itself. If we take \(\xi_{1}={\mathbf {x}}\in R^{n}\) in Lemma 3, then we have the next lemma naturally.
Lemma 4
Since a VIP can be converted into an equivalent optimization problem through a regularized gap function (refer to Theorem 3.1 and Theorem 3.3 in [22]). Here we give the definition of a regularized gap function. Note that this terminology was not used by Fukushima in [22], but the function he defined was really a gap function by his later work [26].
Definition 9
(Facchinei and Pang [1])
Facchinei and Pang discussed many properties of \(g_{\alpha}\); for more details, please refer to the book [1].
Definition 10
(Facchinei and Pang [1])
Lemma 5
(Proposition 4.1 in [22])
Following Fukushima [22], we can give the following lemma.
Lemma 6
Proof
Similar to the proof of Proposition 4.1 in [22] just replace the mapping H with \(H_{\alpha}\). □
From Lemma 6, we can execute the descent methods to find a solution to the VIP related; for more details, one can refer to [22].
3 Main results
We are now in the position to reveal the main results of this paper. Invoked by the definition of stochastic variational inequality problems, we present the definition of uncertain variational inequality problems first.
Hence we now investigate under what circumstances \(f(x)\) can be evaluated in uncertainty theory.
Theorem 1
Proof
Since ξ is an uncertain variable with regular uncertainty distribution \(\Phi(t)\), its inverse uncertainty distribution \(\Phi^{-1}(\alpha)\) continuously exists from Definition 4 and Definition 5. Because \(F({\mathbf {x}},y)\) is continuous on \(R^{n}\times R\), thus \(F({\mathbf {x}},\Phi^{-1}(\alpha))\) is continuous on \(R^{n}\times[0,1]\) by extending the continuity of \(\Phi ^{-1}(\alpha)\) to the endpoint. Then, from Lemma 4 in conjunction with the continuity property of parametric integral, we know that \(f({\mathbf {x}})=E[F({\mathbf {x}},\xi)]=\int_{0}^{1}F({\mathbf {x}},\Phi^{-1}(\alpha))\, \mathrm{d}\alpha\) is continuous on \(R^{n}\). Furthermore, we arrive at the identity (9) from the analysis of matrix and when \(\nabla_{{\mathbf {x}}}F({\mathbf {x}},y)\) is continuous \(R^{n}\times R\), so is \(\nabla_{{\mathbf {x}}}f({\mathbf {x}})=\int_{0}^{1} \nabla_{\mathbf {x}}F({\mathbf {x}},\Phi^{-1}(\alpha))\, \mathrm{d}\alpha\) by the differentiable property of parametric integral. This completes the proof. □
On the other hand, when \(F({\mathbf {x}},y)\) is strictly decreasing, we have the following result.
Corollary 1
Proof
From Lemma 4, what we need to do is to replace \(\Phi^{-1}(\alpha)\) in the proof of Theorem 1 with \(\Phi ^{-1}(1-\alpha)\) since the inverse distribution of uncertain variables in these two situations is different. □
As an extension of Theorem 1 and Corollary 1 to uncertain vector, we present the following result.
Theorem 2
While Theorem 1 and Theorem 2 concern real-valued functions, it is natural for us to extend them to a vector function. Before we start to present the extension, it is necessary to introduce the expected value of an uncertain vector. We give it as follows.
Definition 11
That is, the expected value of an uncertain vector is the vector consisting of the expected value of each component.
Now we are in the position to extend Theorem 2 to vector functions.
Theorem 3
Remark 3
By introducing the expected value of an uncertain variable (vector), from Theorem 3, after transforming a UVIP into a deterministic VIP, the underlying mapping of the reformulation one is continuous and its Jacobian matrix \(\nabla_{\mathbf {x}}f({\mathbf {x}})\) can be detected whether positive definite or not. As long as \(\nabla_{\mathbf {x}}f({\mathbf {x}})\) is positive definite for all \(\mathbf {x}\in S\), we can compute the descent direction by Lemma 6. Then we set \({\mathbf {x}}^{k+1}={\mathbf {x}}^{k}+t^{k}d_{\alpha}^{k}\), \(k=0,1,2,\ldots\) , where \(d_{\alpha }^{k}\) are given by \(d_{\alpha}^{k}=H_{\alpha}({\mathbf {x}}^{k})-{\mathbf {x}}^{k}\) and \(t^{k}\in[0,1]\) which are determined from \(f({\mathbf {x}}^{k}+t^{k}d_{\alpha}^{k})= \min_{0\leqslant t\leqslant1}{f({\mathbf {x}}^{k}+td_{\alpha}^{k})}\). Thus we can find a solution to the UVIP by iterations. For more details, please refer to [22]. In fact, Facchinei and Pang [1] presented more algorithms to find a solution to a finite dimensional variational inequality problem on the basis of gap (merit) functions.
4 An application
As an application of the preceding results, let us present an example from economics in this section. We develop a supply model of newsvendors who are involved in the production of the same kind of products.
Case 1: If \(q_{i}\geqslant\frac{me_{i}}{p_{i}}+ \sum_{j\neq i}o_{ij} (\frac{me_{j}}{p_{j}}-q_{j})\), then \(U_{i}(q_{i},q_{-i})=me_{i}- \sum_{j\neq i}o_{ij}(\frac {mp_{i}e_{j}}{p_{j}}-p_{i}q_{j})-c_{i}q_{i}\).
Case 2: If \(q_{i}<\frac{me_{i}}{p_{i}}+ \sum_{j\neq i}o_{ij} (\frac{me_{j}}{p_{j}}-q_{j})\), then \(U_{i}(q_{i},q_{-i})=(p_{i}-c_{i})q_{i}\).
The optimal condition as expressed in (11) has a nice economic interpretation, which is that a newsvendor will distribute a positive amount of products to the market if the price which the customers are willing to pay is precisely equal to his/her marginal production cost. Note that the production cost function is \(c_{i}q_{i}\) and its marginal production cost is \(c_{i}\), which is the partial derivative of \(c_{i}q_{i}\) with respect to \(q_{i}\). If the newsvendor’s marginal production cost exceeds what the customers are willing to pay, then he/she will transact none with the demand market.
After finishing the procedure of transformation, we can solve variational inequality problems above by line search algorithm which is stated in Section 3.
Remark 4
We just apply our results in the newsvendor’s model. As a matter of fact, there are many problems in practice that can be casted as uncertain variational inequality problems when there exist uncertain variables (vector) such as in electricity supply network [1], user equilibrium [19] and wireless networks [20].
5 Conclusions
In this paper, we discussed a new class of variational inequality problems, namely uncertain variational inequality problems based on the uncertainty theory. It can be regarded as another extension of the classical variational inequality problems besides stochastic variational inequality problems. By introducing the expected value model of uncertain variables (vector), we converted one class of uncertain variational inequality problems into a class of deterministic variational inequality problems, which can be solved by many classical methods such as those presented in [1, 22]. Finally, we applied our results in an economic example.
Declarations
Acknowledgements
The authors gratefully acknowledge the support of the National Natural Science Foundation of China through Grant No. 61273009 and the China Scholarship Council through Grant No. 201406840039. It was accomplished during the first author’s visit to Professor Jinlu Li at Shawnee State University, USA. The authors would like to express their deep gratitude for the warm hospitality from Shawnee State University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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