ARTÍCULO ORIGINAL

doi:10.17230/ingciencia.11.21.1

Least Change Secant Update Methods for Nonlinear Complementarity Problem

Métodos secantes de cambio mínimo para el problema de complementariedad no lineal

Favián Arenas A.¹, Héctor J. Martínez² and Rosana Pérez M.³ 

1 Universidad del Cauca, Popayán, Colombia, farenas@unicauca.edu.co.edu.co.

2 Universidad del Valle, Cali, Colombia, hector.martinez@correounivalle.edu.co.

3 Universidad del Cauca, Popayán, Colombia, rosana@unicauca.edu.co.

Received: 16-12-2013 | Acepted: 20-05-2014 | Online: 30-01-2015

MSC: 90C30, 90C33, 90C53

Abstract

In this work, we introduce a family of Least Change Secant Update Methods for solving Nonlinear Complementarity Problems based on its reformulation as a nonsmooth system using the one-parametric class of nonlinear complementarity functions introduced by Kanzow and Kleinmichel. We prove local and superlinear convergence for the algorithms. Some numerical experiments show a good performance of this algorithm.

Key words: nonsmooth systems; nonlinear complementarity problems; generalized Jacobian; quasi-Newton methods; least change secant update methods; local convergence; superlinear convergence

Resumen

En este trabajo generamos una familia de métodos secante de cambio mínimo para resolver Problemas de Complementariedad no Lineal vía su reformulación como un sistema de ecuaciones no lineales no diferenciable usando una clase de funciones de complementariedad propuesta por Kanzow and Kleinmichel. Bajo ciertas hipótesis demostramos que esta familia proporciona algoritmos local y superlinealmente convergentes. Experimentos numéricos preliminares demuestran un buen desempeño de los algoritmos propuestos.

Palabras clave: sistemas no diferenciables; complementariedad no lineal; métodos cuasi-Newton

1 Introduction

Here,  y ≥ 0 for  y  ⁿ means  y_i ≥ 0 for all  i=1,…,n. The third condition in (1) requires that the vectors  x and  F(x) are orthogonal; for this reason, it is called complementarity condition.

The NCP arises in many applications such as Friction Mechanical Contact problems [1], Structural Mechanics Design problems, Lubrication Elasto-hydrodynamic problems [2], Traffic Equilibrium problems [3], as well as problems related to Economic Equilibrium Models [4]. The importance of NCP in the areas of Physics, Engineering and Economics is due to the fact that the concept of complementarity is synonymous with the notion of system in equilibrium. In recent years, various techniques have been studied to solve the NCP, one of which is to reformulate it as a nonsmooth system of nonlinear equations by using special functions called complementarity functions [5]. A function  φ:2→  such that 

is called a complementarity function.

Geometrically, the equivalence (2) means that the trace of the function  φ obtained by the intersection with the  xy plane is the curve formed by the positive semiaxes  x and  y, which is not differentiable at (0,0). This lack of smoothness on the curve imply the nondifferentiability of the function  φ.

In order to reformulate the NCP as a system of nonlinear equations, it is necessary to consider a complementarity function  φ and to define  Φ:ⁿ→ ⁿ by 

then it follows from lack of smoothness of  φ that the nonlinear system of equations 

is nonsmooth. From the definition of a complementarity function (2) it follows that a vector x_* solves the system (4), if and only if, x_* solves the NCP. Different algorithms have been proposed for solving the reformulation of the NCP by a nonsmooth system of nonlinear equations (4) like nonsmooth Newton methods [6], nonsmooth quasi-Newton methods [7],[8],[9], among others [10], [11],[12],[13].

There are many complementarity functions, but the most used has been the minimum function [14] and the Fischer-Burmeister function[15], defined respectively by

The minimum function is nonsmooth at the points of the form  (a,a), while the Fischer-Burmeister function is not nonsmooth at  (0,0). In 1998, Kanzow and Kleinmichel [15] introduced an one-parametric class of complementarity functions  φ_λ defined by 

where  λ (0,4) and which we will refer to throughout this work as Kanzow function. This function is nonsmooth at  (0,0). For any other vector in  ², the gradient vector of  φ_λ is defined by 

In [16], the author makes a carefully analysis of this function and deduces some important bounds that we will use later. Moreover, In the special case  λ = 2, the function  φ_λ reduces to theFischer-Burmeister function, whereas in the limiting case  λ→ 0, the function  φ_λ becomes a multiple of the minimum function. In what follows, we denote by Φ_λ the function defined in (3) and obtained by the complementarity function  φ_λ.

In this work, we propose a nonsmooth quasi Newton method for solving the NCP using the system  Φ_λ(x)=0; for this method, we prove local convergence. Moreover, we introduce a family of least change secant update for solving the NCP based on the nonsmooth system of equations  Φ_λ(x)=0 and, for these family, we prove local and superlinear convergence under suitable assumptions.

We organize this paper as follows. In Section 2, we reformulate the NCP as a nonsmooth system of equations using the  Φ_λ function and we characterize a subset of the generalized Jacobian of  Φ_λ in  x. In the first part of Section 3, we propose a new algorithm quasi-Newton for solving the nonsmooth system of nonlinear equations  Φ_λ (x)=0 and, for this method, we develop the local convergence theory. In the second part, we introduce a family of least change secant update methods following the theory developed in [17] for this type of methods. We prove, under suitable assumptions, local and superlinear convergence. In Section 4, we analyze numerically, the local performance of the algorithms introduced in the last section, for which we use 8 test problems proposed in [14],[18]. Four of this are applications problems to Economic Equilibrium and Game Theory. Finally, Section 5 contains some remarks on what we have done in this paper and present possibilities for future works.

2 Reformulation of NCP using the Kanzow function

In this section, we reformulate the NCP as a nonsmooth system of equations and from the definition of the generalized Jacobian given in [19], we construct a subset of matrices of the generalized Jacobian of  Φ_λ at  x. Then we show that this subset at a solution of the system  Φ_λ(x) = 0 is a compact set.

Our reformulation of NCP as a system of equations is based on the Kanzow complementarity function  φ_λ defined by (6) and the  Φ_λ function defined in the last section. Exploiting (2) it is readily seen that the NCP is equivalent to the following system of nonsmooth equations 

The most popular method for solving a differentiable system of nonlinear equations  G(x)=0 is Newton's method [8], which require calculating, at each iteration, the Jacobian matrix of  G. There are situations where the derivatives of  G are not available, or are difficult to calculate. For this cases, a less expensive alternative and widely used for solving  G(x)=0 are the quasi-Newton methods [8] which use, at each iteration, a matrix approximation to the Jacobian matrix. Among the latter are the so-called least change secant update methods [10] , which form a family characterized by the fact that, at each iteration, the Jacobian approximation satisfies a secant equation [10] with a minimum variation property relative to some matrix norm. The price of using an approximation to the Jacobian Matrix is reflected in the decrease of the speed of convergence of the respective quasi-Newton method.

When a function is not differentiable as in the case of the function  Φ_λ, the term ''Jacobian matrix'' does not make sense. Fortunately, Frank H. Clarke introduced the concept of Generalized Jacobian that extends the matrix Jacobian concept for some non-differentiable functions [19]. Let  F:ⁿ→ ⁿ be a locally Lipschitzian function. The Generalized Jacobian of  F at  x is the set given by

Since the Kanzow function  φ_λ is locally Lipschitz continuous [16], so is the  Φ_λ function. Thus, the Generalized Jacobian of  Φ_λ(x) exists. In order to build matrices in this set, we consider a sequence of vectors in ⁿ, {y_k}, which converges to x and such that  Φ^'_λ (y_k) exists, then we show that  exists. To classify the indices of the components of  x, we define the set 

The sequence¹ that we will use is

where  {ε_k} is a sequence of positive numbers such that ε_k=0 and the vector  z is chosen such that  z_i ≠ 0 where  i β . Obviously  y_k converges to  x when  k→ ∞. To analyze the differentiability of  Φ_λ in  y_k, we consider two cases. If  i β then  x_i ≠ 0 or  F_i(x) ≠ 0 , by the continuity of  F_i, we can assume  ε_k so small that   ≠ 0 or  F_i(x) ≠ 0, for which  Φ_λ is differentiable at  y_k. If  i β, the  z_i ≠ 0; therefore,   ≠ 0 , which is sufficient for  Φ_λ to be differentiable at  y_k.

By differentiability of  Φ_λ at  y_k, the Jacobian matrix of  Φ_λ at  y_k, exist and its  i th row is given by 

with  χ and  ψ defined by (7) and  {e₁,…,e_n} is the canonical basis of  ⁿ.

For calculating the Φ'_λ (y_k), we consider two cases: If  i β, by continuity of  i th row of  Φ'_λ ( y_k) we have ∇φ λ (y,F_i(y_k))^T is  [H]i where [H]i = ( χ(x_i,F_i(x)) −1 )  + (ψ(x_i,F_i(x)) −1 ) ∇F_i(x)^T. If  i β, from (11), we have 

further, by the Taylor's theorem,

where  ζ^k → x when  k→ ∞. Substituting (12) and (13) in the  i th row of  Φ'_λ (y_k), i β, we obtain 

with[H]i =  ( χ(z_i ,∇F_i(x)^Tz) −1)  + ( ψ(z_i ,∇F_i(x)^Tz) −1 )∇F_i(x)^T. Then, for all  i=1,…,n, the limit when  k→ ∞ of each row of  Φ_λ′(y_k) exist. Therefore,

where 

Because there is an uncountable of ways of choosing the vector  z, we have an uncountable set of matrices  H in  ∂Φ_λ(x) which can be calculated by the above procedure.

Now, we consider the particular case where  x* is a solution of  Φ_λ (x) = 0. If there is any index  i such that  x_i=F_i(x*)=0, then  x* is called a degenerate solution. We will denote the matrices (14) in  x* by  H_*(z). The set of these matrices we will call  . Clearly, for each  z  ⁿ, there is a matrix  H_*(z). Thus,   is an infinite set and further it is a compact set. To verify the compactness, it is sufficient to demonstrate that it is closed, since   ⊆ ∂Φ(x*), which is compact [19], [16].

3 Algorithm and convergence theory

In the first part of this section, we propose a new quasi-Newton algorithm for solving the system  Φ_λ (x)=0 and we develop the local convergence theory for this method. In the second part, we develop a family of least change secant update methods, following [17]. For these family, we prove local and superlinear convergence under suitable assumptions. The following algorithm is the basic quasi-Newton algorithm applied to  Φ_λ (x)=0.

Algorithm 1 Given  x₀ an initial approximation to the solution of the problem and  λ (0,4), compute  x_k+1=x_k−B_k⁻¹Φ_λ (x_k), for  k=1,2,…, where 

Here  {e₁, …,e_n } is a canonical basis of  ⁿ, the matrix  A_k is an approximation of the Jacobian matrix of  F at  x_k (to see Section 6) and z^{k  n} is such that   ≠ 0, if   = F_i(x^k) = 0.

Under the following assumptions, we will prove that the sequence generated by the basic quasi-Newton Algorithm 1is well define and converges linearly to a solution of  Φ_λ (x)=0.

3.1 Local assumptions

H1. There is  x*  ⁿ such that Φ_λ (x*) = 0.

H2. The jacobian matrix of  F is Lipschitz continuous (with constant γ) in a neighborhood of  x*  ⁿ.

H3. The matrices of the set  Z* are nonsingular. From assumption H3 and by the compactness of  , we have that there is a constant  μ such that for all  H_*(z)  ,

3.2 A local convergence theory

The following two Lemmas prepare the ''Theorem of the two neighborhoods''.

Lemma 1 Let  F:ⁿ→ ⁿ, F C¹, such that its Jacobian matrix verifies H2, the matrices  H and  B defined by (14) and (15), respectively, and given positive constants  ϵ and  δ. Then, for each  x  (x* ; ϵ) and  A  (F′(x*) ;δ), there exists a positive constant  θ such that 

See proof in [16].

Lemma 2 Let  r (0,1) and  B the matrix defined by (15). There exist positive constants  ϵ₀ and  δ₀ such that, 

is well defined, and satisfies

Let  r (0,1), > 0 and  δ₀ > 0 such as and  δ₀< , where  ω is the constant (17),  η is the Lipschitz constant of  ∇φ_λ given by [16] and  τ and  μ are defined by Lemma 3.1. We take  x  (x* ; ), A  (F′(x*) ;δ₀),B the matrix associated to  A by the rule (15),  H_* associated to  F′(x*) for the same rule and  H defined by (3.8).

To prove that   is well defined, we must show that  B⁻¹ exist. For this, we consider the inequality 

By the continuity of  F, for all   > 0 exist  > 0 such that , 

We consider the two possibilities for this maximum:

The second part of the proof is to show (19). For this, we subtract  x* in (18), we apply   ·_∞ and perform some algebraic manipulations.

then we obtain,

On the other hand, for  H ∂Φ_λ(x).

The following theorem is analogous to the theorem of the two neighborhoods of differentiable case [10], which guarantees linear convergence of the proposed algorithm. The name of the two neighborhoods is due to, in its assumptions, it requires two neighborhoods, one for the solution which should be the starting point, and the other for the Jacobian matrix of  F at the solution which should be the initial approach.

Theorem 1 Let H1-H3 be verified and let  r (0,1), then there exist positive constants  ϵ₁ and  δ₁ such that, if  x₀−x*_∞ ≤  ϵ₁ and  A_k−F′(x*)_∞ ≤  δ₁, for all  k, then the sequence {x_k} generated by  x_k+1=x_k−B_k⁻¹Φ_λ(x_k), with  B_k the matrix whose rows are defined by (15) is well defined, converges to x* and satisfies 

We consider the function (18). Thus, 

Given  r  (0,1), let  ϵ₁  (0,ϵ₀) and  δ₁  (0, δ₀) where  ϵ₀ and  δ₀ are the positive constants given by the Lema 3.2. We will use induction on  k in the proof of this theorem.

• For k= 0. If x₀−x*∞ ≤ ϵ1 ≤ ϵ₀ andA₀−F′(x*)_∞ ≤ δ₁ ≤ δ₀ then x₁ = (x₀,A₀) is well defined and satisfies 

  

• Induction hypotheses: we assume that for  k = m−1,

  

  then  x_m=x_m−1Φ_λ(x_m−1), is well define, and 

  

  Given that x₀−x_*∞ < ϵ₁, then 

  

  and from the assumption  A_m−F′(x*)_∞ ≤  δ₁, we have for the Lemma 3.2 that  x_m+1 is well defined and satisfies 

  

Therefore, we conclude that (28) is true for all  k=0,1,….

We observe that in the proof of Theorem 3.1 we used the infinity norm. Therefore, if  e_k = x_k−x* is the error related to any other norm, then  e_k ≤ α r^ke₀, where  α is a positive constant that does not depend on  k, and  r is as in Theorem 3.1.

Among the standard theorems of thequasi Newton theory to systems of nonlinear equations is the theorem known as Dennis-Moré condition [21] which gives a sufficient condition for superlinear convergence. The following theorem is analogous to the theorem just mentioned and it will be useful in the next section to prove superlinear convergence of Algorithm 1. In his proof, we use   · =   · ∞ but, we recall that superlinear convergence results are norm-independent.

Theorem 2 Let H1-H3 be verified and that, for some  x₀, the sequence  {x_k} generated by  x_k+1=x_k−Φ_λ(x_k) converges to  x*, with  B_k given by (15) and  H_* = H(x_*). If

where  S_k=x_k+1−x_k, then the sequence  {x_k} converges superlinearly to  x*.

As we mentioned above,  Φ_λ is semismooth in  x*, thus 

By the limit definition, in particular for  , there exists  ϵ > 0, such that if  x_k−x* < ϵ then, 

Applying a norm and the triangle inequality, we obtain

thereby

From (3.2), (37) and (16)

Therefore,

i.e., the sequence {x_k} converges superlinearly to x*.

4 Least change secant update family for solving  Φ_λ (x)=0. 

The quasi-Newton methods differ in how to update the matrix  A_k at each iteration. Among the ''practical'' quasi-Newton algorithm are those that are called least change secant methods, in which the updating of  A_k, named  A_k+1, must satisfy the secant equation [10] given by  A_k+1(x_k+1−x_k) = F(x_k+1)−F(x_k) and its change (measured in some norm) relative to A_k must be minimum. Requiring that secant equation and a minimum change are satisfied between two consecutive updates makes the sequence of matrices  {A_k } have a property known as bounded deterioration [10] [8], which guarantees that the matrices of the sequence remain in a neighborhood of  F′(x*). This is essential to demonstrate local and linear convergence. Thus, at each iteration of the least change secant algorithm, the vectors  x_k and  x_k+1 defined the set  V by

Given that we need the matrix in  V ''nearest'' A_k, it is natural to think of the orthogonal projection of this matrix on  V, named  P_V (A_k) = P_xk,x_k+1 (A_k). Given that 

Thus, A_k+1 =  P_V(A_k).

Different least change secant updates are obtained by varying the matrix norm ^n×n or the subspace  S, producing the family of least change secant update methods. For example, ''Good'' Broyden update,''Bad'' Broyden update [22], Schubert update [23] andSparse Schubert update [23].

Algorithm 2 Assume that  x₀ and  A₀ are arbitrary. x_k+1 and  A_k+1 for  k=0,1,..., are generated as follows: 

In order to develop the theory of convergence of the least change secant update methods generated by Algorithm 2we will assume an additional Assumption.

H4. For all x,z in a neighborhood x*, there are  A  V(x,z) and  α₁ >  0 such that 

where  σ(x,z)=max{x−x*,z−x*}.

5 Additional convergence results

In the next lemma, we show that a matrix generated using the rule (45) to update the matrix  A_k may deteriorate, but in a controlled way.

Lemma 5  Let H1-H4 be verified and let  A₊ be the orthogonal projection of  A on the set  V(x,z) and   the orthogonal projection of  F′(x) on  V(x,z) then  A₊ −F′(x) ≤ A−F′(x)+α₂σ(x,z), where  α₂ >  0 and  σ(x,z)=max{x−x*,z−x*}.

Lemma 5. 2 Let assumptions H1-H4 be verified. Then there exists  c > 0 such that 

whenever the vectors  x and  y belong to a neighborhood of  x*, with y−x* ≤ x−x* and the matrix  A in a neighborhood of  F′(x*).

The two previous lemmas (see proofs in [16] and [24], respectively) and assumptions H1-H4 are central to ensuring the following result.

Theorem 5. 1 Let H1 - H4 be verified and that the sequence  {A_k} is generated by (45). Given  r (0,1), there exist positive constants   and   such that if  x₀−x* ≤   and  A₀ −F′(x*) ≤  , the sequence  {x_k} generated by  x_k+1 = x_k−Φ(x_k) is well defined, converges to  x* and, for all  k=1,2,…, 

Let   (0,δ₁). We can choose   (0,ϵ₁) with 

1. If   <  ϵ₁, the constant   which depends on  ϵ₁ because   >  δ₁ − · 

2. If   >  ϵ₁, we can choose any value in  (0,ϵ₁) as  .

1. For  k=0, if x₀−x* ≤  < ϵ₁ and A₀−F′(x*) ≤  < δ₁ then from Lemma 3.2,  x₁=x₀−B₀⁻¹Φ(x₀) is well defined and satisfies  x₁−x* ≤ rx₀−x*.

2. We assume that for  k=m−1, if x_m−1−x* ≤  and  A_m−1−F′(x*) ≤ ,  x_m = x_m−1−B₀⁻¹Φ(x_m−1) is well defined, and 

3. From (51) and (47) 

4. From (47), 

where

and  c_i is the constant of Lemma 5.2. Thus, from Lemma 3.2,  x_m+1 is well defined and satisfies

Lemma 5. 3 We assume the Assumptions H1 - H4 are verified and let the sequence  { A_k} be generated by (45). There are positive constants   y   such that, if x₀−x* ≤  and  A₀−F′(x*) ≤ , and the sequence {x_k} is generated by   x_k+1x_k−B_k⁻¹Φ( x_k), with  B_k defined by (15), then 

With this result (See proof in [16]), we can derive sufficient condition to have super linear convergence as shown by the next Theorem.

Theorem 5.2. Let the Assumptions H1-H4 and let the sequences  {x_k} and  { A_k} be generated by the Algorithm 2 and  x_k=x*. If

The proof follows in a straightforward way from Theorems 4.2 and 4.3.

6 Some numerical experiments

In this section, we analyze numerically the local behavior of the family of least change secant update methods introduced in Section 2. For this, we compare our algorithms with the Generalized Newton method proposed in [6]. The Algorithm 3 is a Generalized Newton type method. It uses at each iteration, the matrix  Hk (defined in the previous section) which uses the Jacobian Matrix of  F.

The Algorithm 4, which is a least secant change type method is based on the Algorithm 2 that we proposed at Section 3. For updating the matrix  A_k, in each iteration, we use the four formulas: ''Good'' and ''bad'' Broyden, Schubert and Sparse Schubert, whereby we have four versions of Algorithm 4.

For all the tests, we use the software MATLAB®. We use 8 test problems for nonlinear complementarity, four of which we chose from a list proposed, [14] and which are considered ''hard problem''. These are Kojima-Shindo (application to Economic Equilibrium [25]), Kojima-Josephy, Nash-Cornout (application to the Game Theory harker) and Modified Mathiesen (application to Walrasian Economic Equilibrium [26]) problems. We generate the four remaining problems like in [27]; for this, we define  F : ⁿ→ⁿ by, 

We use the same stopping criteria proposed in [28]. We choose the parameter  λ:=λ_min for using in the Algorithms 3 and 4 as follows

We vary  λ in the interval  (0,4) from  λ = 10⁻³ to  λ = 3.999 with increments of  10⁻³.

We use the generalized Newton method (Algorithm 3) with each of these values of  λ.

We called  λ_min to the value of  λ for which the generalized Newton converges in fewer iterations.

Table 1 presents the results of our numerical tests. Its columns contains the following information: Problem means the problem name, n is the dimension problem, λ_min is the value of λ for which the Algorithm 3 converges in fewer iterations. We also include a column with the algorithm and the secant update used. Thus,  GN means Generalized Newton;  SSU, BBU, SU and  GBU means Algorithm 4 with the Sparse Schubert Update, ''Bad'' Broyden Update, Schubert Update and the ''Good'' Broyden Update, respectively. A − sign means divergence.

From Table 1, we observe that, for these preliminary numerical tests, the Algorithm 4 that we proposed for solving the NCP has good local behavior. In particular, we highlight the Modified Mathiesen problem, in which each method converges with the same number of iterations but to different solutions to the problem.

7 Conclusions

In this paper, we propose a quasi-Newton method for solving the nonlinear complementarity problem when this is reformulated as a nonlinear system of equations. This method can be useful when the derivatives of the system are very expensive or difficult to obtain. Moreover, we generated a family of least change secant update methods that, under certain hypotheses, converge local and superlinearly to the solution of the problem.

Some numerical experiments shows a good local performance of this algorithm, but it is necessary more numerical tests using others well-known LCSU methods such as Column Updating method [29] Inverse Column Updating method [30]. It is necessary to incorporate a globalization strategy to the algorithm proposed and to develop theoretical and numerical analysis of the global algorithm.

Acknowledgements

Footnotes:

1 We consider the same sequence of [20] which is used for the theoretical developments with the Fischer function.

2 Let  · a matrix norm induced in ^{n ×n}, A and  C  ^n×n. If  C is non singular and  I_n−C⁻¹A <  1 then  A is non singular and  A⁻¹ ≤ · 

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