A Smoothing SQP Method for Mathematical Programs with Linear Second-Order Cone Complementarity Constraints

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1 A Smoothing SQP Method for Mathematical Programs with Linear Second-Order Cone Complementarity Constraints Hiroshi Yamamura, Taayui Ouno, Shunsue Hayashi and Masao Fuushima June 14, 212 Abstract In this paper, we focus on the mathematical program with second-order cone (SOC complementarity constraints, which contains the well-nown mathematical program with nonnegative complementarity constraints as a subclass. For solving such a problem, we propose a smoothing-based sequential quadratic programming (SQP methods. We first replace the SOC complementarity constraints with equality constraints using the smoothing natural residual function, and apply the SQP method to the smoothed problem with decreasing the smoothing parameter. We show that the proposed algorithm possesses the global convergence property under the Cartesian P property and the nondegeneracy assumptions. We finally observe the effectiveness of the algorithm by means of numerical experiments. Key words: mathematical programs with equilibrium constraints; sequential quadratic programming; second-order cone complementarity Mathematics Subject Classification: 65K5, 9C3, 9C33 1 Introduction In this paper, we focus on the following mathematical program with second-order cone (SOC complementarity constraints, abbreviated as MPSOCC: Minimize f(x, y x,y,z subject to Ax b, z = Nx + My + q, K y z K, (1.1 where f : R n+m R is a continuously differentiable function, A R p n, b R p, N R m n, M R m m and q R m are given matrices and vectors, denotes the perpendicularity, and K is the Cartesian product of second-order cones, that is, K := K m 1 K m 2 K m l R m 1 R m 2 R m l = R m with { { } u = (u1, u K mi = 2 R R mi 1 u 2 u 1 (mi 2, R + = {u R u } (m i = 1. Throughout the paper, we suppose that m i 2 for each i. Mathematical program with equilibrium constraints (MPEC [15] has been studied extensively, since it finds wide application such as design problems in engineering, the equilibrium problems in economics, and gametheoretic multi-level optimization problems. Particularly, equilibrium constraints in MPECs are often written as linear or nonlinear complementarity constraints. Such an MPEC is also called a mathematical program with complementarity constraints (MPCC. When m i = 1 for all i, i.e., K = R m +, MPSOCC (1.1 reduces to the MPCC, for which there have been proposed many algorithms. For example, Fuushima and Tseng [12] proposed an active set algorithm, and proved that any accumulation point of the sequence generated by the algorithm Technical report 212-5, July 14, 212. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto , Japan (yamamura, {t_ouno, shunhaya, fuu}@amp.i.yoto-u.ac.jp. 1

2 is a B-stationary point under the uniform linear inequality constraint qualification (LICQ on the ε-feasible set. Luo, Pang, and Ralph [15] proposed a piece-wise sequential quadratic programming (SQP algorithm, and showed that the generated sequence converges to a B-stationary point locally superlinearly or quadratically under the LICQ and the second order sufficient conditions. Fuushima, Luo, and Pang [1] proposed an SQPtype algorithm, and showed that the sequence generated by the algorithm globally converges to a B-stationary point under the nondegeneracy condition at the limit point. Problems with SOC constraints also attract much attention of many researchers. One of the typical problems is the second-order cone program (SOCP [1]. The SOCP has a lot of applications such as the antenna array weight design, the finite response impulse (FIR filter design, the portfolio optimization, and the magnetic shield design optimization. Moreover, SOCP includes many classes of problems such as linear program (LP, convex quadratic program (QP, etc. The second-order cone complementarity problem (SOCCP [5, 7, 11] is another type of problems involving SOC constraints. Fuushima, Luo and Tseng [11] studied smoothing functions for the Fischer-Burmeister function and the natural residual function with respect to the SOC complementarity condition. Using those smoothing functions, Hayashi, Yamashita and Fuushima [14] proposed a globally and quadratically convergent algorithm based on the smoothing and regularization methods. As an application of the SOCCP, Nishimura, Hayashi and Fuushima [16] studied the SOCCP reformulation of the robust Nash equilibrium problem in an N-person non-cooperative game. As mentioned in the last two paragraphs, there have been many researches on the MPECs with nonnegative complementarity constraints and the optimization/complementarity problems with SOC constraints. However, there are only a few studies on MPECs with SOC complementarity constraints. For example, Yan and Fuushima [2] proposed a smoothing method for solving such problems. To show convergence of the algorithm, they assume that smoothed subproblems are solved exactly. However, it can hardly be expected in practice. To overcome such a difficulty, we propose to combine an SQP-type method with the smoothing method. The proposed method replaces the SOC complementarity condition of MPSOCC (1.1 with a certain vector equation by using a smoothed natural residual function, thereby yielding convex quadratic programming subproblems which can be solved efficiently by any state-of-the-art method such as the active method and interior point method. Although our method may be viewed as an extension of the SQP method in [1], the convergence analysis is quite different since it exploits the particular properties of the natural residual associated with the SOC complementarity condition. This paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we reformulate MPSOCC (1.1 as a nonlinear programming problem by replacing the second-order cone complementarity constraints by equivalent nonsmooth equality constraints. In Section 4, we introduce a smoothing technique to deal with the nonsmooth constraints. In Section 5, we propose an SQP-type algorithm for solving problem (1.1 and show that the proposed method is well-defined under the Cartesian P property. In Section 6, we show that the proposed algorithm possesses the global convergence property under the nondegeneracy assumptions. In Section 7, we give some numerical examples. In Section 8, we conclude the paper with some remars. Throughout the paper, we use the following notations. For a given vector z R m, z i denotes the i-th element of z R m, while z i R m i denotes the i-th column subvector conforming to the given Cartesian structure of K. For subvectors z 1 R m1, z 2 R m2,..., z l R m l, we often let (z 1, z 2,..., z l denote the column vector ((z 1,, (z l R m1+ +m l. For a vector z R m, we denote z 1 := m i=1 z i, z := z z and z := max 1 i m z i. For a matrix M R m m, M denotes the operator norm defined by M = max x =1 Mx. For matrices M, N R m m, M ( N denotes that M N is a positive (semi-definite matrix. We denote the interior and the boundary of K by int K and bd K, respectively. *1 For vectors y, z R m, y K z and y K z mean y z K and y z int K, respectively. We denote the nonnegative cone in R m and its interior by R m + := {z R m z i (i = 1, 2,..., m} and R m ++ := {z R m z i > (i = 1, 2,..., m}, respectively. Finally, for a set C R m and a vector z C, we denote the normal cone [18] of C at z by N C (z. 2 Preliminaries 2.1 Spectral factorization and natural residual The proposed algorithm relies on the fact that the SOC complementarity condition K y z K can be *1 Note that int K = {(z 1, z 2 R R m 1 z 1 > z 2 } and bd K m = {(z 1, z 2 R R m 1 z 1 = z 2 }. In addition, for K = K m 1 K m l, we have int K = int K m 1 int K m l and bd K = K \ int K. 2

3 rewritten as a system of equations by means of the natural residual. To be specific, we first recall the spectral factorization of a vector with respect to the SOC, K m. Definition 2.1. For any vector z := (z 1, z 2 R R m 1, we define the spectral factorization with respect to K m as z = λ 1 c 1 + λ 2 c 2, where λ 1 and λ 2 are the spectral values given by and c 1 and c 2 are the spectral vectors given by 1 c j = 2 λ j = z 1 + ( 1 j z 2, j = 1, 2, ( 1, ( 1 j z 2 z 2 if z (1, ( 1j v if z 2 = respectively, where v R m 1 is an arbitrary vector such that v = 1. j = 1, 2, By using the spectral factorization, we can write the Euclidean projection onto K m explicitly as follows [11]: P K m(z := argmin z K m z z = max{, λ 1 }c 1 + max{, λ 2 }c 2, where λ j and c j (j = 1, 2 are the spectral values and the spectral vectors of z, respectively. Now, let us define the natural residual for the SOC complementarity condition by using the Euclidean projection. Definition 2.2. Let y := (y 1, y 2,, y l and z := (z 1, z 2,, z l R m1 R m2 R m l = R m be arbitrary vectors. Then, the natural residual function Φ : R m R m R m with respect to K = K m 1 K m 2 K m l is defined as where It can be shown [11] that Φ(y, z := y P K (y z (2.1 ϕ 1 (y 1, z 1 =. ϕ l (y l, z l R m 1 R m l, ϕ i (y i, z i = y i P K m i (y i z i, i = 1, 2,..., l. ϕ i (y i, z i = K mi y i z i K mi, Φ(y, z = K y z K. The following proposition states the property of function Φ(y, z when y z / bd (K K. Proposition 2.1. Let y, z R m be chosen so that y z / bd (K K. Then, the function Φ : R m R m R m defined by (2.1 is continuously differentiable at (y, z, and the following equality holds: where I m R m m denotes the identity matrix. y Φ(y, z + z Φ(y, z = I m, Proof. It suffices to consider the case where K = K m. Let λ 1, λ 2 R be the spectral values of y z defined as in Definition 2.1. Note that, from y z / bd (K m K m, we have λ 1, λ 2. Then, from [14, Proposition 4.8], the Clare subdifferential P K m(y z is explicitly given as I m (λ 1 >, λ 2 >, λ 2 P K m(y z = I m + W (λ 1 <, λ 2 >, λ 2 λ 1 O (λ 1 <, λ 2 <, 3

4 where W := 1 ( r1 r2 2 r 2 r 1 r 2 r2, (r 1, r 2 := (y 1 z 1, y 2 z 2. y 2 z 2 Thus P K m is differentiable at y z. This fact readily implies the continuous differentiability of Φ at (y, z since Φ(y, z = y P K m(y z. We next show the second half of the proposition. By an easy calculation, we have y Φ(y, z = I m P K m(y z. Similarly, we have z Φ(y, z = P K m(y z. Hence we obtain the desired equality. 2.2 Cartesian P and P matrices In this subsection, we introduce the Cartesian P and the Cartesian P matrices. The concept of the Cartesian P (P matrix is a natural extension of the well-nown P (P matrix [8]. Although the Cartesian P (P matrix can be defined not only for the SOC but also for the semidefinite cone [6] and the symmetric cone [13], we restrict ourselves to the case of the SOCs. Definition 2.3. Suppose that the Cartesian structure of K R m is given as K := K m 1 K m 2 K m l. Then, M R m m is called (a a Cartesian P matrix if, for every nonzero z = (z 1,..., z l R m = R m1 R m l, there exists an index i {1,..., l} such that (z i (Mz i ; (b a Cartesian P matrix if, for every nonzero z = (z 1,..., z l R m = R m 1 R m l, there exists an index i {1,..., l} such that (z i (Mz i >. Here, (Mz i R mi denotes the i-th subvector of Mz R m conforming to the Cartesian structure of K. Notice that the definition of the Cartesian P (P property depends on the Cartesian structure of K. In what follows, we assume that the Cartesian structure of K is always given as K = K m 1 K m 2 K m l. The definition of the classical P (P matrix corresponds to the case where K = R m +. It is easily seen that every Cartesian P (P matrix is a P (P matrix [17]. The following proposition implies that the Cartesian P (P property is preserved under a nonsingular blocdiagonal transformation. Proposition 2.2. Let M R m m be any matrix, and H i R m i m i (i = 1, 2,..., l be arbitrary nonsingular matrices. Let the matrix M R m m be defined by M :=H MH, H := H 1... H l. Then, the following statements hold. (a If M is a Cartesian P matrix, then M is a Cartesian P matrix. (b If M is a Cartesian P matrix, then M is a Cartesian P matrix. Proof. We first show (b. Let z = (z 1,..., z l R m = R m1 R m l be an arbitrary nonzero vector. We 4

5 show that there exists an i {1, 2,..., l} such that (z i (M z i >. Note that H 1 H (M z i = 1... M... z H l H l i H 1 H = 1 z 1... M. H l H l z l i l H1 M 1 H z =1 =. l M l H z = ( H l H i =1 l M i H z. where ( i and ( i denote the i-th subvector and the (i, -th bloc entry, respectively, conforming to the Cartesian structure of K. Hence, we have =1 i (z i (M z i = (z i H i l M i H z = ( H i z i l =1 =1 M i H z = ((Hz i (MHz i. Since M is a Cartesian P matrix and Hz from the nonsingularity of H, we have (z i (M z i = ((Hz i (MHz i > for some i. Hence, M is a Cartesian P matrix. We omit the proof of (a since it can be shown in a similar manner to (b. 3 Reformulation of MPSOCC and relationship between the KKT conditions and B-stationary points In the previous section, we introduce the natural residual function Φ and observed that second-order cone complementarity condition K y z K can be represented as Φ(y, z = equivalently. In this section, we rewrite MPSOCC (1.1 as the following problem where the SOC complementarity constraint is replaced by the equivalent equality constraint involving the natural residual function Φ: Minimize f(x, y x,y,z subject to Ax b, z = Nx + My + q, Φ(y, z =. (3.1 We also call this problem MPSOCC. MPSOCC (3.1 is a nonsmooth optimization problem since Φ is not differentiable everywhere. However, as is claimed by the next proposition, Φ is continuously differentiable at any (y, z satisfying the following nondegeneracy condition: Definition 3.1 (Nondegeneracy. Suppose that (y, z R m R m satisfies the SOC complementarity condition K y z K. Moreover, decompose y and z as y = (y 1, y 2,..., y l and z = (z 1, z 2,..., z l R m 1 R m 2 R m l = R m conforming to the Cartesian structure of K. Then, (y, z is said to be nondegenerate if, for every i = 1, 2,..., l, one of the following three conditions holds: (i y i int K m i, z i = ; (ii y i =, z i int K m i ; (iii y i bd K m i \ {}, z i bd K m i \ {}, (y i z i =. Proposition 3.1. Let (y, z R m R m satisfy the nondegeneracy condition. Then, Φ is continuously differentiable at (y, z. 5

6 Proof. The nondegeneracy condition readily yields y z / bd (K K. Hence, Φ is continuously differentiable at (y, z by Proposition 2.1. Now, let X := {x R n Ax b}, and let w := (x, y, z R n R m R m be a feasible point of MPSOCC (1.1 that satisfies the nondegeneracy condition. By Proposition 3.1, MPSOCC (3.1 can be viewed as a smooth optimization problem in a small neighborhood of w. Then, the Karush-Kuhn-Tucer (KKT conditions on w are represented as x f(x, y N T y f(x, y + M T u + y Φ(y, z v N X (x {} 2m, I z Φ(y, z Φ(y, z =, Ax b, z = Nx + My + q, (3.2 where {} 2m := {} {} {} R 2m, and u R l, v R m and η R m are Lagrange multipliers. We next consider the stationarity of MPSOCC (1.1 or (3.1. So far, several inds of stationary points have been studied in the literature of MPECs, e.g., see [19]. Among them, a Bouligand- or B-stationary point is the most desirable, since it is directly related to the first order optimality condition. Specifically, a B-stationary point for MPSOCC (1.1 is defined as follows: Definition 3.2 (B-stationarity. Let F R n+2m denote the feasible set of MPSOCC (1.1. We say that w := (x, y, z F is a B-stationary point of MPSOCC (1.1 if ( f(x, y, N F (w holds. In what follows, we show that a point satisfying KKT conditions (3.2 is a B-stationary point. purpose, we give three useful lemmas. For this Lemma 3.1. [18, Proposition 6.41] Let C := C 1 C 2 C s for nonempty closed sets C i R n i. Choose ζ := (ζ 1, ζ 2,..., ζ s C 1 C 2 C s. Then, we have N C (ζ = N C1 (ζ 1 N C2 (ζ 2 N Cs (ζ s. Lemma 3.2. [18, Chapter 6-C] For a continuously differentiable function F : R p R q, let D := {ζ R p F (ζ = }. Choose ζ D arbitrarily. If F (ζ has full column ran, then we have N D (ζ = F (ζr q := {ζ R p ζ = F (ζv, v R q }. Lemma 3.3. [18, Theorem 6.14] For a continuously differentiable function F : R p R q and a closed set C R p, let D := {ζ C F (ζ = }. Choose ζ D arbitrarily. Then we have N D (ζ F (ζr q + N C (ζ. Now, we show that the KKT conditions (3.2 are sufficient conditions for w = (x, y, z to be B-stationary. Proposition 3.2. Let w := (x, y, z be a feasible point of MPSOCC (3.1. Suppose that the nondegeneracy condition holds at (y, z. If w satisfies the KKT conditions (3.2, then w is a B-stationary point of MPSOCC (1.1. Proof. Let X := {x R n Ax b} and Y := {(y, z R m R m Φ(y, z = }. We first note that the nondegeneracy at (y, z implies the continuous differentiability of Φ at (y, z from Proposition 2.1. Choose v R m such that Φ(y, zv =. From Proposition 2.1, we then have y Φ(y, zv = and z Φ(y, zv = (I y Φ(y, zv =, which readily imply v =, and thus Φ(y, z has full column ran. Therefore, by Lemma 3.2 with p = q := m, D := Y and F := Φ, we have Then, it holds that N Y (y, z = Φ(y, zr m. (3.3 N X Y (w = N X (x N Y (y, z = N X (x Φ(y, zr m, (3.4 where the first equality follows from Lemma 3.1 and the second equality follows from (3.3. Now, let F R n+2m denote the feasible set of MPSOCC (3.1, i.e., F = {(x, y, z X Y Nx + My z + q = }. Then, from Lemma 3.3, we have N F (w ( N, M, I R m + N X Y (w. (3.5 Combining (3.4 with (3.5, we obtain N F (w ( N, M, I R m + N X (x Φ(y, zr m, which together with the KKT conditions (3.2 implies ( f(x, y, N F (w. point. Thus, w is a B-stationary 6

7 4 Smoothing function of natural residual The natural residual function Φ given in Definition 2.2 is not differentiable everywhere, and therefore, we cannot employ a derivative-based algorithms such as Newton s method to solve MPSOCC (3.1. To overcome such a difficulty, we will utilize a smoothing technique. Definition 4.1. Let Ψ : R m R m be a nondifferentiable function. Then, the function Ψ µ : R m R m parametrized by µ > is called a smoothing function of Ψ if it satisfies the following properties: For any µ >, Ψ µ is differentiable on R m ; for any z R m, it holds that lim µ + Ψ µ (z = Ψ(z. A smoothing function of the natural residual function can be constructed by means of the Chen-Mangasarian (CM function ĝ : R R [11]. Definition 4.2. A differentiable convex function ĝ : R R + is called a CM function if lim ĝ(α =, lim α (ĝ(α α =, < α ĝ (α < 1 (α R. (4.1 Notice that, if function p µ : R R is defined by p µ (α := µĝ(α/µ with a CM function ĝ and a positive parameter µ, then it becomes a smoothing function for p(α := max{, α}. Thans to this fact, we can next provide a smoothing function P µ for the projection operator P K. Definition 4.3. Let z R m be an arbitrary vector decomposed as z = (z 1, z 2,, z l R m1 R m2 R m l = R m conforming to the given Cartesian structure of K. For an arbitrary CM function ĝ : R R, let g : R m R m be defined as g(z := g 1 (z 1. g l (z l, (4.2 g i (z := ĝ(λ i1 c i1 + ĝ(λ i2 c i2, where λ ij R and c ij R mi ((i, j {1, 2,, l} {1, 2} are the spectral values and the spectral vectors of subvectors z i with respect to K m i, respectively. Then, the smoothing function P µ : R m R m of P K is given as P µ (z := µg(z/µ. Now, by using the above smoothing function P µ, we can define the smoothing function for the natural residual Φ. Definition 4.4. Let µ > be arbitrary. Let g : R m R m, g i : R m i R m i (i = 1, 2,..., l, and P µ : R m R m be defined as in Definition 4.3. Then, the smoothing function Φ µ : R m R m for the natural residual Φ is given as Φ µ (y, z := y P µ (y z ( y z = y µg µ ( y y 1 µg 1 1 z 1 µ =.. (4.3 ( y y l µg l l z l µ Before closing this subsection, we provide the following propositions which will be used in the subsequent analyses. Proposition 4.1. Let P µ : R m R m be defined as in Definition 4.3, and choose z / bd (K K arbitrarily. Let {z } R m and {µ } R ++ be arbitrary sequences such that z z and µ as. Then, we have P K (z = lim P µ (z. (4.4 7

8 Proof. For simplicity, we consider the case where K = K m. Let z and z be decomposed as z = λ 1 c 1 + λ 2 c 2 and z = λ 1c 1 + λ 2c 2, where λ i, λ i R are spectral values, and ci, c i Rm (i = 1, 2 are spectral vectors of z and z, respectively. Since z / bd (K m K m, P K m is differentiable at z and P K m(z is given as I m (λ 1 >, λ 2 >, λ P K m(z = 2 I m + W (λ 1 <, λ 2 >, λ 2 λ 1 O (λ 1 <, λ 2 <, where W := 1 ( r1 r2 2 r 2 r 1 r 2 r2, (r 1, r 2 := (z 1, z 2 z 2, z := (z 1, z 2 R R m 1. On the other hand, by [11, Proposition 5.2], P µ (z is written as ĝ (z1 /µ I m (z2 =, where P µ (z c µ (z2 = b µ z2 c µ z2 z2 a µ I m 1 + (b µ a µ z 2 z2 a µ = ĝ(λ 2/µ ĝ(λ 1/µ λ 2 /µ λ 1 /µ, b µ = 1 2 c µ = 1 ( ( ( λ ĝ 2 λ ĝ 1 2 µ µ z 2 2 (z 2, ( ( ( λ ĝ 2 λ + ĝ 1, µ µ, z := (z 1, z 2 R R m 1, and ĝ is defined as in Definition 4.3. Note that, from the definition of ĝ, we have ĝ(α α, ĝ( α, ĝ (α 1 and ĝ ( α as α. Then, it follows that 1 ( < λ 1 λ 2 1 ( < λ 1 λ 2 ( < λ 1 λ 2 lim a µ = λ 2 /(λ 2 λ 1 (λ 1 < < λ 2, lim b µ = 1/2 (λ 1 < < λ 2, lim c µ = 1/2 (λ 1 < < λ 2. (λ 1 λ 2 < (λ 1 λ 2 < (λ 1 λ 2 < From this fact, it is not difficult to observe that P K m(z = lim P µ (z. Proposition 4.2. [11, Proposition 5.1] Let Φ µ : R m R m R m and Φ : R m R m R m be defined by (2.1 and (4.3, respectively. Let ρ := ĝ(. Then, for any y, z R n and µ > ν >, we have ρ(µ νe K Φ ν (y, z Φ µ (y, z K, ρµe K Φ(y, z Φ µ (y, z K, where e := (e 1, e 2,..., e l R m 1 R m 2 R m l with e i := (1,,,..., R m i for i = 1, 2,..., l. Proposition 4.3. [11, Corollary 5.3 and Proposition 6.1] Let Φ µ : R m R m R m and Φ : R m R m R m be defined by (2.1 and (4.3, respectively. Let g : R m R m be defined by (4.2. Then, the following statements hold. (a Function g is continuously differentiable and g(z = diag( g 1 (z 1,..., g l (z l R m m is symmetric for any z R m, where the latter matrix denotes the bloc-diagonal matrix with bloc-diagonal elements g i (z i, i = 1, 2,..., l. (b For any y, z R m, we have ( y z y Φ µ (y, z = I m g, z Φ µ (y, z = g µ where I m R m m denotes the identity matrix. ( y z µ, 8

9 (c For any y, z R m, we have O y Φ µ (y, z I m, O z Φ µ (y, z I m, ( y z O g I m. µ Proposition 4.4. Let Φ µ : R m R m R m and Φ : R m R m R m be defined by (2.1 and (4.3, respectively. Let ρ := ĝ(. Then, Φ µ (y, z Φ(y, z 2ρµ for any (µ, y, z R ++ R m R m. Proof. For simplicity, we only consider the case where K = K m. Let Φ(y, z Φ µ (y, z = λ 1 c 1 + λ 2 c 2, where λ i R and c i R m (i = 1, 2 are the spectral values and spectral vectors of Φ(y, z Φ µ (y, z. Since e = c 1 + c 2 and ρµe K m Φ(y, z Φ µ (y, z K m from Proposition 4.2, we have ρµ(c 1 + c 2 K m λ 1 c 1 + λ 2 c 2 K m, which implies < λ 1 λ 2 ρµ. Hence, we obtain Φ(y, z Φ µ (y, z = λ 1 c 1 + λ 2 c 2 λ 1 c 1 + λ 2 c 2 2ρµ, where the first inequality is due to the triangle inequality and < λ 1 λ 2, and the last inequality follows from c 1 = c 2 = 1/ 2 and λ 1 λ 2 ρµ. This completes the proof. Proposition 4.5. Let Φ µ : R m R m R m be defined by (4.3. Then, for any µ > ν > and (y, z R m R m, it holds that Φ ν (y, z 1 Φ µ (y, z 1 mρ(µ ν, where ρ = ĝ(. Proof. We first assume K = K m. From Proposition 4.2, we have ρ(µ νe (Φ ν (y, z Φ µ (y, z K m, (4.5 Φ ν (y, z Φ µ (y, z K m, (4.6 where e = (1,,..., R m. Moreover, for any w = (w 1, w 2,..., w m K m, we have since w 1 w w2 m. Therefore, for each i = 1, 2,..., m, we have w 1 w i (i = 1,..., m, (4.7 ρ(µ ν ( Φ ν (y, z Φ µ (y, z 1 ( Φ ν (y, z Φ µ (y, z i Φ ν (y, z i Φ µ (y, z i, (4.8 where the first inequality holds from (4.3 and (4.5, the second equality holds from (4.6 and (4.7, and the last equality holds from the triangle inequality. Summing up (4.8 for all i, we obtain the desired conclusion. When K = K m 1 K m l, we can prove it in a similar way. 5 Algorithm In this section, we propose an SQP type algorithm for MPSOCC (1.1. The SQP method solves a quadratic programming (QP problem in each iteration to determine the search direction. This method is nown as one of the most efficient methods for solving nonlinear programming problems. In the remainder of the paper, to apply the SQP method, we mainly consider MPSOCC (3.1 equivalent to MPSOCC (1.1. We should notice, however, that the SQP method cannot be applied directly to MPSOCC (3.1, since Φ(y, z is not differentiable everywhere. We thus consider the following problem where the smooth equality constraint Φ µ (y, z = replaces Φ(y, z = in each iteration Minimize f(x, y x,y,z subject to Ax b, z = Nx + My + q, Φ µ (y, z =. (5.1 9

10 Given a current iterate (x, y, z satisfying Ax b and z = Nx + My + q, we then generate the search direction (dx, dy, dz by solving the following QP subproblem, which consists of quadratic and linear approximations of the objective and constraint functions of problem (5.1 with µ = µ, respectively, at (x, y, z : Minimize dx,dy,dz ( dx f(x, y + 1 dx dy dy 2 dz B subject to Adx b Ax, ( N M Im y Φ µ (y, z z Φ µ (y, z dx dy dz dx ( dy = dz Φ µ (y, z where B R (n+2m (n+2m is a positive definite symmetric matrix. In the numerical experiments in Section 7, B will be updated by using the modified Broyden-Fletcher-Goldfarb-Shanno (BFGS formula. Note that the Karush-Kuhn-Tucer (KKT conditions of QP (5.2 can be written as x f(x, y dx N A y f(x, y + B dy + M u + y Φ µ (y, z v + η =, dz I m z Φ µ (y, z ( dx ( (5.3 N M Im y Φ µ (y, z z Φ µ (y, z dy = Φ dz µ (y, z, (b Ax Adx η, where (η, u, v R p R m R m denotes the Lagrange multipliers. For simplicity of notation, we denote w := (x, y, z R n R m R m, dw := (dx, dy, dz R n R m R m. Also, we define the l 1 penalty function by, (5.2 θ µ,α (w := f(x, y + α Φ µ (y, z 1, (5.4 where α > is the penalty parameter. Note that this function has the directional derivative θ µ,α(w; dw for any w and dw. Algorithm 1. Step : Choose parameters δ (,, β (, 1, ρ (, 1, σ (, 1, µ (,, α 1 (, and a symmetric positive definite matrix B R (n+2m (n+2m. Choose w = (x, y, z R n R m R m such that Nx + My + q = z and Ax b. Set :=. Step 1: Solve QP subproblem (5.2 to obtain the optimum dw = (dx, dy, dz and the Lagrange multipliers (η, u, v. Step 2: If dw =, then let w +1 := w, α := α 1 and go to Step 3. Otherwise, update the penalty parameter by { α α := 1 if α 1 v + δ, max{ v (5.5 + δ, α 1 + 2δ} otherwise. Then, set the step size τ := ρ L, where L is the smallest nonnegative integer satisfying the Armijo condition Let w +1 := w + τ dw, and go to Step 3. θ µ,α (w + ρ L dw θ µ,α (w + σρ L θ µ,α (w ; dw. (5.6 Step 3: Terminate if a certain criterion is satisfied. Otherwise, let µ +1 := βµ and update B to determine a symmetric positive definite matrix B +1. Return to Step 1 with replaced by

11 In the remainder of this section, we establish the well-definedness of Algorithm 1. We first show the feasibility of QP subproblem (5.2. In general, a QP subproblem generated by the SQP method may not be feasible, even if the original nonlinear programming problem is feasible. However, in the present case, we can show that QP subproblem (5.2 is always feasible under the Cartesian P property of the matrix M. To this end, the following lemma will be useful. Lemma 5.1. Let M R m m be a Cartesian P matrix. Let H i R mi mi (i = 1, 2,..., l be positive definite matrices with m = l i=1 m i, and H R m m be a bloc diagonal matrix with bloc diagonal elements H i (i = 1,..., l. Then, H + M is nonsingular. Proof. The matrix H + M can easily be shown to be a Cartesian P matrix, which is nonsingular. The next proposition shows the feasibility and solvability of QP subproblem (5.2. In the proof, the matrix plays an important role. ( M I D := m y Φ µ (y, z z Φ µ (y, z Proposition 5.1 (Feasibility of QP subproblem. Let M be a Cartesian P matrix, and {w } be a sequence generated by Algorithm 1. Then, (i Ax b and z = Nx +My +q hold for all, and (ii QP subproblem (5.2 is feasible and hence has a unique solution for all. Proof. Since (i can be shown easily, we only show (ii. Since the objective function of QP (5.2 is strongly convex, it suffices to show the feasibility. We first show that the matrix D defined by (5.7 is nonsingular. Note that, by Proposition 4.3(c, z Φ µ (y, z is nonsigular. Let D be the Schur complement of the matrix z Φ µ (y, z with respect to D, that is, D := M + ( z Φ µ (y, z 1 y Φ µ (y, z ( ( y z 1 ( ( y z = M + g I m g ( = M + diag µ g i ( y i, z i, µ µ 1 I mi where y i, and z i, denote the i-th subvectors of y and z, respectively, conforming to the Cartesian structure of K, and each equality follows from Proposition 4.3. Since M is a Cartesian P matrix and ( g i ((y i, z i, /µ 1 I mi R mi mi is positive definite from Proposition 4.3 (c, Lemma 5.1 ensures that D is nonsingular, and hence D is nonsingular. It then follows from (i that ( ( dy dx =, = D 1 dz Φ µ (y, z comprise a feasible solution to (5.2. This completes the proof. The following proposition shows that the search direction dw produced in Step 1 of Algorithm 1 is a descent direction of the penalty function θ µ,α defined by (5.4. It guarantees the well-definedness of the line search in Step 2 in the sense that there exists a finite L satisfying the Armijo condition (5.6. Proposition 5.2 (Descent direction. Let {w } and {dw } be sequences generated by Algorithm 1. Then, we have (a θ µ,α (w ; dw = x f(x, y dx + y f(x, y dy α Φ µ (y, z 1, (b θ µ,α (w ; dw (dw B dw for each. Moreover, if Φ µ (y, z, then the inequality in (b holds strictly. l i=1, (5.7 11

12 Proof. We first show (a. Let J +, J, and J {1, 2,..., m} be the index sets defined by J + = {j Φ µ (y, z j > }, J = {j Φ µ (y, z j = }, J = {j Φ µ (y, z j < }, where Φ µ (y, z j R denotes the j-th component of Φ µ (y, z R m. Then, we have θ µ,α (w ; dw = f(x, y ( dx dy + α j J + [ Φµ (y, z ] ( dy j dz [ + α Φµ (y, z ] ( dy [ j dz α Φµ (y, z ] ( dy j dz j J j J where [ Φ µ (y, z ] j denotes the j-th column vector of Φ µ (y, z. Since [ Φµ (y, z ] ( dy j dz = Φ µ (y, z j from the constraint of QP subproblem (5.2, we have, (5.8 θ µ,α (w ; dw = x f(x, y dx + y f(x, y dy α Φ µ (y, z 1. (5.9 We next show (b. Taing the inner product of dw = (dx, dy, dz and both sides of the first equality in the KKT conditions (5.3 with dw = dw, η = η, u = u, v = v for the subproblem (5.2, we obtain ( dx f(x, y dy + (dw B dw + (u (Ndx + Mdy dz ( dy + (v Φ µ (y, z dz + (η Adx =. (5.1 Moreover, from the constraints of the subproblem (5.2 and the KKT conditions (5.3, we have and Ndx + Mdy dz =, (5.11 ( dy Φ µ (y, z dz = Φ µ (y, z, (5.12 = (η (b Ax Adx = (η Adx + (η (b Ax (η Adx, (5.13 where the inequality is due to η and b Ax from (5.3. Substituting (5.11 (5.13 into (5.1, we have ( dx f(x, y dy + (dw B dw (v Φ µ (y, z. Furthermore, from (5.9, we obtain θ µ,α (w ; dw (dw B dw + (v Φ µ (y, z α Φ µ (y, z 1 = (dw B dw + (vj α [ Φ µ (y, z ] j j J + + (vj + α [ Φ µ (y, z ] j j J (dw B dw, where the last inequality follows from α > v and the definitions of J + and J. Moreover, if Φ µ (y, z, then the last inequality holds strictly since J + J. This completes the proof of (b. 12

13 6 Convergence analysis In this section, we study the convergence property of the proposed algorithm. To begin with, we mae the following assumption. Assumption 1. Let sequences {w } and {B } be produced by Algorithm 1. (a {w } is bounded. (b There exist constants γ 1, γ 2 > such that γ 1 d 2 d B d γ 2 d 2 for any d R n+2m and. (c There exists a constant c > such that D 1 c for any, where D is the matrix defined by (5.7. Assumption 1 (b means that {B } is bounded and uniformly positive definite. Assumption 1 (c holds if and only if any accumulation point of {D } is nonsingular. The next proposition provides a sufficient condition under which Assumption 1 (c holds. Proposition 6.1. Suppose that M is a Cartesian P matrix and Assumption 1 (a holds. Then, Assumption 1 (c holds. Proof. Let (E y, E z be an arbitrary accumulation point of {( y Φ µ (y, z, z Φ µ (y, z }. Then, it suffices to show that the matrix ( M Im D := is nonsingular. By Proposition 4.3, E y and E z satisfy the following three properties: (a E y + E z = I m ; (b E y I m and E z I m ; E y E z (c E y and E z are symmetric and have the bloc-diagonal structure conforming to the Cartesian structure of K = K m 1 K m l. From (a and (b, there exists an orthogonal matrix H R m m, HE y H = diag(α i m i=1, HE z H = diag(1 α i m i=1, α i 1 (i = 1, 2,..., m, (6.1 where α i (i = 1, 2,..., m are the eigenvalues of E y. Moreover, from (c, H has the same bloc-diagonal structure as E y and E z. Hence, M := HMH is a Cartesian P matrix by Proposition 2.2. Now, let D R 2m 2m be defined as ( ( ( H H D := D H H = M I m diag(α i m i=1 diag(1 α i m i=1 and let (ζ, η R m R m be an arbitrary vector such that D ( ζη =. Then, we have, (6.2 Mζ = η, (6.3 α i ζ i + (1 α i η i = (i = 1, 2,..., m. (6.4 If α i =, then we have ( Mζ i = η i = from (6.3 and (6.4. If α i = 1, then we have ζ i = from (6.4. If < α i < 1, then we have ζ i ( Mζ i = ζ i η i = α i (1 α i 1 ζ 2 i. Thus, for all i, we have ζ i( Mζ i. Since M is a Cartesian P matrix and every Cartesian P matrix is a P matrix, we must have ζ = η =. Hence, D is nonsingular. From (6.2 and the nonsingularity of H, matrix D is also nonsingular. The following three lemmas play crucial roles in establishing the convergence theorem for the algorithm. 13

14 Lemma 6.1. Let {w } be a sequence generated by Algorithm 1, and dw := (dx, dy, dz R n R m R m be the unique optimum of QP subproblem (5.2 for each. Let {(µ, τ } R ++ R ++ be a sequence converging to (, and α > be a fixed scalar. Suppose that {w } satisfies Assumption 1(a. In addition, assume that {dw } has an accumulation point, and w := (x, y, z and dw := (dx, dy, dz are arbitrary accumulation points of {w } and {dw }, respectively. Then we have provided y z / bd (K K and dw. ( θµ,α(w + τ dw θ µ,α(w lim sup θ µ τ,α(w ; dw Proof. Taing subsequences if necessary, we may suppose w w and dw dw. Moreover, we have θ µ,α(w ; dw = x f(x, y dx + y f(x, y dy α Φ µ (y, z 1 as shown in (5.9. Thus, we have only to show lim sup ( θµ,α(w + τ dw θ µ,α(w From the mean-value theorem and the continuity of f, we have Therefore, it suffices to show that τ ( dx f(x, y α Φ(y, z dy 1. f(x + τ dx, y + τ dy f(x, y lim = f(x, y τ lim sup ( dx. dy Φ µ (y + τ dy, z + τ dz j Φ µ (y, z j τ Φ(y, z j (6.5 for each j. By Definition 5 and Proposition 4.3, we have y Φ µ (y, z = I m P µ (y z and z Φ µ (y, z = P µ (y z, which together with the constraints of QP subproblem (5.2 yield Hence, we have Φ µ (y, z = y Φ µ (y, z dy + z Φ µ (y, z dz = (I P µ (y z dy + P µ (y z dz = dy P µ (y z (dy dz. (6.6 Φ µ (y + τ dy, z + τ dz Φ µ (y, z = τ dy ( P µ (y + τ dy (z + τ dz P µ (y z = τ ( Φ µ (y, z + P µ (y z (dy dz P µ (y z + τ (dy dz + P µ (y z = τ Φ µ (y, z + τ δ, (6.7 where the first equality is due to (4.3, the second equality follows from (6.6, and δ := (δ 1, δ 2,..., δ m R m is given by δj := P µ (y z j (dy dz τ 1 ( Pµ (y z + τ (dy dz j P µ (y z j. (6.8 To show (6.5, we consider three cases: (i Φ(y, z j =, (ii Φ(y, z j > and (iii Φ(y, z j <. In case (i, we first notice that Φ µ (y + τ dy, z + τ dz j Φ µ (y, z j τ Φ µ (y + τ dy, z + τ dz j Φ µ (y, z j τ = Φ µ (y, z j δ j, (6.9 14

15 where the equality follows from (6.7. By applying the mean-value theorem in (6.8, we can find ζ j (, 1 for each such that δ j = ( P µ (y z P µ ( y z + ζ j τ (dy dz j (dy dz. (6.1 Since Proposition 4.1 and the boundedness of {dw } imply lim P µ (y z P µ (y z + ζ j τ (dy dz =, (6.11 we obtain lim δj =. Moreover, by Proposition 4.4, we have lim Φ µ (y, z j = Φ(y, z j =. Then, by letting in (6.9, we obtain (6.5. In case (ii and case (iii, we have and Φ µ (y + τ dy, z + τ dz j Φ µ (y, z j = Φ µ (y j, z j + δ j Φ µ (y + τ dy, z + τ dz j Φ µ (y, z j = Φ µ (y j, z j δ j, respectively. Then, a similar argument to that in case (i leads to the desired inequality (6.5. Lemma 6.2. Let {w } be a sequence generated by Algorithm 1. Suppose that Assumption 1 holds. Then, we have the following statements. (i {dw } and {(u, v } are bounded. (ii There exists such that α = α for all. (iii The sequences {θ µ,α (w } and {θ µ,α (w +1 } converge to the same limit. Proof. We first prove (i. Let ( dỹ d z ( := D 1 Φ µ (y, z (6.12 and d w := (, dỹ, d z, where D is defined by (5.7. Then, d w is a feasible point of QP subproblem (5.2. Note that the objective function of QP subproblem (5.2 is rewritten as 1 2 (dw B 1 g B (dw B 1 g + constant, where g := ( f(x, y,. Since dw is the optimum of QP subproblem (5.2, we have (d w B 1 g B (d w B 1 This together with Assumption 1 (b implies g (dw B 1 g B (dw B 1 g. γ 2 d w B 1 g 2 γ 1 dw B 1 g 2. (6.13 Now, notice that {d w } is bounded from (6.12, Assumption 1 (a, (c and Proposition 4.4. In addition, {B 1 } and {g } are also bounded from Assumption 1 (a, (b. Thus, we have the boundedness of {dw } from (6.13. On the other hand, from the first equality of the KKT conditions (5.3, we have ( u v = (D 1 (( y f(x, y + B dw, where B is the 2m (n + 2m matrix consisting of the last 2m rows of B. This equation together with Assumption 1 and the boundedness of {dw } yields the boundedness of {(u, v }. We next prove (ii. From the update rule (5.5, we can easily see that {α } is nondecreasing. Moreover, if v > α 1 δ, (6.14 then we have α = max{ v + δ, α 1 + 2δ} α 1 + 2δ, that is, α increases at least by 2δ at a time. Let ˆK := { v > α 1 δ}. If ˆK =, then α as and hence { v } is unbounded from (6.14. However this contradicts (i. Thus we have (ii. 15

16 We finally show (iii. Since we have (ii, there exist α and such that α = α for all. In what follows, we suppose. Since µ +1 µ, Proposition 4.5 together with (5.4 implies θ µ+1,α(w +1 + αmρµ +1 θ µ,α(w +1 + αmρµ (6.15 θ µ,α(w + αmρµ, (6.16 where the last inequality follows from the Armijo condition (5.6 and Proposition 5.2 (b. From (6.16, {θ µ,α(w +αmρµ } is a monotonically nonincreasing sequence. In addition, {θ µ,α(w +αmρµ } is bounded, since {θ µ,α(w } is bounded from Assumption 1 (a. Therefore, {θ µ,α(w + αmρµ } is convergent. This fact together with lim µ = and (6.16 yields that {θ µ,α(w +1 } and {θ µ,α(w } must converge to the same limit. Finally, we show that a sequence generated by the algorithm globally converges to B-stationary point of MPSOCC (3.1 under the assumption that any accumulation point w = (x, y, z satisfies y z / bd (K K, which is equivalent to the nondegeneracy condition given in Definition 3.1 if y and z satisfy the SOC complementarity condition. Theorem 6.1. Let {w } be a sequence generated by Algorithm 1. Suppose that {w } satisfies Assumption 1. Let w = (x, y, z be an arbitrary accumulation point of {w }. If (y, z satisfies y z / bd (K K, then w is a B-stationary point of MPSOCC (1.1. Proof. By Lemma 6.2(ii, there exists some constant α > and such that α = α for all. In this proof, we suppose without loss of generality that α = α holds for all. We first show that lim dw =. (6.17 From Proposition 5.2 (b and Assumption 1 (b, we have which together with Armijo condition (5.6 yields Hence, from Lemma 6.2 (iii, we obtain θ µ,α(w ; dw (dw B dw γ 1 dw 2, (6.18 θ µ,α(w +1 θ µ,α(w + στ θ µ,α(w ; dw θ µ,α(w γ 1 στ dw 2. lim τ dw 2 =. Now, assume for contradiction that (6.17 does not hold. Then, there exists an infinite index set K {, 1,...} such that and hence lim dw >, (6.19 K lim τ =. K Let l be the smallest nonnegative integer L satisfying (5.6, i.e., ρ l = τ. Then, from the definition of l, we have θ µ,α(w + ρ l 1 dw > θ µ,α(w + σρ l 1 θ µ,α(w ; dw, which implies ξ := θ µ,α(w + ρ l 1 dw θ µ,α(w ρ l 1 θ µ,α(w ; dw > (1 σθ µ,α(w ; dw. (6.2 By Lemma 6.1 together with lim, K ρ l 1 = and y z / bd (K K, we have lim sup, K ξ. Moreover, we have from (6.18 (1 σθ µ,α(w ; dw (1 σγ 1 dw 2. (

17 From (6.2 and (6.21, we must have lim, K dw =. However, this contradicts (6.19, and hence we have (6.17. Next, we show that w satisfies the KKT conditions (3.2 of MPSOCC (3.1. Let {(η, u, v } be the sequence of multipliers corresponding to {dw }. Then, {(u, v } is bounded from Lemma 6.2 (i. Hence, there exist vectors u, v and an index set K such that lim, K (w, u, v = (w, u, v. By (5.3 and letting X := {x R n Ax b} we have ζ N X (x {} 2m, (6.22 ( dx ( N M Im y Φ µ (y, z z Φ µ (y, z dy = dz Φ µ (y, z, (6.23 b Ax Adx, (6.24 where ζ := x f(x, y dx N y f(x, y + B dy + M u + y Φ µ (y, z v, (6.25 dz I m z Φ µ (y, z and (6.22 follows from the first equation of (5.3 with N X (x = {A η η, η (Ax b = }. By letting K tend to in (6.23 and (6.24, we have b Ax, Φ(y, z =. (6.26 Note that (6.26 implies x X. In addition, note that {ζ } K is a convergent sequence satisfying (6.22, since {B } is bounded from Assumption 1 and lim Φ µ (y, w = Φ(y, w from the nondegeneracy condition. Then, these facts together with (6.17 and the closedness of the point-to-set map N X ( yield x f(x, y N T y f(x, y + M T u + y Φ(y, z v = lim N, K I m z Φ(y, z ζ X (x {} 2m. This together with (6.26 means that w satisfies the KKT conditions (3.2 of MPSOCC (3.1. By Proposition 3.2, w is a B-stationary point of MPSOCC ( Numerical experiments In this section, we implement Algorithm 1 for solving problem (1.1 and report some numerical results. The program is coded in Matlab 28a and run on a machine with an IntelRCore2 Duo E685 3.GHz CPU and 4GB RAM. In Step of the algorithm, we set the parameters as δ := 1, α 1 := 1, σ := 1 3, ρ :=.9. The choice of smoothing parameters {µ }, i.e., µ and β, and a starting point w vary with the experiment. We let B be the identity matrix, and update B by the modified BFGS formula: B +1 := B B s (B s (s B s + ζ (ζ (s ζ with s = w +1 w and ζ = θ ζ +(1 θ B s, where ζ := w L µ+1 (w +1, u, v, η w L µ (w, u, v, η, L µ denotes the Lagrangian function defined by L µ (w, u, v, η := f(x, y + Φ µ (y, zv + (Nx + My + q zu + (Ax bη, and θ is determined by 1 if (s ζ.2(s B s θ :=.8(s B s (s (B s ζ otherwise. 17

18 In Step 1, we use the quadprog solver in Matlab Optimization Toolbox for solving the QP subproblems. In Step 3, we terminate the algorithm if the following condition is satisfied: Φ(y, z + dw 1 7. (7.1 The rationale for using (7.1 is as follows. If Φ(y, z =, i.e., K y z K holds, then w = (x, y, z is feasible to MPSOCC (1.1, since the remaining constraints Ax b and z = Nx + My + q always hold from Proposition 5.1. Moreover, if dw =, then w satisfies the KKT conditions (3.2 of MPSOCC (3.1. Thus, by Theorem 6.1, Φ(y, z + dw = indicates that w is a B-stationary point under the assumption y z / bd (K K, which is equivalent to the nondegeneracy condition in Definition 3.1 if K y z K. Hence, (7.1 is appropriate for a stopping criterion of the algorithm. As the CM function, we choose ĝ(α := ((α /2 + α/2. Experiment 1 In the first experiment, we solve the following test problem of the form (1.1: Minimize x 2 + y 2 subject to Ax b, z = Nx + My + q, K y z K, (7.2 where (x, y, z R 1 R m R m, and each element of A R 1 1, N R m 1 is randomly chosen from [ 1, 1]. Moreover, each element of b R 1 is randomly chosen from [, 1]. In addition, M R m m is a positive semidefinite symmetric matrix generated by M = M 1 M1 +.1I, and M 1 R m m is a matrix whose entries are randomly chosen from [ 1, 1]. The vector q R m is set to be q := ξ z Mξ y with ξ y R m and ξ z R m, whose components are randomly chosen from [ 1, 1]. We choose different Cartesian structures for K, and generate 5 problems for each K. In applying Algorithm 1, we set an initial point w = (x, y, z := (, ξ y, ξ z R 1 R m R m, so that Ax b and z = Nx + My + q are satisfied. We choose smoothing parameters {µ } as µ := 1.8. The obtained results are shown in Tables 1 and 2, where (K ν κ := K ν K ν K ν R νκ and each column represents the following: ite: the average number of iterations among 5 test problems for each K; cpu(s: the average cpu-time in second among 5 test problems for each K; non(%: percentage of test problems whose solutions obtained by Algorithm 1 satisfy the nondegeneracy condition in Definition 3.1. Recall that convergence to a B-stationary point is proved under the nondegeneracy condition. Hence, the value of non represents the percentage of problems for which the algorithm successfully finds B-stationary points. From Table 1, we can observe that ite does not change so much although the values of cpu(s tends to be larger as m increases. From Table 2, we can see that non(% tends to be less than 1 if K includes K 1 or K 2. Indeed, when K = (K 1 1 and (K 2 5, the values of non(% is 74 and 86, respectively, whereas it becomes 1 when the dimension of all SOCs in K is larger than 1. 18

19 m K ite cpu(s non(% 1 K K K K K K K K K K Table 1: Results for problems with a single SOC complementarity constraint (Experiment 1 m K ite cpu(s non(% 1 K (K K 5 K 2 K (K K 5 K 2 (K 1 2 K 5 (K (K (K Table 2: Results for problems with multiple SOC complementarity constraints (Experiment 1 Experiment 2. In the second experiment, we apply Algorithm 1 to a bilevel programming problem with a robust optimization problem in the lower-level. Bilevel programming has wide application such as networ design and production planning [2, 9]. On the other hand, robust optimization is nown to be a powerful methodology to treat optimization problems with uncertain data [3, 4]. In this experiment, we solve the following problem: with Minimize x (x,y R 4 R Cy i=1 x i subject to x i 5 (i = 1, 2, 3, 4, 1 x 1 + 2x 2 + x 4 3, 1 x 2 + x 3 x 4 2, y solves P (x, P (x : Minimize max y R 4 x U (x x y + 1 r 2 y My, where r is an uncertainty parameter, U r (x R 4 is an uncertainty set defined by U r (x := { x R 4 x x r}, and M :=, C := For solving problem (7.3, we introduce an auxiliary variable γ R to reformulate the lower-level minimax problem P (x as the following SOCP: 1 Minimize (γ,y R R 4 2 y My + x y + rγ ( γ subject to K y 5. (7.3 19

20 Furthermore, the above SOCP can be rewritten as the following SOC complementarity problem: ( ( γ r K 5 K 5. y My + x Thus, we can convert problem (7.3 to the following problem: Minimize x (x,y,z,γ R 4 R 4 R 5 R Cy i=1 x i subject to x i 5 (i = 1, 2, 3, 4, 1 x 1 + 2x 2 + x 4 3, 1 x 2 + x 3 x 4 2, z = ( r My + x, K 5 ( γ z K y 5, (7.4 which is of the form (1.1. For the sae of comparison, problem (7.4 is solved not only by Algorithm 1, but also by the smoothing method [2], which is described as follows: Smoothing method Step. Choose a positive sequence {τ l } such that τ l. Set l :=. Step 1. Find a stationary point w l = (x l, y l, z l of the smoothed problem (5.1 with µ = τ l. Step 2. If w l is feasible for MPSOCC (1.1, then stop. Otherwise, set l := l + 1 and go to Step 1. Each algorithm is implemented in the following way: In Step of Algorithm 1, we set smoothing parameters µ :=.8 ( and a starting point x := (1, 1, 1, 1, (γ, y, z := (,, R R 4 R 5. The choice of the other parameters is the same as in Experiment 1. In Step of the smoothing method, we set τ l :=.8 l for l. In Step 1, for solving problem (5.1 with µ = τ l, we apply Algorithm 1 with slight modification, where the smoothing parameter µ is fixed to τ l for all and the termination criterion is replaced by dw 1 7. In Step 2, we stop the smoothing method when Φ(y l, z l 1 7 is satisfied. *2 We then test both the methods to problem (7.4 with r =.2,.4,.6,.8 and.1. The obtained results are shown in Tables 3 and 4, whose columns represent the following: (x, y, γ : the value of x, y, γ obtained by Algorithm 1; (λ 1, λ 2: spectral values of ( γ y +z with respect to K 5 defined as in Definition 2.1, where z := (r, M( γ y + x ; ite out : the number of outer iterations; QP: the number of QP-subproblems (5.2 solved in each trial. From Table 3, for all r, we can observe (λ 1, λ 2 >, which means ( γ y + z int K 5, i.e., the nondegeneracy condition holds at the obtained solution. Hence, Algorithm 1 finds a B-stationary point of problem (7.4 successfully. From Table 4, we cannot find a significant difference between the values of ite out for the two methods. However, it is observed that the value of QP in the smoothing method tends to be much larger than that in Algorithm 1. Indeed, when r =.2, the smoothing method has QP = 218, which is almost five times larger than QP = 44 in Algorithm 1. This fact suggests that the smoothing method needs to solve a number of QP-subproblems in Step 1 for solving each smoothed problem (5.1 with fixed µ, while Algorithm 1 only solves one QP subproblem (5.2 for each smoothed problem (5.1. As a result, the computational cost in the smoothing method tends to be larger than Algorithm 1. *2 The remaining constraints Ax l b and z l = Nx l + My l + q are automatically satisfied since w l is feasible to the smoothed problem (5.1. 2

21 r x y γ (λ 1, λ 2.2 (.9236,.9618,.382,. (.6152,.112,.915, (.4, (.9495,.9747,.253,. (.617,.116,.95, (.8, (.9754,.9877,.123,. (.6189,.192,.985, (.12, (1.21, 1.7,.,.7 (.6218,.184,.121, (.16, (1.416, 1.139,.,.139 (.6356,.1123,.144, (.2, Table 3: Results for Algorithm 1 (Experiment 2 Algorithm 1 smoothing method r cpu(s ite out QP cpu(s ite out QP Table 4: Comparison of Algorithm 1 and the smoothing method 8 Conclusion In this paper, we have considered the mathematical program with second-order cone (SOC complementarity constraints. We have proposed an algorithm based on the smoothing and the sequential quadratic programming (SQP methods, in which we replace the SOC complementarity constraints with smooth equality constraints by means of the natural residual and its smoothing function, and apply the SQP method while decreasing the smoothing parameter gradually. We have shown that the proposed algorithm possesses the global convergence property under the Cartesian P and the nondegeneracy assumptions. We have further confirmed the efficiency of the algorithm through numerical experiments. References [1] F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming, 95 (23, pp [2] J. F. Bard, Practical Bilevel Optimization, Kluwer Academic Publishers, The Netherlands (1998. [3] A. Ben-Tal and A. Nemirovsi, Robust convex optimization, Mathematics of Operations Research, 23 (1998, pp [4] A. Ben-Tal and A. Nemirovsi, Robust solutions of uncertain linear programs, Operations Research Letters, 25 (1999, pp [5] J.-S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with secondorder cones, Mathematical Programming, 11 (24, pp [6] X. Chen and H. D. Qi, Cartesian P-property and its applications to the semidefinite linear complementarity problem, Mathematical Programming, 16 (26, pp [7] X. D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems, Computational Optimization and Applications, 25 (23, pp [8] R. W. Cottle, J.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, New Yor, [9] S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers, The Netherlands (22. 21