Linear complementarity problems on extended second order cones

Transcription

1 Linear complementarity problems on extended second order cones arxiv: v3 [math.oc] 14 Sep 217 S. Z. Németh School of Mathematics, University of Birmingham Watson Building, Edgbaston Birmingham B15 2TT, United Kingdom L. Xiao School of Mathematics, University of Birmingham Watson Building, Edgbaston Birmingham B15 2TT, United Kingdom September 15, 217 Abstract In this paper, we study the linear complementarity problems on extended second order cones ESOCLCP. We convert the ESOCLCP into a mixed complementarity problem on the nonnegative orthant. We state necessary and sufficient conditions for a point to be a solution of the converted problem. We also present solution strategies for this problem, such as the Newton method and Levenberg-Marquardt algorithm. Finally, we present some numerical examples. 1 Introduction Although research in cone complementarity problems goes back a few decades only, the underlying concept of complementarity is much older, being firstly introduced by Karush in 1939 [1]. It seems that the concept of complementarity problems was first considered by Dantzig and Cottle in a technical report [2], for the nonnegative orthant. In 1968, Cottle and Dantzig [3] restated the linear programming problem, the quadratic programming problem and the bimatrix game problem as a complementarity problem, which inspired the research in this field see [4 8]. The definition of the classical complementary problem [9] x, y, and x,y =, denotes the componentwise order induced by the nonnegative orthant and, the canonical scalar product in R m, was later extended to more general cones K, as follows: x K, y K, and x,y =, 21 AMS Subject Classification: 9C33, 9C25. Key words and phrases: complementarity problem, extended second order cone, conic optimization 1

2 K is the dual of K [1]. The complementarity problem is a cross-cutting area of research which has a wide range of applications in economics, finance and other fields. Earlier works in cone complementarity problems present the theory for a general cone and the practical applications merely for the nonnegative orthant only similarly to the books [8, 11]. These classical applications are related to equilibrium in economics, engineering, physics, finance and traffic. The classical applications to economics are Walrasian price equilibrium models, price oligopoly models, Nash-Cournot production/distribution models, models of invariant capital stock, Markov perfect equilibria, models of decentralised economy and perfect competition equilibrium, models with individual markets of production factors. Classical applications to engineering and physics are frictional contact problems, elastoplastic structural analysis and nonlinear obstacle problems. A classical application to finance is the discretisation of the differential complementarity formulation of the Black-Scholes models for the American options [12]. A classical application to congested traffic networks is the prediction of steady-state traffic flows. In the recent years several applications have emerged the complementarity problems are defined by cones essentially different from the nonnegative orthant such as positive semidefinite cones, second order cones and direct product of these cones for mixed complementarity problems containing linear subspaces as well. Recent applications of second order cone complementarity problems are in elastoplasticity [13, 14], robust game theory [15, 16] and robotics [17]. All these applications come from the Karush-Kuhn-Tucker conditions of second order conic optimization problems. Németh and Zhang extended the concept of second order cones in [18] to extended second order cones. Their extension seems the most natural extension of second order cones. Sznajder showed that the extended second order cones in [18] are irreducible cones i.e., they cannot be written as a direct product of simpler cones and calculated the Lyapunov rank of these cones [19]. The applications of second order cones and the elegant way of extending them suggest that the extended second order cones will be important from both theoretical and practical point of view. Although conic optimization problems with respect to extended second order cones can be reformulated as conic optimization problems with respect to second order cones, we expect that for several such problems using the particular inner structure of the second order cones provides a more efficient way of solving them than solving the transformed conic optimization problem with respect to second order cones. Indeed, such a particular problem is the projection onto an extended second order cone which is much easier to solve directly than solving the reformulated second order conic optimization problem [2]. Until now the extended second order cones of Németh and Zhang were used as a working tool only for finding the solutions of mixed complementarity problems on general cones [18] and variational inequalities for cylinders whose base is a general convex set [21]. The applications above for second order cones show the importance of these cones and motivates considering conic optimization and complementarity problems on extended second order cones. As a first step, in this paper, we study the linear complementarity problems on extended second order cones ESOCLCP. We find that an ESOCLCP can be transformed to a mixed implicit, mixed implicit complementarity problem on the nonnegative orthant MixCP. We will give the conditions for which a point is a solution of the reformulated MixCP problem, and in this way we provide conditions for a point to be a solution of ESOCLCP. The paper is structured as follows: In Section 2, we illustrate the main terminology and definitions used in this paper. In Section 3, we introduce the notion of mixed implicit complementarity problem as an implicit complementarity problem on the direct product of a cone and a Euclidean space. In Section 4, we reformulate the linear complementarity problem as a mixed implicit, mixed implicit complementarity problem on the nonnegative orthant. Our main result is Theorem 1, which discusses the connections between an ESOCLCP and mixed implicit, mixed implicit complementarity problems. In particular, under some mild 2

3 conditions, given the definition of Fischer-Burmeister regularity and of the stationarity of a point, we prove in Theorem 2 that a point can be the solution of a mixed complementarity problem if it satisfies specific conditions related to regularity and stationarity Theorem 2. This theorem can be used to determine whether a point is a solution of a mixed complementarity problem converted from ESOCLCP. In Section 5, we use Newton s method and Levenberg-Marquardt algorithm to find the solution for the aforementioned MixCP. In the final section, we provide an example of a linear complementarity problem on an extended second order cone. Based on the above, we convert this linear complementarity problem into a mixed complementarity problem on the nonnegative orthant, and use the aforementioned algorithms to solve it. A solution of this mixed complementarity problem will provide a solution of the corresponding ESOCLCP. 2 Preliminaries Let m,k,l,ˆl be nonnegative integers such that m = k +l. Recall the definitions of the mutually dual extended second order cone Lk,l and Mk,l in R m R k R l : Lk,l = {x,u R k R l : x u e, x }, 1 Mk,l = {x,u R k R l : e x u,x }, 2 e = 1,...,1 R k. If there is no ambiguity about the dimensions, then we simply denote Lk,l and Mk,l by L and M, respectively. Denote by, the canonical scalar product in R m and by the corresponding Euclidean norm. The notation x y means that x,y =, x,y R m. Let K R m be a nonempty closed convex cone and K its dual. Definition 1 The set is called the complementarity set of K. CK := {x,y K K : x y} Definition 2 Let F : R m R m. Then, the complementarity problem CPF,K is defined by: The solution set of CPF,K is denoted by SOL-CPF,K: CPF,K : x,fx CK. 3 SOL-CPF,K = {x R m : x,fx CK}. If T is a matrix, r R m and F is defined by Fx = Tx + r, then CPF,K is denoted by LCPT, r, K and is called linear complementarity problem. The solution set of LCPT, r, K is denoted by SOL-LCPT, r, K. Definition 3 Let G,F : R m R m. Then, the implicit complementarity problem ICPF,G,K is defined by ICPF,G,K : Gx,Fx CK. 4 The solution set of ICPF,G,K is denoted by SOL-ICPF,G,K: SOL-ICPF,G,K = {x R m : Gx,Fx CK}. Let m,k,l be nonnegative integers such that m = k + l, Λ R k be a nonempty closed convex cone and K = Λ R l. Denote by Λ the dual of Λ in R k and by K the dual of K in R k R l. It is easy to check that K = Λ {}. 3

4 Definition 4 Consider the mappings F 1 : R k R l R k and F 2 : R k R l Rˆl. The mixed complementarity problem MixCPF 1,F 2,Λ is defined by F 2 x,u = MixCPF 1,F 2,Λ : 5 x,f 1 x,u CΛ. The solution set of MixCPF 1,F 2,Λ is denoted by SOL-MixCPF 1,F 2,Λ: SOL-MixCPF 1,F 2,Λ = {x R m : F 2 x,u =,x,f 1 x,u CΛ}. Definition 5 [8, Definition ] A matrix Π R n n is said to be an S matrix if the system of linear inequalities Πx, x has a solution. Proposition 1 Consider the mappings F 1 : R k R l R k, F 2 : R k R l R l. Define the mappings F : R k R l R k R l by Fx,u = F 1 x,u,f 2 x,u,. Then, x,u SOL-CPF,K x,u SOL-MixCPF 1,F 2,Λ. Proof. The result follows immediately from K = Λ {} and the definitions of CPF,K and MixCPF 1,F 2,Λ. Definition 6 [22, Schur complement] The notation of the Schur complement for a matrix Π = P Q R S, with P nonsingular, is Definition 7 [23, Definition 4.6.2] Π/P = S RP 1 Q. i Let I be an open subset with I R m and f : I R m. We say that f is Lipschitz function, if there is a constant λ > such that fx fx λ x x x,x I. 6 ii We say that f is locally Lipschitz if for every x I, there exists ε > such that f is Lipschitz on I B ε x, B ε x = {y R m : y x ε}. 3 Mixed implicit complementarity problems Let m,k,l,ˆl be nonnegative integers such that m = k+l, Λ R k be a nonempty closed convex cone and K = Λ R l. Denote by Λ the dual of Λ in R k and by K the dual of K in R k R l. Definition 8 Consider the mappings F 1,G 1 : R k R l R k and F 2 : R k R l Rˆl. The mixed implicit complementarity problem MixICPF 1,F 2,G 1,Λ is defined by F 2 x,u = MixICPF 1,F 2,G 1,Λ : 7 G 1 x,u,f 1 x,u CΛ. The solution set of MixICPF 1,F 2,G 1,Λ is denoted by SOL-MixICPF 1,F 2,G 1,Λ: SOL-MixICPF 1,F 2,G 1,Λ = {x R m : F 2 x,u =,G 1 x,u,f 1 x,u CΛ}. 4

5 Proposition 2 Consider the mappings F 1,G 1 : R k R l R k, F 2,G 2 : R k R l R l. Define the mappings F,G : R k R l R k R l by Fx,u = F 1 x,u,f 2 x,u, Gx,u = G 1 x,u,g 2 x,u, respectively. Then, x,u SOL-ICPF,G,K x,u SOL-MixICPF 1,F 2,G 1,Λ. Proof. The result follows immediately from K = Λ {} and the definitions of ICPF,G,K and MixICPF 1,F 2,G 1,Λ. 4 Main results Proposition 3 Let x,y R k and u,v R l \{}. i x,,y,v CL if and only if e y v and x,y CR k +. ii x,u,y, CL if and only if x u and x,y CR k +. iii x,u,y,v := x,u,y,v CL if and only if there exists a λ > such that v = λu, e y = v and x u e,y CR k +. Proof. Items i and ii are an easy consequence of the definitions of L, M and the complementarity set of a nonempty closed convex cone. Item iii follows from Proposition 1 of [2]. For the sake of completeness, we will reproduce its proof here. First assume that there exists λ > such that v = λu, e y = v and x u e,y CR p +. Thus, x,u L and y,v M. On the other hand, x,u,y,v = x y +u v = u e y λ u 2 = u v λ u 2 =. Thus, x,u,y,v CL. Conversely, if x,u,y,v CL, then x,u L, y,v M and = x,u,y,v = x y +u v u e y +u v u v +u v. This implies the existence of a λ > such that v = λu, e y = v and x u e y =. It follows that x u e,y CR p +. Theorem 1 Denote z = x,u, ẑ = x u,u, z = x t,u,t and r = p,q with x,p R k, u,q R l and t R. Let T = C A D B with A Rk k, B R k l, C R l k and D R l l. The square matrices T, A and D are assumed to be nonsingular. i Suppose u =. We have z SOL-LCPT,r,L x SOL-LCPA,p,R k + and e Ax+p Cx+q. ii Suppose Cx+Du+q =. Then, z SOL-LCPT,r,L z SOL-MixCPF 1,F 2,R k + and x u, F 1 x,u = Ax+Bu+p and F 2 x,u =. 5

6 iii Suppose u and Cx+Du+q. We have z SOL-LCPT,r,L z SOL-MixICPF 1,F 2,G 1,R k +, F 2 x,u = u C +ue A x+ue Bu+p+ u Du+q, G 1 x,u = x u e and F 1 x,u = Ax+Bu+p. iv Suppose u and Cx+Du+q. We have z SOL-LCPT,r,L ẑ SOL-MixCPF 1,F 2,R k +, F 2 x,u = u C +ue A x+ u e+ue Bu+p+ u Du+q and F 1 x,u = Ax+ u e+bu+p. v Suppose u, Cx+Du+q and u C +u ea is a nonsingular matrix. We have z SOL-LCPT,r,L ẑ SOL-ICPF 1,F 2,R k +, u C F 1 u = A +ue A 1 ue Bu+p+ u Du+q +Bu+p and F 2 u = u C +ue A 1 ue Bu+p+ u Du+q. vi Suppose u, Cx+Du+q. We have z SOL-LCPT,r,L t >, such that and z MixCP F 1, F 2,R k +, F 1 x,u,t = Ax+te+Bu+p tc +ue F 2 x,u,t = A x+te+ue Bu+p+tDu+q t 2 u 2. 8 Proof. i We have that z SOL-LCPT,r,L is equivalent to x,,ax+p,cx+q CL or, by item i of Proposition 3, to x,ax+p CR k + and e Ax+p Cx+q. ii We have that z SOL-LCPT,r,L is equivalent to x,u,ax+bu+p, CL or, by item ii of Proposition 3, to x,ax+bu+p CR k + and x u, or to z SOL-MixCPF 1,F 2,R k + and x u, F 1 x,u = Ax+Bu+p and F 2 x,u =. 6

7 iii Suppose that z SOL-LCPT,r,L. Then, x,u,y,v CL, y = Ax+Bu +p and v = Cx + Du+q. Then, by item iii of Proposition 3, we have that λ > such that Cx+Du+q = v = λu, 9 e Ax+Bu+p = e y = v = Cx+Du+q = λ u, 1 G 1 x,u,f 1 x,u = x u e,ax+bu+p = x u e,y CR k + 11 From equation 9 we obtain u Cx+Du+q = λ u u, which by equation 1 implies u Cx+Du+q = ue Ax+Bu+p, which after some algebra gives F 2 x,u =. 12 From equations 11 and 12 we obtain that z SOL-MixICPF 1,F 2,G 1. Conversely, suppose that z SOL-MixICPF 1,F 2,G 1. Then, and u v+ue y = u Cx+Du+q+ue Ax+Bu+p = F 2 x,u = 13 x u e,y = x u e,ax+bu+p = G 1 x,u,f 1 x,u CR k +, 14 v = Cx+Du+q and y = Ax+Bu+p. Equations 14 and 13 imply Equations 15 and 16 imply v = λu, 15 λ = e y/ u >. 16 e y = v. 17 By item iii of Proposition 3, equations 15, 17 and 14 imply x,y,u,v CL and therefore z SOL-LCPT, r, L. iv It is a simple reformulation of item iii by using the change of variables x,u x u e,u. v It is a simple reformulation of item iv by using that u C + u ea is a nonsingular matrix. vi Suppose that z SOL-LCPT,r,L. Then, x,u,y,v CL, y = Ax+Bu +p and v = Cx + Du + q. Let t = u, Then, by item iii of Proposition 3 we have that λ > such that Cx+Du+q = v = λu, 18 e Ax+Bu+p = e y = v = Cx+Du+q = λt, 19 z, F 1 x,u,t = x te,ax+bu+p = x te,y CR k +, 2 z = x t,u,t. From equation 18, we get tcx +Du+q = tλu, which, by equation 19, implies tcx+du+q = ue Ax+Bu+p, which after some algebra gives F 2 x,u,t =. 21 From equations 2 and 21 we obtain that z SOL-MixCP F 1, F 2,R k +. 7

8 Note that the itemvi makes F 1 x,u,t and F 2 x,u,t become smooth functions by adding the variable t. The smooth functions therefore make the smooth Newton s method applicable to the mixed complementarity problem. Solving the MixCP is much easier than solving the original LCP on extended second order cones. In order to ensure the existence of the solution of MixCP, we introduce the scalar Fischer-Burmeister C-function see [24, 25]. ψ a,b = a 2 +b 2 a+b a,b R 2. It is obvious that ψ 2 a,b is a continuously differentiable function on R2. The equivalent -based equation formulation for the MixCP problem is: with the associated merit function: = F MixCP x,u,t = ψx 1, F 1x,u,t 1. ψx k, F 1x,u,t k, 22 F 2 x,u,t θ MixCP x,u,t = 1 2 FMixCP x,u,t T F MixCP x,u,t. WecontinuebycalculatingtheJacobianmatrixfortheassociatedmeritfunction. Ifi 1,...,k is such that z i, F 1 i,, then the differential with respect to z = x,u,t Rm+1 is F MixCP i = x i z 1 2 x 2 i Fi ei + F 1x,u,t i 1 F 1x,u,t i 2, + 1 x,u,t x 2 i Fi + z 1 x,u,t e i denotes the i-th canonical unit vector. The differential with respect to z j with j i is F MixCP i = F 1x,u,t i z j 1 F 1x,u,t i 2, x 2 i Fi + z j 1 x,u,t It is obvious that the differential with respect to z j with j > k, is equal to zero. Note that i if z i, F 1 i FMixCP =,, then will be a generalised gradient of a composite function, i.e. z a closed unit ball B,1. However, this case will not occur in our paper. As for the term F 2 x,u,t with i k +1,...m+1, the Jacobian matrix is much more simple, since F MixCP i z = F 2 ix,u,t. z Therefore, the Jacobian matrix for the associated merit function is: D A = a +D b J x F1 x,u,t D b J u,t F1 x,u,t, J x F2 x,u,t J u,t F2 x,u,t x i D a = diag x 1 F, D 2 i + F 1 i b = diag 1 ix,u,t 1, i = 1,...,k. x,u,t2 x 2 i + F 1 ix,u,t2 8

9 Define the following index sets: { C i : x i, F } 1 i,x i Fi 1 x,u,t = R {1,...,k}\C P {i R : x i >, F } i1 x,u,t > N R\P complementarity index residual index positive index negative index Definition 9 A point x,u,t R m+1 is called -Regular for the merit function θ MixCP or for the MixCP F1, F 2,R k + if its partial Jacobian matrix of F MixCP x,u,t with respect to x, J x F1 x,u,t is nonsingular and if for w R k,w with there exists a nonzero vector v R k such that w C =, w P >, w N <, v C =, v P, v N, 23 and w Πx,u,t/J T x F1 x,u,t v, 24 J Πx,u,t x F1 x,u,t J u,t F1 x,u,t R m+1 m+1, J x F2 x,u,t J u,t F2 x,u,t and Πx,u,t/J x F1 x,u,t is the Schur complement of J x F1 x,u,t in Πx,u,t. In our case, for the MixCP F1, F 2,R k +, the Jacobian matrices are: and J F 1 x,u,t à B, J F 2 x,u,t C D, tc +ue à = A, B = B Ae, C A =, e D = Ax+te+Bu+pI +diage Bu+tD Cx+2tCe+ue Ae+Du 2u. 2t In our case, if the Jacobian matrix block J x F1 x,u,t = A is nonsingular, then the Schur complement Πx,u,t/J x F1 x,u,t is Πx,u,t/J x F1 x,u,t = D Cà 1 B. 25 Proposition 4 If the matrices à and D are nonsingular for any z R m+1, then the Jacobian matrix A for the associated merit function is nonsingular. 9

10 Proof. It is easy to check that D A = a +D b à D b B. C D A is a nonsingular matrix if and only if the sub-matrix D a +D b à and its Schur complement arenonsingular, andtheyarenonsingularifandonlyifthematricesãand D arenonsingular. The following theorem is [8, Theorem 9.4.4]. For the sake of completeness, we provide a proof here. Theorem 2 A point x,u,t R m+1 is a solution of the MixCP F 1, F 2,R k if and only if x,u,t is an regular point of θ MixCP and x,u,t is a stationary point of F MixCP. Proof. Suppose that z = x,u,t SOL-MixCP F 1, F 2,R k. Then, it follows that z is a global minimum and hence a stationary point of θ MixCP. Thus, x, F 1 z CR k +, and we have P = N =. Therefore the regularity of x holds since x = x C, because there is no nonzero vector x satisfying conditions 23. Conversely, suppose that x is regular and z = x,u,t is a stationary point of θ MixCP. It follows that θ MixCP =, i.e.: A F MixCP D = a +D b J x F1 z J x F2 z F MixCP D b J u,t F1 z J u,t F2 z =, x D a = diag i x 1 F 1 D i 2 + F 1 iz 2 b = diag iz 1 x i 2 + F 1 iz 2 i = 1,...,k. Hence, for any w R m+1, we have w D a +D b J x F1 z J x F2 z F MixCP D b J u,t F1 z J u,t F2 z =. 26 Assume that z is not a solution of MixCP. Then, we have that the index set R is not empty. Define v D b F MixCP. We have Take w with v C =, v P >, v N <. w C =, w P >, w N <. From the definition of D a and D b, we know that D a F MixCP Therefore, and D b F MixCP have the same sign. w D a F MixCP By the regularity of J F 1 z, we have = wc D af MixCP C +wp D af MixCP P +wn D af MixCP N >. 27 w J F 1 z D a F MixCP = w J F 1 z w. 28 The inequalities 27 and 28 together contradict condition 26. Hence R =. It means that z is a solution of MixCP F 1, F 2,R k. 1

11 5 Algorithms For solving a complementarity problem, there are many different algorithms available. The common algorithms include numerical methods for systems of nonlinear equations such as Newton s method [26], the interior point method Karmarkar s Algorithm [27], the projection iterative method [28], and the multi-splitting method [29]. In the previous chapters, we have already provided sufficient conditions for using regularity and stationarity to identify a solution of the MixCP problem. In this chapter, we are trying to find a solution of LCP by finding the solution of MixCP which is converted from LCP. One convenient way to do this is using the Newton s Method as follows: Algorithm Newton s method: Given initial data z R m+1, and r = 1 7. Step 1: Set k =. Step 2: If F MixCP z k r, then STOP. Step 3: Find a direction d k R m+1 such that F MixCP z k +A z k d k =. Step 4: Set z k+1 := z k +d k and k := k +1, go to Step 2. If the Jacobian matrix A is nonsingular, then the direction d k R m+1 for each step can be found. The following theorem, which is based on an idea similar to the one used in [3], proves that such a Newton s Method can efficiently solve the LCP on extended second order cone i.e. solve the problem within polynomial time, by finding the solution of the MixCP: Theorem 3 Suppose that the Jacobian matrix A is nonsingular. Then, Newton s method for MixCP F 1, F 2,R k + converges at least quadratically to z SOL-MixCP F 1, F 2,R k +, if it starts with initial data z sufficiently close to z. Proof. Suppose that the starting point z is close to the solution z, and suppose that A is a Lipschitz function. There are ρ >,β 1 >,β 2 >, such that for all z with z z < ρ, there holds A 1 z < β 1, and Az k Az β 2 z k z. By the definition of the Newton s method, we have because F MixCP so z k+1 z = z k z A 1 z k F MixCP z k = A 1 z k [ Az k z k z F MixCP z k F MixCP z ], z = when z SOL-MixCP. By Taylor s theorem, we have F MixCP z k F MixCP z = 1 A z k +sz z k x k z ds, Az k z k z F MixCP z k F MixCP z 1 [ = Az k A z k +sz z k ] dsz k z 1 Az k A z k +sz z k ds z k z 1 z k z 2 β 2 sds = 1 2 β 2 z k z 2. 11

12 Also, we have z z < ρ, that is z k+1 z 1 2 β 1β 2 z k z 2. Another widely-used algorithm is presented by Levenberg and Marquardt in[31]. Levenberg- Marquardt algorithm can approach second-order convergence speed without requiring the Jacobian matrix to be nonsingular. We can approximate the Hessian matrix by: and the gradient by: Hence, the upgrade step will be Hz = A zaz, Gz = A zf MixCP z. z k+1 = z k [ A z k Az k +µi ] 1 A z k F MixCP z k. As we can see, Levenberg-Marquardt algorithm is a quasi-newton s method for an unconstrained problem. When µ equals to zero, the step upgrade is just the Newton s method using approximated Hessian matrix. The number of iterations of Levenberg-Marquardt algorithm to find a solution is higher than that of Newton s method, but it works for singular Jacobian as well. The greater the parameter µ is, the slower the calculation speed becomes. Levenberg- Marquardt algorithm is provided as follows: Algorithm Levenberg-Marquardt: Given initial data z R m+1, µ =.5, and r = 1 7. Step 1: Set k =. Step 2: If F MixCP z k r, stop. Step 3: Find a direction d k R m+1 such that Az k F MixCP z k + [ A z k Az k +µi ] d k =. Step 4: Set z k+1 := z k +d k and k := k +1, go to Step 2. Theorem 4 [32] Without the nonsingularity assumption on the Jacobian matrix A, Levenberg- Marquardt Algorithm for MixCP F 1, F 2,R k + converges at least quadratically to z SOL-MixCP F 1, F 2,R k +, if it starts with initial data z sufficiently close to z. The proof is omitted. 12

13 6 A numerical example In this chapter, we will provide a numerical example for LCP on extended second order cones. Let L3, 2 be an extended second order cone defined by 1. Following the notation in Theorem 1, let z = x,u, ẑ = x u,u, z = x t,u,t and r = p,q = 55, 26,5, 19, 26 with x,p R 3, u,q R 2, and t R. Consider T = A B C D = with A R 3 3, B R 3 2, C R 2 3 and D R 2 2. It is easy to show that square matrices T, A and D are nonsingular. By item vi of Theorem 1, we can reformulate this LCP problem as a smooth MixCP problem. We will use the Levenberg-Marquardt algorithm to find the solution of the -based equation formulation 22 of MixCP problem. The convergence point is: z = x t,u,t =, , 341, 146, ,, We need to check the regularity of z. It is easy to show that the partial Jacobian matrix of F 1 z J x F1 z = à = is nonsingular. Moreover, we have that x t =, ,, F1 z =,,, and therefore x t, F 1 z =. That is, x, F 1 z CR 3 +, so the index sets P = N =. The matrix à is invertible. In addition, we can calculate that the Schur complement of Π z with respect to J x F1 z : Π z /J x F1 z = D Cà 1 B =

14 The regularity of x holds as there is no nonzero vector x satisfying conditions 23. Then, we compute the gradient of the merit function, which is A F MixCP D = a +D b J x F1 z J x F2 z F MixCP D b J u,t F1 z J u,t F2 z = =. Hence, z is a stationary point of F MixCP. By Theorem 2, we can conclude that z is the solution of the MixCP problem. By the item vi of Theorem 1, we have that z = x,u 1271 is the solution of LCPT,r,L problem. = 3582, , , 148, 724, Final remarks In this paper, we studied the method of solving a linear complementarity problem on an extended second order cone. By checking the stationarity and regularity of a point, we can verify whether it is a solution of the mixed complementarity problem or not. Such conversion of a linear complementarity problem to a mixed complementarity problem reduces the complexity of the original problem. The connection between a linear complementarity problem on an extended second order cone and a mixed complementarity problem on a nonnegative orthant will be useful for our further research about applications to practical problems, such us portfolio selection and signal processing problems. References [1] W. Karush. Minima of functions of several variables with inequalities as side conditions. Master thesis, University of Chicago, [2] G. B. Dantzig and R. W. Cottle. Positive semi- definite matrices and mathematical programming. Technical report, California Univ Berkeley Operations Research Center, [3] R. W. Cottle and G. B. Dantzig. Complementary pivot theory of mathematical programming. Linear algebra and its applications, 11:13 125,

15 [4] O. L. Mangasarian. Linear complementarity problems solvable by a single linear program. Mathematical Programming, 11:263 27, [5] C. B. Garcia. Some classes of matrices in linear complementarity theory. Mathematical Programming, 51:299 31, [6] J. M. Borwein and M. A. H. Dempster. The linear order complementarity problem. Mathematics of Operations Research, 143: , [7] F. Alizadeh and D. Goldfarb. Second-order cone programming. Mathematical programming, 951:3 51, 23. [8] F. Facchinei and J.-S. Pang. Finite-dimensional variational inequalities and complementarity problems. Vol. II. Springer-Verlag, New York, 23. [9] J.-S. Facchinei, F.and Pang. Finite-dimensional variational inequalities and complementarity problems. Vol. I. Springer-Verlag, New York, 23. [1] S. Karamardian. Generalized complementarity problem. J. Optimization Theory Appl., 8: , [11] Igor Konnov. Equilibrium models and variational inequalities, volume 21. Elsevier, 27. [12] P. Jaillet, D. Lamberton, and B. Lapeyre. Variational inequalities and the pricing of american options. Acta Applicandae Mathematicae, 213: , 199. [13] K. Yonekura and Y. Kanno. Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion. Optim. Eng., 132: , 212. [14] L. L. Zhang, J. Y. Li, H. W. Zhang, and S. H. Pan. A second order cone complementarity approach for the numerical solution of elastoplasticity problems. Comput. Mech., 511:1 18, 213. [15] G. Luo, X. An, and J. Xia. Robust optimization with applications to game theory. Appl. Anal., 888: , 29. [16] R. Nishimura, S. Hayashi, and M. Fukushima. Robust Nash equilibria in N-person noncooperative games: uniqueness and reformulation. Pac. J. Optim., 52: , 29. [17] R. Andreani, A. Friedlander, M. P. Mello, and S. A. Santos. Box-constrained minimization reformulations of complementarity problems in second-order cones. J. Global Optim., 44:55 527, 28. [18] S. Z. Németh and G. Zhang. Extended Lorentz cones and mixed complementarity problems. Journal of Global Optimization, 623: , 215. [19] R. Sznajder. The Lyapunov rank of extended second order cones. Journal of Global Optimization, 663: , 216. [2] O. P. Ferreira and S. Z. Németh. How to project onto extended second order cones. arxiv: v2, 216. [21] S. Z. Németh and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 1683: , 216. [22] F. Zhang. The Schur complement and its applications, volume 4. Springer Science & Business Media,

16 [23] H. H. Sohrab. Basic real analysis, volume 231. Springer, 23. [24] A. Fischer. A special newton-type optimization method. Optimization, 243-4: , [25] A. Fischer. A newton-type method for positive-semidefinite linear complementarity problems. Journal of Optimization Theory and Applications, 863:585 68, [26] K. E. Atkinson. An introduction to numerical analysis. John Wiley & Sons, 28. [27] Narendra Karmarkar. A new polynomial-time algorithm for linear programming. In Proceedings of the sixteenth annual ACM symposium on Theory of computing, pages ACM, [28] O. L. Mangasarian. Solution of symmetric linear complementarity problems by iterative methods. Journal of Optimization Theory and Applications, 224: , [29] D. P. O Leary and R. E. White. Multi-splittings of matrices and parallel solution of linear systems. SIAM Journal on algebraic discrete methods, 64:63 64, [3] D. G. Luenberger and Y. Ye. Linear and nonlinear programming, volume 228. Springer, 215. [31] D. W. Marquardt. An algorithm for least-squares estimation of nonlinear parameters. Journal of the society for Industrial and Applied Mathematics, 112: , [32] N. Yamashita and M. Fukushima. On the rate of convergence of the Levenberg-Marquardt method. In Topics in numerical analysis, pages Springer,