Some results on the filter method for nonlinear complementary problems

Recent studies show that the filter method has good numerical performance for nonlinear complementary problems (NCPs). Their approach is to reformulate an NCP as a constrained optimization solved by filter algorithms. However, they can only prove that the iterative sequence converges to the KKT point of the constrained optimization. In this paper, we investigate the relation between the KKT point of the constrained optimization and the solution of the NCP. First, we give several sufficient conditions under which the KKT point of the constrained optimization is the solution of the NCP; second, we define regular conditions and regular point which include and generalize the previous results; third, we prove that the level sets of the objective function of the constrained optimization are bounded for a strongly monotone function or a uniform P-function; finally, we present some examples to verify the previous results.


Introduction
Consider the following nonlinear complementarity problem (NCP): find x ∈ R n such that x ≥ 0, F(x) ≥ 0, and x T F(x) = 0, (1.1) where F : R n → R n is continuously differentiable everywhere and the superscript T denotes the transpose operator. When F(x) = Mx + q, with M ∈ R n×n and q ∈ R n , the NCP becomes a linear complementarity problem. Nonlinear complementarity problems have many important applications in engineering and equilibrium modeling [10,12,18,28], and many numerical methods are developed to solve NCPs [1,6,17,29,32,38]. Based on NCP functions some researchers try to solve NCPs by reformulating them as an unconstrained optimization [5,9,11,14]. Under some assumptions, the solutions of NCPs are obtained by solving these problems. Subsequently, derivative-free algorithms for NCPs are presented [3-5, 11, 14, 17, 19, 23, 25, 37, 40].
In the last 18 years, the filter method [7,8,13,15,16,30,[33][34][35] has been regarded as an efficient constrained optimization method. The advantage of this method is that trial points are accepted if they improve the objective function or improve the constraint violation instead of a combination of those two measures defined by a merit function. Recently, some authors [21,22,27,31,36,39] have naturally reformulated the NCP as an inequality constrained optimization: subject to F j (x) ≥ 0, j ∈ {1, 2, . . . , n}, (1.2b) x j ≥ 0, j ∈ {1, 2, . . . , n}, (1.2c) and obtained good numerical performance by filter algorithms. However, they can only prove that the iterative sequence converges to the KKT point of the constrained optimization. The relation between the KKT point of constrained optimization (1.2a)-(1.2c) and the solution of NCP (1.1) has not been studied. What are the conditions for the KKT point of (1.2a)-(1.2c) to be the solution of (1.1)? This is an interesting question which should be answered.
From the above discussion, we shall study the relation between the solution of the NCP and the KKT point of (1.2a)-(1.2c) and propose several sufficient (and necessary) conditions for the KKT point of (1.2a)-(1.2c) to be the solution of the NCP. This work explains the relation between an optimization and an NCP and provides the theoretical basis of filter algorithms for the NCP. The paper is outlined as follows. In Sect. 2, we recall some definitions and basic facts; we give several sufficient conditions in Sect. 3; the definitions of the regular point and regular conditions are proposed in Sect. 4; the boundedness of level sets is discussed in Sect. 5; some numerical results are presented to verify previous results in Sect. 6.
Notation: Given F : R n → R n , the ith component function is denoted by F i (x). F (x) is the Jacobian of F at x ∈ R n . ∇F(x) = [∇F 1 (x), . . . , ∇F n (x)] denotes the transpose Jacobian of F at x. e i ∈ R n denotes the ith column of the identity matrix I n . x • y = (x 1 y 1 , . . . , x n y n ) T .

Preliminaries
In this section, we recall some background concepts and materials.

Definition 1 ([14]) A function
(2) strictly monotone if, for all x, y ∈ R n with x = y, (3) strongly monotone (with modulus ω > 0) if, for all x, y ∈ R n , Obviously, strongly monotone functions are strictly monotone, and strictly monotone functions are monotone.

Lemma 1 ([14]) For a continuously differentiable function F
Note that the converse direction in Lemma 1 (2) is not correct in general. A solution x * ∈ R n of the NCP is said to be nondegenerate if x * i + F i (x * ) > 0 for all i ∈ I, and degenerate otherwise. 14]) Assume that F : R n → R n is a continuous and strongly monotone function. Then NCPs have at most one solution.

Definition 2 ([11])
A matrix M ∈ R n×n is said to be a P 0 -matrix if all its principal minors are nonnegative.

Lemma 3 ([11]) A matrix M ∈ R n×n is a
(1) P 0 -matrix if every of its principal minors is nonnegative; (2) P-matrix if every of its principal minors is positive; has 0 as its unique solution.
It is obvious that every P-matrix is also a P 0 -matrix, and it is known that every P-matrix is an R 0 -matrix. We shall also need the following characterization of P 0 (P)-matrices.

Lemma 4 ([11])
A matrix M ∈ R n×n is a P 0 (P)-matrix if and only if, for every nonzero vector x, there exists an index i such that x i = 0 and x i (Mx) i ≥ (>)0.
for every x and y in R n with x = y, there is an index i such that (2) P-function if, for every x and y in R n with x = y, there is an index i such that (3) uniform P-function if there exists a positive constant ω > 0 such that, for every x and y in R n , there is an index i such that It is obvious that every monotone function is a P 0 -function, every strictly monotone function is a P-function, and every strongly monotone function is a uniform Pfunction. Furthermore, it is known that the Jacobian of every continuously differentiable P 0 -function is a P 0 -matrix and that if the Jacobian of a continuously differentiable function is a P-matrix for every x, then the function is a P-function. If F is affine (that is, if if M is a P-matrix (note that in the affine case, the concepts of uniform P-function and P-function coincide).

Sufficient conditions
In this paper, NCP (1.1) is transformed into the following equivalent inequality and nonnegative constrained optimization: The KKT conditions of (3.1a)-(3.1c) are where μ, ν ∈ R n are the vectors of multipliers corresponding to inequalities.
Next, we introduce some index sets: and further partition the index set C as follows: Proof Suppose that x * is a KKT point of (3.1a)-(3.1c), which implies that (3.3) holds. Suppose the contrary, i.e., (x * ) = 0. Premultiplying (3.3) by Since ∇F(x * ) is positive semidefinite, we obtain that There are four cases for μ * and ν * : The global minima of the constrained optimization is obviously a KKT point of (3.1a)-(3.1c).

Corollary 1 Suppose that the mapping F is a monotone function, then x
Proof Since the mapping F is a monotone function, the Jacobian F (x * ) of the mapping F is a positive semidefinite matrix.
Proof Suppose that x * is a KKT point of (3.1a)-(3.1c), then we know from Lemma 7 that Suppose the contrary, i.e., that x * is not the solution of the NCF, then R = ∅. There are four cases for μ * and ν * which are the same as those in Theorem 1. Without loss of generality we have that Since ∇F(x * ) is a P-matrix, for every nonzero vector x, there exists an index i such that It follows from (3.6) and (3.7) that The global minima of the constrained optimization is obviously a KKT point of (3.1a)-(3.1c).

Corollary 2 Suppose that the mapping F is a P-function, then x
Proof Since the mapping F is a P-function, the Jacobian F (x * ) of the mapping F is a Pmatrix.

Regular conditions
In this section, we give some sufficient (and necessary) conditions for the KKT point of (3.1a)-(3.1c) to be the solution of the NCP. We call these conditions regular conditions which can be considered as a generalization of previous conclusions. First, we give the definitions of regular point and regular conditions.
there exists a vector y ∈ R n such that and Moreover, a point x ∈ R n is called strictly regular if, for every vector z ∈ R n (z ≤ 0 does not hold) with there exists a vector y ∈ R n such that and . Then x * solves the NCP if and only if x * is regular (or strictly regular).
Proof If x * ∈ R n is a solution of the NCP, then R = ∅ and z = z C , and hence there is no vector z (z ≤ 0 does not hold).
Suppose that x * is regular and a KKT point of (3.1a)-(3.1c). From Lemma 7 we obtain that Without loss of generality we have that Consequently, we have for any y ∈ R n . Assume that x is not a solution of the NCF. Then R = ∅ and choose which satisfies (4.1). Since x * is regular, there exists a vector y ∈ R n such that (4.2) and (4.3) hold. With some computation, we obtain that and which contradicts (4.6). Hence, x * must be a solution of the NCP.
Suppose that x * is strictly regular and a KKT point of (3.1a)-(3.1c). Similar to the above proof, we obtain that and which contradicts (4.6). Hence, x * must be a solution of the NCP.
Remark 3 Theorem 1 is the corollary of Theorem 3. Assume that x * is a KKT point of (3.1a)-(3.1c) and is not a solution of the NCP, i.e., R = ∅. Since ∇F(x * ) is positive semidefinite, for the nonzero vector z T ∇F(x * )z ≥ 0 holds. Choose y = z, and it is easy to see that y satisfies (4.2), (4.7), and (4.3), i.e., and y T ∇F(x * )z = z T ∇F(x * )z ≥ 0. By Theorem 3, x * is a regular point and must be a solution of the NCP.
Remark 4 Theorem 2 is the corollary of Theorem 3. Assume that x * is a KKT point of (3.1a)-(3.1c) and is not a solution of the NCP, i.e., R = ∅. Since the Jacobian F (x * ) of the mapping F is a P-matrix, we have that, for the nonzero vector Choose y to be the vector whose components are all 0 except for its ith component, which is equal to z i . It is easy to see that y satisfies (4.4), (4.8), and (4.5), i.e., and y T ∇F(x * )z = z i (∇F(x * )z) i > 0. By Theorem 3, x * is a regular point and must be a solution of the NCP.

1) if and only if it is a KKT point of (3.1a)-(3.1c).
Proof Assume that x * is a KKT point of (3.1a)-(3.1c) and is not a solution of the NCP, i.e., R = ∅. By assumptions we have that, for the nonzero vector , there exists an index i ∈ R ∪ C 3 such that z i = 0 and z i (∇F(x * )z) i ≥ 0. If i ∈ C 3 , z i < 0, otherwise, z i > 0. Choose y to be the vector whose components are all 0 except for its ith component, which is equal to z i . It is easy to see that y satisfies (4.2), (4.7), and (4.3), i.e., and y T ∇F(x * )z = z i (∇F(x * )z) i ≥ 0. By Theorem 3, x * is a regular point and must be a solution of the NCP. If x * solves NCP (1.1), it is a KKT point of (3.1a)-(3.1c).

Corollary 4
Suppose that the Jacobian F (x * ) of the mapping F is a P 0 -matrix and μ * C 1 = 0, then x * solves NCP (1.

1) if and only if it is a KKT point of (3.1a)-(3.1c).
Proof Assume that x * is a KKT point of (3.1a)-(3.1c) and is not a solution of the NCP, i.e., R = ∅. Since the Jacobian F (x * ) of the mapping F is a P 0 -matrix and μ * C 1 = 0, we have that, for the nonzero vector there exists an index i ∈ R ∪ C 3 such that z i = 0 and z i (∇F(x * )z) i ≥ 0. If i ∈ C 3 , z i < 0, otherwise, z i > 0. The rest of the proof is the same as that of Corollary 3. By Theorem 3, x * is a regular point and must be a solution of the NCP. If x * solves NCP (1.1), it is a KKT point of (3.1a)-(3.1c).

Boundedness of level sets
In this section, we prove that the level sets of the objective function of (3.1a)-(3.1c) are bounded for a strongly monotone function or a uniform P-function.  Let

Theorem 4 Suppose that the mapping F is a strongly monotone function or a uniform
There are two cases to be considered.
(1) If F is a strongly monotone function, we get Due to the boundedness of the sequence y k and the continuity of However, 1 2 , which is a contradiction. (2) If F is a uniform P-function, there exists an index i 0 ∈ {1, 2, . . . , n} such that The second case is impossible because the left-hand side of the inequality is positive. Thus, Since i∈J (x k i ) 2 = 0, we obtain Similar to the proof of case (1), we get that lim k→∞ |F i 0 (x k )| = ∞, i 0 ∈ J, i.e., Remark 5 Theorem 4 implies that there exists at least one accumulation point of a sequence remaining in L(x 0 ).

Some examples
In this section, we present several examples which are tested by a filter algorithm to verify the previous results. We intend to modify a globally convergent filter algorithm [16] to solve the NCP. Consider the following optimization: There are two merit functions in the new algorithm In order to prevent the algorithm from cycling, the algorithm maintains a filter F = (θ , ) ∈ R 2 : θ ≥ θ (x 0 ) .
The search direction d k is obtained by the QP subproblem: where B k denotes the approximation of the Hessian ∇ 2 xx (x k ) of the Lagrangian function After a search direction d k has been computed, a step size α k is determined in order to obtain the next iterate x k+1 = x k + α k d k . We say that a trial point x k (α k,l ) = x k + α k,l d k is acceptable to the filter if and only if is a dwindling function. We say that a trial point where γ θ , γ ∈ (0, 1). But this could result in convergence to a feasible but non-optimal point. In order to prevent this, we change to a different sufficient reduction criterion If condition (6.7) holds, the trial point x k (α k,l ) is required to satisfy the Armijo condition x k (α k,l ) ≤ (x k ) + η m k (α k,l ), (6.8) where η ∈ (0, 1 2 ). If condition (6.7) for α k does not hold, the filter is augmented for a new iteration using the updated formula We now formally state the new filter algorithm for the NCP.

Remark 6
The global convergence of Algorithm 1 is similar to that in [16]. For details, see [16].
In the following, some numerical results are given on an HP i5 personal computer with 4G memory. The selected parameter values are: = 10 -6 , γ θ = 0.5, γ = 0.5, δ = 1, s = 3.2, s θ = 1.5, η = 0.3, τ 1 = τ 2 = 0.5, and φ(α) = α 4 3 . The computation terminates when the stopping criterion d k + θ (x k ) ≤ is satisfied. We use the Matlab function quadprog to solve the QP(x k ) subproblem. NIT and NF stand for the numbers of iterations and function evaluations, respectively. Gap stands for the absolute value of x T F(x) at the final iteration.  Table 1. Algorithm 1 can compete with Nie's filter algorithm [27]. Because the Jacobian F (x) = M of F(x) is positive definite, F is strictly monotone. By Theorem 1 or Corollary 1, x * is a solution of the NCP (a regular point).  Table 2. Algorithm 1 can compete with Su's filter algorithm [31]. Because the Jacobian F (x) = M of F(x) is approximately positive semidefinite as n → ∞, F is approximately monotone.
Because the Jacobian F (x) = M of F(x) is a P-matrix, F is a P-function. By Theorem 2 or Corollary 2, x * is a solution of the NCP (a regular point). Data and images on the convergence rate of Algorithm 1 (Example 3, n = 8) are shown in Table 4 and Fig. 1, where data1 and data2 denote x kx * and x k+1 -x * x k -x * , respectively. From Table 4 and Fig. 1 we see that which means that Algorithm 1 converges Q-superlinearly.
There are three cases: (1) The sequence {x k } generated by Algorithm 1 converges to a KKT point (0, 0, 0, 2) of (3.1a)-(3.1c) which is not a solution of the NCP. If x * = (0, 0, 0, 2), we have that is not a P 0 -matrix and F is not a P 0 -function.
(2) The sequence {x k } generated by Algorithm 1 converges to a degenerate solution ( whose eigenvalues are 11.502, 0.12753, 1.9147, and -8.5447. This indicates that it is not a positive semidefinite matrix. Because it is not a P 0 -matrix and F is not a P 0 -function. But x * is a solution of the NCP (a regular point). , whose eigenvalues are 11.835, -2.6848, 0.84972, and 1. This indicates that it is not a positive semidefinite matrix. Because D 1,2 = 6 5 2 0 = -10 < 0, it is not a P 0 -matrix and F is not a P 0 -function. But x * is a solution of the NCP (a regular point).
We note that most of the sequences converge to a degenerate solution. Although the Jacobian of F is not a P 0 -matrix and F is not a P 0 -function in the last two cases, x * are still the solutions of the NCP (regular points).    it is not a P 0 -matrix and F is not a P 0 -function. But x * is a solution of the NCP (a regular point). The results of Example 5 are given in Table 6. From Table 6 we see that Algorithm 1 performs better than Chen's algorithm [2]. Different algorithms with the same starting point converge to different solutions. The advantage of the filter method is that it has two merit functions, which makes the requirements for trial points more relaxed and easy to accept the superlinear steps.

Conclusion
In this paper, we analyze the relation between the constrained optimization reformulation and the NCP which is not involved in filter algorithms [21,22,27,31,36,39]. First, we give several sufficient conditions under which the KKT point of the constrained optimization is the solution of the NCP. Second, we define regular conditions and regular point which include and generalize the previous results. Third, we prove that the level sets of the objective function of (3.1a)-(3.1c) are bounded for a strongly monotone function or a uniform P-function. Finally, we present some examples to verify the previous results.
The above work explains the principle of the filter method for NCPs and promotes the development of the theory and algorithm. In the future, we will consider the following problems: the influence of different value functions [20,24] on the algorithm and the possibility of other conditions.