Advances in Nonlinear Analysis and OptimizationView this Special Issue
On Set-Valued Complementarity Problems
This paper investigates the set-valued complementarity problems (SVCP) which poses rather different features from those that classical complementarity problems hold, due to tthe fact that he index set is not fixed, but dependent on . While comparing the set-valued complementarity problems with the classical complementarity problems, we analyze the solution set of SVCP. Moreover, properties of merit functions for SVCP are studied, such being as level bounded and error bounded. Finally, some possible research directions are discussed.
1. Motivations and Preliminaries
The set-valued complementarity problem (SVCP) is to find such that where is a set-valued mapping. The set-valued complementarity problem plays an important role in the sensitivity analysis of complementarity problems  and economic equilibrium problems . However, there has been very little study on the set-valued complementarity problems compared to the classical complementarity problems. In fact, the SVCP (1) can be recast as follows, which is denoted by SVNCP to find , such that where and are a set-valued mapping. To see this, if letting then (1) reduces to (2). Conversely, if and , then (2) takes the form of (1).
The SVNCP given as in (2) provides an unified framework for several interesting and important problems in optimization fields described as below. (i)Nonlinear complementarity problem , which is to find , such that This corresponds to and for all . In other words, the set-valued complementarity problem reduces to the classical complementarity problem under such case. (ii)Extended linear complementarity problem [3, 4], which is to find , such that where and are a polyhedron. This corresponds to and . In particular, when , it further reduces to the horizontal linear complementarity problem and to the usual linear complementarity problem, in addition to being an identify matrix. (iii)Implicit complementarity problem , which is to find and , such that where . This can be rewritten as This is clearly an SVNCP where and . (iv)Mixed nonlinear complementarity problem, which is to find , and such that This is an SVNCP where it corresponds to . Note that the mixed nonlinear complementarity problem is a natural extension of Karush-Kuhn-Tucker (KKT) conditions for the following nonlinear programming: To see this, we first write out the KKT conditions as follows: where , , and . Then, letting , , and implies that the KKT system (10) becomes a mixed complementarity problem.
Besides the above various complementarity problems, SVNCP has a close relation with the Quasi-variational inequality, a special of the extended general variational inequalities [6, 7], and min-max programming, which is elaborated as below. (v)Quasi-variational inequality . Given a point-to-point map from to itself and a point-to-set map from into subsets of , the Quasi-variational inequality QVI is to find a vector , such that It is well-known that QVI reduces to the classical nonlinear complementarity problem when is independent of , say, for all . Now, let us explain why it is related to SVNCP. To this end, given , we define , and let Clearly, which says by taking in (12). Note that because . Next, we will claim that for all . It is enough to consider the case where . Under such case, by taking in (12) with being an arbitrarily positive scalar, we have . Since can be made sufficiently large, it implies that . Thus, we obtain . In summary, under such case, QVI becomes which is an SVNCP. (vi)Min-max programming , which is to solve the following problem: where is a continuously differentiable function, and is a compact subset in . First, we define . Although is not necessarily Frechet differentiable, it is directional differentiable (even semismooth), see . Now, let us check the first-order necessary conditions for problem (15). In fact, if is a local minimizer of (15), then which is equivalent to where means the active set at , that is, . At our first glance, the formula (17) is not related to SVNCP. Nonetheless, we will show that if is convex and the function is concave over , then the first-order necessary conditions form an SVNCP, see below proposition.
Proposition 1. Let be nonempty, compact, and convex set in . Suppose that, for each , the function is concave over . If is a local optimal solution of (15), then there exists , such that
Proof. Note first that for each the inner problem is a concave optimization problem, since is concave and is convex. This ensures that , which denotes the optimal solution set of (19), is convex as well. Now we claim that the function is concave over . Indeed, for and , we have where we use the fact that (since is convex) and for all . On the other hand, applying the Min-Max Theorem [10, Corollary ] to (17) yields Hence, for arbitrary , we can find , such that that is, In particular, plugging in in (24) implies Since is bounded and is closed, we can assume, without loss of generality, that as . Thus, taking the limit in (25) gives Now, let . It follows from (24) that which implies that by letting , and hence for all , that is, . This together with (26) means that . Thus, (18) holds.
From all the above, we have seen that SVNCP given as in (2) covers a range of optimization problems. Therefore, in this paper, we mainly focus on SVNCP. Due to its equivalence to SVCP (1), our analysis and results for SVNCP can be carried over to SVCP (1). This paper is organized as follows. In Section 1, connection between SVNCP and various optimization problems is introduced. We recall some background materials in Section 2. Besides comparing the set-valued complementarity problems with the classical complementarity problems, we analyze the solution set of SVCP in Section 3. Moreover, properties of merit functions for SVCP are studied in Section 4, such as level bounded and error bound. Finally, some possible research directions are discussed.
A few words about the notations used throughout the paper. For any , the inner product is denoted by or . We write (or ) if (or ) for all . Let be the vector with all components being , and let be the -row of identity matrix. Denote . While SVNCP() meaning the set-valued nonlinear complementary problem (2), SVLCP() denotes the linear case, that is, , where and . For a continuously differentiable function , we denote the Jacobian matrix of partial derivatives of at with respect to by , whereas the transposed Jacobian is denoted by . For a mapping , define Given a set-valued mapping , define We say that is outer semicontinuous at , if and inner semicontinuous at if We say that is continuous at if it is both outer semicontinuous and inner semicontinuous at . For more details about these functions, please refer to [9, 11]. Throughout this paper, we always assume that the set-valued mapping is closed valued; that is, is closed for all [11, Chapter 1].
2. Focus on SVLCP
It is well known that various matrix classes paly different roles in the theory of linear complementarity problem, such as -matrix, -matrix, -matrix, and -matrix, see [1, 12] for more details. Here we recall some of them which will be needed in the subsequent analysis.
Definition 2. A matrix is said to be an -matrix if there exists , such that
Note that is an -matrix, if and only if the classical linear complementarity problem LCP is feasible for all , see [12, Prop. 3.1.5]. Moreover, the above condition in Definition 2 is equivalent to see [13, Remark 2.2]. However, such equivalence fails to hold for its corresponding cases in set-valued complementarity problem. In other words, is not equivalent to It is clear that (34) implies (35). But, the converse implication does not hold, which is illustrated in Example 3.
Example 3. Let
Proof. We only need to show (35)(34). Let , and denote by the th row of . Hence . With this, suppose is an arbitrary but fixed point, we know that for any , there exists such that . Since is inner semi-continuous, for any , there exists satisfying . This implies Then, taking the lower limit yields where the equality follows from the continuity of , which is ensured by the continuity of . Because is arbitrary, and is an arbitrary sequence converging to , we obtain which says is lower semi-continuous. This further implies that is, If satisfies (35), then which is equivalent to On the other hand, , and for . By taking small enough, we know satisfies (34). Thus, the proof is complete.
There is another point worthy of pointing out. We mentioned that the classical linear complementarity problem LCP is feasible for all if and only if is a -matrix; that is, there exists , such that Is there any analogous result in the set-valued set? Yes, we have an answer for it in Theorem 5 below.
Theorem 5. Consider the set-valued linear complementarity problem SVLCP. If there exists , such that then SVLCP is feasible for all being bounded from below.
Proof. Let be any mapping from to being bounded from below, that is, there exists , such that . Suppose that and satisfy (45), which means
Then, for any , we have . In particular, we observe the following: (1)if taking , then there exists , such that ;(2)if taking , then there exists with , such that .
Repeating the above process yields a sequence , such that and . Since , it ensures the existence of , such that . Taking large enough to satisfy gives . Then, it implies that and hence which says is a feasible point of SVLCP.
Definition 6. A matrix is said to be a -matrix if all its principal minors are positive, or equivalently [12, Theorem 3.3.4],
From [12, Corollary 3.3.5], we know that every -matrix is an -matrix. In other words, if satisfies (49), then the following system is solvable: Their respective corresponding conditions in set-valued complementarity problem are Example 7 shows that the aforementioned implication is not valid as well in set-valued complementarity problem.
Example 7. Let
For , we have , and hence . For , we know (since ) which says , and hence . Therefore, condition (51) is satisfied. But, condition (52) fails to hold because or . Hence, implies that or , which contradicts with .
Definition 8. A matrix is said to be semimonotone if
For the classical linear complementarity problem, we know that is semimonotone if and only if LCP with has a unique solution (zero solution), see [12, Theorem 3.9.3]. One may wonder whether such fact still holds in set-valued case. Before answering it, we need to know how to generalize concept of semi-monotonicity to its corresponding definition in the set-valued case.
Definition 9. The set of matrices is said to be (a)strongly semimonotone if for any nonzero , (b)weakly semimonotone if for any nonzero ,
Unlike the classical linear complementarity problem case, here are parallel results regarding set-valued linear complementarity problem which strong (weak) semi-monotonicity plays in.
Theorem 10. For the SVLCP, the following statements hold. (a)If the set of matrices is strongly semi-monotone, then for any positive mapping , that is, for all , SVLCP has zero as its unique solution.(b)If SVLCP with has zero as its unique solution, then the set of matrices is weakly semi-monotone.
Proof. (a) It is clear that, for any positive mapping , is a solution of SVLCP(). Suppose there is another nonzero solution , that is, , such that
It follows from (55) that there exists , such that and , and hence , which contradicts condition (57).
(b) Suppose is not weakly semi-monotone. Then, there exists a nonzero , for all , for all . Choose . Let for all and Therefore, for all . According to the above construction, we have that is, the nonzero vector is a solution of SVLCP, which is a contradiction.
Example 11. Let
For any nonzero , we have . If we plug in , by a simple calculation, satisfies which means that SVLCP has a nonzero solution. We also notice that the set-valued mapping is even continuous in Example 11.
So far, we have seen some major difference between the classical complementarity problem and set-valued complementarity problem. Such phenomenon undoubtedly confirms that it is an interesting, important, and challenging task to study the set-valued complementarity problem, which, to some extent, is the main motivation of this paper.
To close this section, we introduce some other concepts which will be used later too. A function is level bounded, if the level set is bounded for all . The metric projection of to a closed convex subset is denoted by , that is, . The distance function is defined as .
3. Properties of Solution Sets
Recently, many authors study other classes of complementarity problems, in which another type of vector is involved, for example, the stochastic complementarity problem [14–17], to find , such that where is a random vector in a given probability space and the semi-infinite complementarity problem  to find , such that which we denote it by SINCP. In addition, the authors introduce the following two complementarity problems in  to find , such that where These two problems are denoted by NCP and NCP, respectively. Is there any relationship among their solutions sets? In order to further describing such relationship, we adapt the following notations:(i)SOL() means the solution set of SVNCP, (ii)SOL() means the solution set of SVLCP, (iii) means the solution set of SINCP, (iv)SOL means the solution set of NCP(), (v)SOL means the solution set of NCP().
Besides, for the purpose of comparison, we restrict that is fixed; that is, there exists a subset in , such that for all .
It is easy to see that the solution set of SINCP is , but that of SVNCP is , where . Hence, the solution set of SINCP() is included in that of SVNCP(). In other words, we have The inclusion (67) can be strict as shown in Example 12.
Example 12. Let and . Then, we can verify that , whereas .
Example 13. and . (a)Let and . Then, , .(b)Let and . Then, and .
Example 14. and . (a)Let and . Then, and .(b)Let and . Then, and .
Similarly, Example 15 shows that the solution set of NCP and NCP cannot be included each other.
Example 15. and . (a)Let and . Then, and .(b)Let and . Then, and .
In spite of these, we obtain some results which describe the relationship among them.
Theorem 16. Let for all . Then, we have (a);(b);(c).
For further characterizing the solution sets, we recall that for a set-valued mapping , its inverse mapping (see [9, Chapter 5]) is defined as
Theorem 17. For SVNCP(), we have
Proof. In fact, the desired result follows from where the second equality is due to the definition of inverse mapping given as above.
4. Merit Functions for SVNCP and SVLCP
It is well known that one of the important approaches for solving the complementarity problems is to transfer it to a system of equations or an unconstrained optimization via NCP functions or merit functions. Hence, we turn our attention in this section to address merit functions for SVNCP and SVLCP.
A function is called an NCP function if it satisfies For example, the natural residual and the Fischer-Burmeister function are popular NCP-functions. Please also refer to  for a detailed survey on the existing NCP-functions. In addition, a real-valued function is called a merit (or residual) function for a complementarity problem if for all and if and only if is a solution of the complementarity problem. Given an NCP-function , we define Then, it is not hard to verify that the function given by is a merit function for SVNCP(). Note that the merit function (74) is rather different from the traditional one, because the index set is not a fixed set, but dependent on . We say that a merit function has a global error bound with a modulus if For more information about the error bound, please see  which is an excellent survey paper regarding the issue of error bounds.
Theorem 18. Assume that there exists a set , such that for all , and that for each , is a global error bound of with the modulus , that is, In addition, if then provides a global error bound for SVNCP() with the modulus .
Proof. Noticing that if for all , then It then follows from Theorem 17 that Therefore, Thus, the proof is complete.
One may ask when condition (77) is satisfied? Indeed, the condition (77) is satisfied if (i) is a finite set; (ii) where is continuous, and for each the matrix is a -matrix. In this case the modulus takes an explicitly formula, that is, see [21, 22]. Hence, we see that is well defined because is continuous, and is compact.
For simplification of notations, we write instead of . We now introduce the following definitions which are similar to (29):
Definition 19. For SVLCP, the set of matrices is said to have the limit- property if
In the case of a linear complementarity problem, that is, is a fixed single-point set, Definition 19 coincides with that of -matrix.
Theorem 20. For SVLCP, suppose that there exists a bounded set , such that for all , and and are continuous on . If the set of matrices has the limit- property, then the merit function is level bounded.
Proof. We argue this result by contradiction. Suppose there exists a sequence satisfying , and is bounded. Then, where we assume the minimizer is attained at , whose existence is ensured by the compactness of , since is closed and is bounded. Taking a subsequence if necessary, we can assume that and are both convergent in which and represent their corresponding limit point. Thus, we have Now, taking the limit in (85) yields where we have used the fact that , because is continuous, and is bounded. This contradicts (84) since is a nonzero vector.
Note that the condition (84) is equivalent to which is also equivalent to saying that each matrix for is a -matrix.
Theorem 21. For SVLCP, suppose that there exists a compact set , such that for all , and and are continuous on . If is level bounded, then the following implication holds
Proof. Suppose that there exist a nonzero vector , and , such that
Similar to the argument as in Theorem 5, there exists a sequence with and . Hence,
Next, we proceed the arguments by discussing the following two cases.
Case 1. For , we have from (90). Since is finite due to the compactness of and the continuity of , for sufficiently large. Therefore, we obtain
Case 2. For , by a simple calculation, we have where the inequality in the latter case comes from the fact that . Thus, This contradicts the level boundedness of since .
The above conclusion is equivalent to say that for each , the matrix is a -matrix. Finally, let us discuss a special case where the set-valued mapping has an explicit form, for example, , where and . Then, the solution set can be further characterized.
Theorem 22. If , then where is defined as and .
Proof. Noting that the problem (2) is to find and , such that namely, to find and satisfying In other words, Then, the desired result follows.
The foregoing result indicates that the set-valued complementarity problem is different from the classical complementarity problem, since it restricts that some components of the solution must be positive or zero, which is not required in the classical complementarity problems.
Moreover, the set-valued complementarity problem can be further reformulated to be an equation, that is, finding and to satisfy the following equation where . Note that when is a closed convex set, then