First-order optimality conditions for mathematical programs with second-order cone complementarity constraints

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1 First-order optimality conditions for mathematical programs with second-order cone complementarity constraints Jane J. Ye Jinchuan Zhou Abstract In this paper we consider a mathematical program with second-order cone complementarity constraints SOCMPCC. The SOCMPCC generalizes the mathematical program with complementarity constraints MPCC in replacing the set of nonnegative reals by second-order cones. There are difficulties in applying the classical Karush- Kuhn-Tucker condition to the SOCMPCC directly since the usual constraint qualification such as Robinson s constraint qualification never holds if it is considered as an optimization problem with a convex cone constraint. Using various reformulations and recent results on the exact formula for the proximal/regular and limiting normal cone, we derive necessary optimality conditions in the forms of the strong-, Mordukhovich- and Clarke- S-, M- and C- stationary conditions under certain constraint qualifications. We also show that unlike the MPCC, the classical KKT condition of the SOCMPCC is in general not equivalent to the S-stationary condition unless the dimension of each second-order cone is not more than 2. Finally, we show that reformulating an MPCC as an SOCMPCC produces new and weaker necessary optimality conditions. Key words: mathematical program with second-order cone complementarity constraints, necessary optimality conditions, constraint qualifications, S-stationary conditions, M-stationary conditions, C-stationary conditions. AMS subject classification: 9C3, 9C33, 9C46. Introduction In this paper we consider the following mathematical program with second-order cone complementarity constraints SOCMPCC or MPSOCC SOCMPCC min fz s.t. hz =, gz, K i G i z H i z K i, i =,, J, where a b means that the vector a is perpendicular to vector b. Throughout the paper we assume that f : R n R, g : R n R p, h : R n R q, G i : R n R m i, H i : R n R m i are The first author work was partially supported by NSERC. The second author s work was supported by National Natural Science Foundation of China 248, Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada V8W 2Y2 janeye@uvic.ca. Department of Mathematics, School of Science, Shandong University of Technology, Zibo 25549, P.R. China jinchuanzhou@63.com.

2 SOCMPCC 2 all continuously differentiable and K i is an m i -dimensional second-order cone defined as K i := {x = x, x 2 R R m i x x 2 }, where denotes the Euclidean norm and when m i =, K i stands for the set of nonnegative reals R +. In particular, SOCMPCC with all m i = for i =,, J coincides with the mathematical program with complementarity constraints MPCC which has received a lot of attention in the last twenty years or so [9, 3]. The generalization from MPCC to SOCMPCC has many important applications. We briefly review two of them. In practice it is more realistic to assume that an optimization problem involves uncertainty. An approach to optimization under uncertainty is robust optimization. For example, consider a robust bilevel programming problem where for a fixed upper level decision variable x, the lower level problem is replaced by its robust counterpart: P x : min y {fx, y, ζ : gx, y, ζ ζ U}, where U is some uncertainty set in the space of the data. It is well-known see [2] that if the uncertainty set U is given by a system of conic quadratic inequalities, then the deterministic counterpart of the problem P x is a second-order cone program. If this second-order cone program can be equivalently replaced by its Karush-Kuhn-Tucker KKT condition, then it yields an SOCMPCC. Another application of SOCMPCC is in modelling an inverse quadratic programming problem over the second-order cone, in which the parameters in a given second-order cone quadratic programming problem need to be adjusted as little as possible so that a known feasible solution becomes optimal see [28] for details. It is known that if an MPCC is treated as a nonlinear program with equality and inequality constraints, then Mangasarian-Fromovitz constraint qualification MFCQ fails to hold at each feasible point of the feasible region; see [26, Proposition.]. This causes great difficulties in applying classical theories and algorithms in nonlinear programs directly to MPCCs. To remedy this problem, several variants of stationary conditions such as the strong S-, Mordukhovich M-, Clarke C- stationary conditions have been proposed and constraint qualifications under which a local minimizer is an S-, M-, C-stationary point have been studied; see e.g., [7, 23] for a detailed discussion. For mathematical program with semidefinite cone complementarity constraints SDCMPCC, the matrix analogue of the MPCC, it was shown in [6] that Robinson s CQ, which is the usual constraint qualification for an optimization problem with a convex cone constraint, fails to hold at each feasible point and the corresponding S-, M-, C-stationary conditions were proposed and the constraint qualifications under which a local minimizer is an S-, M-, C-stationary point have been studied. The same difficulties exist for SOCMPCC. Notice that the cone complementarity constraint K Gz Hz K, where G, H : R n R m and K is the m-dimensional second-order cone, amounts to the following convex cone constraints: Gz, Hz, Gz K, Hz K. In this paper we show that if SOCMPCC is regarded as an optimization problem with a convex cone constraint, then Robinson s CQ fails to hold at each feasible point of SOCM- PCC.

3 SOCMPCC 3 So far there are only a few papers devoted to the study of SOCMPCC [8, 4, 8, 9, 2, 27, 28, 29] and [8, 9, 2, 27, 28, 29] mainly study numerical algorithms which are not the main purpose of this paper. To the best of our knowledge, the problem SOCMPCC was studied for the first time by Outrata and Sun in [4]. The approach taken was to consider the cone complementarity constraint as Gz Hz, Gz gphπ K, where gphπ K is the graph of the metric projection operator onto the second-order cone K. By computing the limiting normal cone to gphπ K or equivalently the limiting coderivative of the metric projection Π K, an M-stationary condition was shown to be necessary for optimality under the condition that there is no abnormal multipliers see [4, Theorem 6]. The same reformulation was further taken in Zhang, Zhang and Wu [27] to define M- and S-stationary conditions in terms of the regular and the limiting coderivative of the metric projection onto the second-order cone respectively see [27, Definitions 3.6 and 3.7]. Moreover a B-stationary condition is defined in [27, Definition 3.3] and it was shown that under the SOCMPCC-LICQ, the B-stationarity is equivalent to the S-stationarity [27, Lemma 3.2]. Moreover in [27, Definition 3.5] the C-stationary condition was proposed to be the nonsmooth KKT condition involving the Clarke generalized gradient for problem SOCMPCC where the cone complementarity constraint is reformulated as a nonsmooth equation constraint: Gz Π K Gz Hz =. However these optimality conditions are not in forms that are analogues to the S-, M- and C- stationary conditions for MPCCs and they are not explicit due to the existence of coderivatives or Clarke subdifferential of the metric projection onto the second-order cone in these formulas. Notice that the second-order cone complementarity constraint can be reformulated as a nonconvex cone constraint: Gz, Hz Ω, where Ω := {x, y x K, y K, x T y = } is called the second-order cone complementarity set or complementarity cone since it is a cone. Note that Ω is nonconvex due to the existence of complementarity conditions. If the exact expression for the regular and the limiting normal cones of second-order cone complementarity sets can be derived, then the corresponding stationary conditions would be the suitable generalization of the S- and M-stationary conditions. The first attempt in this direction was initiated by Liang, Zhu and Lin in [8] where they tried to derive exact expressions for the regular and the limiting normal cones of the second-order cone complementary set by using the relationships between the metric projection operator and the second-order cone complementary set. Unfortunately, there are some gaps in their expressions of the regular and the limiting normal cones, mainly on the boundary points, which result in gaps in their proposed expressions for the S-, M-, and C-stationary conditions. In a recent paper [25], we fill in this gap and establish the correct exact expressions for the regular and limiting normal cone of the second-order cone complementary set. Furthermore, we show that the regular and the proximal normal cones to the second-order cone complementary set coincide with each other. Using these exact expressions for the regular and the limiting normal cone of the second-order cone complementary set, in this paper we propose S-, M-,

4 SOCMPCC 4 and C-stationary conditions for SOCMPCC in a form that are analogues to the S-, M- and C-stationary conditions for MPCCs. It is well-known that for MPCC, the classical KKT condition is equivalent to the S- stationary condition see e.g. [7]. For SDCMPCC it was shown in [6] that in general the classical KKT condition is stronger than the S-stationary condition but these two conditions may not be equivalent. It is natural to ask the question whether or not the classical KKT condition is equivalent to the S-stationary condition for SOCMPCC. In this paper we show that for SOCMPCC, in general the classical KKT condition is a stronger condition than the S-stationary condition while these two concepts coincide when the dimension of each second-order cone K i is not more than 2. Moreover an example is given to illustrate that an S-stationary point may not be a classical KKT point when one of the second-order cone K i has dimension greater than 2. Since in general the classical KKT condition and the S-stationary condition are different, we introduce a new stationary point concept called K-stationary point, which is equivalent to the classical KKT point. Furthermore we have derived an exact expression for the set of all multipliers satisfying the K-stationary condition and shown that it is just a subset of the regular normal cone of the second-order cone complementarity set. We summarize our main contributions as follows: We show that Robinson s CQ fails to hold at every feasible point of SOCMPCC if the SOCMPCC is treated as an optimization problem with a convex cone constraint. We obtain the precise description for the S-, M-, and C-stationary conditions in the forms that are analogues to the associated stationary conditions for MPCCs and show that they are necessary for optimality under the corresponding Clarke calmness conditions. We also show that the S-stationary condition is a necessary optimality condition for a local minimum if the SOCMPCC LICQ holds. Moreover we show that for the case where all mappings are affine and the dimension of each second-order cone is less or equal to 2, a local minimal solution of SOCMPCC must be an M-stationary point without any further constraint qualification. We derive the relationships between various stationary conditions and shown that in general the K-stationary condition is stronger than the S-stationary condition but not equivalent and these two concepts coincide when the dimension of all K i is less or equal to 2. We obtain the relationship between various Clarke calmness conditions for the general optimization problem with symmetric cone complementarity constraints. Such results are new even for the case of MPCCs. We establish the relationship of various stationary points between MPCC and its SOCMPCC reformulation. We organize our paper as follows. Section 2 contains the preliminaries. In Section 3, we show that Robinson s CQ never holds if SOCMPCC is considered as an optimization problem with a convex cone constraint. The K-stationary condition is introduced and studied in this section. In Sections 4, 5 and 6, we give the explicit expressions for the S-, M- and C-stationary conditions and propose some constraint qualifications for them to be necessary for optimality. Section 7 gives the connections among various stationary

5 SOCMPCC 5 conditions and various Clarke calmness conditions. In Section 8 we reformulate MPCC as SOCMPCC and obtain some new and weaker necessary optimality conditions for MPCCs. The following notations will be used throughout the paper. We denote by I and O the identity and zero matrix of appropriate dimensions respectively. For a matrix A, we denote by A T its transpose. The inner product of two vectors x, y is denoted by x T y or x, y. For x = x, x 2 R R m, we write its reflection about the x axis as ˆx := x, x 2. Denote by Rx the set {tx t R}. R + x and R ++ x where R + := [, and R ++ :=, are similarly defined. For a set C, denote by intc, clc, bdc, coc, C c its interior, closure, boundary, convex hull, and complement, respectively. The polar cone of a vector v is v := {x x T v }. Given a point x R n and ε >, B ε x denotes an open ball centered at x with radius ε while B denotes the open unit ball center at the origin of an appropriate dimension. For a differentiable mapping H : R n R m and a vector x R n, we denote by J Hx the Jacobian matrix of H at x and Hx := J Hx T. The graph of a set-valued mapping Φ : R n R m, is denoted by gphφ, i.e., gphφ := {z, v R n R m v Φz}. 2 Preliminaries In this section we review some basic concepts in variational analysis and then specialize it to the second-order cone and the second-order cone complementarity set. 2. Background in variational analysis First we summarize some background materials on variational analysis which will be used throughout the paper. Detailed discussions on these subjects can be found in [4, 5,, 2, 6]. Let C be a nonempty subset of R n. Given x clc, the proximal normal cone of C at x is defined as N π Cx := {v R n M >, such that v, x x M x x 2 x C} and the regular/fréchet normal cone is N C x := {v R n v, x x o x x x C}, where o means that oα/α as α. The limiting/mordukhovich normal cone is defined as the outer limit of either the proximal normal cone or the regular normal cone, i.e., N C x := { lim i ζ i ζ i N π Cx i, x i x, x i C} = { lim i ζ i ζ i N C x i, x i x x i C}. Proposition 2. Change of coordinates [6, Exercise 6.7] Let F : R n R m be smooth and set D R m. Suppose that F x has full column rank m at a point x F := {x R n F x D}. Then N F x = { F x y y N D F x }, N F x = { F x y y N D F x }.

6 SOCMPCC 6 Let Φ : R n R m be a set-valued map and x, y gphφ. The regular coderivative and the limiting Mordukhovich coderivative of Φ at x, y are the set-valued mappings defined by D Φx, y v := {u R n u, v N gphφ x, y }, D Φx, y v := {u R n u, v N gphφ x, y }, respectively. We omit y in the coderivative notation if the set-valued map Φ is single-valued at x. For a single-valued Lipschitz continuous map Φ : R n R m, the Bouligand-subdifferential B Φ is defined as B Φx = { lim k J Φx k x k x, Φ is diffferentiable at x k }. It is known that co B Φx = c Φx, the Clarke generalized Jacobian of Φ at x see [4]. Moreover if Φ is a continuously differentiable single-valued map, then D Φx = D Φx = Φx. 2.2 Background in variational analysis associated with the second-order cone Let K be the m-dimensional second-order cone. The topological interior and the boundary of K are respectively. intk = {x, x 2 R R m x > x 2 } and bdk = {x, x 2 R R m x = x 2 }, Proposition 2.2 For any x, y bdk\{}, the following equivalence holds: x T y = y = kˆx with k = y /x > y = kˆx with k R ++. Proof. Suppose that x, y bdk\{} and x T y =. Then x = x 2 >, y = y 2 >, x T y = x y + x T 2 y 2 =, 2 which implies that x T 2 y 2 = x y = x 2 y 2. Hence there exists a positive constant k such that y 2 = kx 2. It follows from 2 that k = y /x and hence y = kˆx. The rest of the proof follows from [8, Lemma 2.3]. The exact formula of the regular normal cone and limiting normal cone of Ω have been established in [25]. Proposition 2.3 [25, Theorem 3.] Let x, y be in the m-dimensional second-order cone complementarity set Ω. Then {u, v u R m, v = } if x =, y intk; {u, v u =, v R m } if x intk and y = ; {u, v u x, v y, x N Ω x, y = û + y v Rx} if x, y bdk\{}, x T y = ; {u, v u ŷ, v R ŷ} if x =, y bdk\{}; {u, v u R ˆx, v ˆx } if x bdk\{}, y = ; {u, v u K, v K} if x =, y =.

7 SOCMPCC 7 Proposition 2.4 [25, Theorem 3.3] Let x, y Ω where Ω is the m-dimensional secondorder cone complementarity set. Then N Ω x, y = N {u, v u R m, v = } if x =, y intk; Ω x, y = {u, v u =, v R m } if x intk, y = ; {u, v u x, v y, x û + y v Rx} if x, y bdk\{}. For x =, y bdk\{}, N Ω x, y = {u, v u R m, v = or u ŷ, v Rŷ or u, ŷ, v R ŷ}; for x bdk\{}, y =, N Ω x, y = {u, v u =, v R m or u Rˆx, v ˆx or u R ˆx, v, ˆx }; for x = y =, N Ω x, y = {u, v u K, v K or u R m, v = or u =, v R m or u R ξ, v ξ or u ξ, v R ξ or u ξ, v ˆξ, αû + αv Rξ, α [, ], for some ξ C} where C := {, w w R m, w = }. 3 Failure of Robinson s CQ and the classical KKT condition Note that G i z, H i z K i implies that G i z T H i z for i =,..., J. Hence SOCM- PCC can be rewritten as an optimization problem with a convex cone constraint: K-SOCMPCC min fz s.t. gz, hz =, Gz, Hz, Gz, Hz K K, where Gz := G z,, G J z, Hz := H z,, H J z, and K := K K 2 K J. We denote by τ := J j= m j. For a general optimization problem with a cone constraint such as K-SOCMPCC, the following Robinson s CQ is considered to be a usual constraint qualification: h i z i =,..., q are linearly independent, h i z T d =, i =,..., q, d R n such that gz + gz T d intr p +, Hz Gz + Gz Hz T d <, Gz + Gz T d int K, Hz + Hz T d int K. It is well-known that the MFCQ never holds for MPCCs. We now show that Robinson s CQ never holds for the K-SOCMPCC.

8 SOCMPCC 8 Proposition 3. For K-SOCMPCC, Robinson s CQ fails to hold at every feasible solution of SOCMPCC. Proof. Any feasible solution z of SOCMPCC must be a solution to the following mathematical program with a convex cone constraint: min Gz, Hz s.t. Gz K, Hz K. By the Fritz John necessary optimality condition, there exist λ, λ G R τ, λ H R τ not all equal to zero such that = λ G, H z + Gz λ G + Hz λ H, λ G N KGz, λ H N KHz. It is clear that,,, λ, λ G, λ H is a singular Lagrange multiplier of K-SOCMPCC. By [3, Propositions 3.6 ii and 3.9iii], a singular Lagrange multiplier exists if and only if Robinson s CQ does not hold. Therefore we conclude that Robinson s CQ does not hold at z for K-SOCMPCC. For a feasible point z of SOCMPCC, define the following index sets I g z := {i g i z = }, I G z := {i G i z = }, I + G z := {i G iz intk i }, I H z := {i H i z = }, I + H z := {i H iz intk i }, B H z := {i H i z bdk i \{}}, B G z := {i G i z bdk i \{}}. For simplicity we may omit the dependence of z in the above index sets and denote G i z, H i z by G i, H i and Ĝiz, Ĥiz by Ĝi, Ĥi, respectively, if there is no confusion. Now we introduce a new concept of stationary point for SOCMPCC, called K-stationary point, and we show that the K-stationary condition 3 is equivalent to the classical KKT conditions 4. Definition 3. K-stationary point Let z be a feasible solution of SOCMPCC. We say that z is a K-stationary point of SOCMPCC if there exists a multiplier λ g, λ h, λ G, λ H such that the following K-stationary condition holds: fz + gz λ g + hz λ h + J G i z λ G i + J H i z λ H i =, λ g, gz T λ g =, λ G i R m i, λ H i = if i I G I + H, λ G i =, λ H i R m i if i I + G I H, λ G i RĜiz, λ H i RĤiz if i B G B H, λ G i K i + R + H i z, λ H i R Ĥ i z if i I G B H, λ G i R Ĝ i z, λ H i K i + R + G i z if i B G I H, λ G i K i, λ H i K i if i I G I H. Definition 3.2 We say that K-SOCMPCC is Clarke calm at a feasible solution z if there exist positive ε and µ such that, for all r, s, t, p εb, for all z B ε z F K r, s, t, p, one has fz fz + µ r, s, t, p, 3

9 SOCMPCC 9 where F K r, s, t, p := { z hz + r =, gz + s, Gz, Hz + t, Gz, Hz + p K K }. Theorem 3. Let z be a local optimal solution of SOCMPCC. Suppose that the problem K-SOCMPCC is Clarke calm at z. Then z is a K-stationary point. Moreover the K- stationary condition is equivalent to the classical KKT condition 4. Proof. Since the problem K-SOCMPCC is Clarke calm at z, by the classical necessary optimality condition see e.g. [6, Theorem 2.2], there exists λ g, λ h, a, b, γ R p R q R τ R τ R such that the classical KKT condition holds: fz + gz λ g + hz λ h + J G i z a i + J H i z b i +γ G T Hz =, λ g, gz T λ g =, G i z K i, a i K i, G i z T a i =, i =,, J, H i z K i, b i K i, H i z T b i =, i =,, J, γ. We now show the equivalence of the classical KKT condition 4 and the K-stationary condition 3. Let λ G := a + γhz and λ H := b + γgz. We first show that λ g, λ h, λ G, λ H satisfies 3. Consider the following cases. i I G I + H. Then G iz =, H i z intk i. By 4, b i = and hence λ H i = b i + γg i z =. i I + G I H. Similar to Case we can show that λ G i =. i B G B H. Then H i z, G i z bdk i \{} and H i z G i z. Since a i G i z and a i K i by 4, then a i R + Ĝ i z. Similarly, b i R + Ĥ i z by b i H i z and b i K i. It follows from Proposition 2.2 that H i z R ++ Ĝ i z and G i z R ++ Ĥ i z. So λ G i = a i + γh i z RĜiz and λ H i = b i + γg i z RĤiz. i I G B H. Then G i z =, H i z bdk i \{}. Since a i, b i K i and b i H i z by 4, then b i R + Ĥ i z. Hence λ G i = a i + γh i z K i + R + H i z and λ H i = b i + γg i z = b i R Ĥ i z. i B G I H. Similarly to the above case, we have λ G i = a i +γh i z = a i R Ĝ i z and λ H i = b i + γg i z K i + R + G i z. i I G I H. Then G i z = H i z =. By 4, we have λ G i = a i K i and λ H i = b i K i. Hence λ g, λ h, λ G, λ H satisfies 3. Conversely, take λ g, λ h, λ G, λ H satisfying 3. Let a := λ G γhz and b := λ H γgz where γ >. We now show that λ g, λ h, a, b, γ satisfies 4 if γ is sufficiently large. Consider the following cases. 4

10 SOCMPCC i I G I + H. Then a i = λ G i γh i z = γ [ λ G i γ H i z ] and b i = λ H i γg i z = λ H i =. Since H i z intk i, λ G i /γ H iz K i when γ is sufficiently large. Hence a i K i and b i = K i. i I + G I H. Similar to Case, we can show that a i = K i and b i K i. i B G B H. Then G i z, H i z bdk i \{}. By Proposition 2.2, G i z = kĥiz for some k >, which in turn implies that H i z = Ĝiz /k. By 3, λ G i = t Ĝ i z and λ H i = t 2 Ĥ i z for some t, t 2 R. Hence a i = γh i z λ G i = γ/k t Ĝiz K i and b i = γg i z λ H i = γk t 2 Ĥiz K i provided that γ is sufficiently large. In addition, a i, G i z = t γ/k Ĝiz, G i z = and b i, H i z = t 2 γk Ĥiz, H i z =. i I G B H. It follows from 3 that λ G i = t H i z ξ i and λ H i = t 2 Ĥ i z for some t, t 2 and ξ i K i. Hence a i = γh i z λ G i = γ t H i z +ξ i K i as γ t. Similarly, b i = λ H i γg i z = λ H i = t 2 Ĥ i z K i. In addition, a i, G i z = since G i z = and b i, H i z = t 2 Ĥ i z, H i z =. i B G I H. The argument is similar to the above case. i I G I H. Then G i z = H i z = and a i = λ G i K i and b i = λ H i K i. Hence λ g, λ h, a, b, γ satisfies 4. 4 S-stationary conditions For MPCC, it is known see Ye [2, Theorem 3.2] that the S-stationary condition is equivalent to the stationary condition derived by using the proximal normal cone of the complementarity set. In this vector case, the regular normal cone is the same as the proximal normal cone. For SDCMPCC, it was shown that the regular normal cone is the same as the proximal normal cone and the S-stationary condition is defined by using the proximal normal cone [6]. Similarly, in [25] it was verified that the regular normal cone coincides with the proximal normal cone for the second-order cone complementarity set and hence we can define the S-stationary condition below. First we introduce the concept of weak W- stationary points. Note that when the dimension m i 2, the condition G i z ˆλG i + H i z λ H i RG i z is redundant and can be omitted. Definition 4. W-stationary point Let z be a feasible solution of SOCMPCC. We say that z is a weak stationary point of SOCMPCC if there exist a multiplier λ g, λ h, λ G, λ H such that fz + gz λ g + hz λ h + J G i z λ G i + J H i z λ H i =, λ g, gz T λ g =, λ G i R m i, λ H i = if i I G I + H, λ G i =, λ H i R m i if i I + G I H, λ G i G i z, λ H i H i z, G i z ˆλG i + H i z λ H i RG i z if i B G B H. 5

11 SOCMPCC Definition 4.2 S-stationary point Let z be a feasible solution of SOCMPCC. We say that z is a strong stationary point of SOCMPCC if there exist a multiplier λ g, λ h, λ G, λ H such that = fz + gz λ g + hz λ h + J G i z λ G i + J H i z λ H i, λ g, gz T λ g =, λ G i, λh i N Ωi G i z, H i z, i =,..., J, or equivalently such that 5 and the following condition hold: λ H i R Ĥ i z, λ G i, Ĥiz if i I G z B H z, λ G i R Ĝ i z, λ H i, Ĝiz if i B G z I H z, λ G i K i, λ H i K i if i I G z I H z. Definition 4.3 Let z be a feasible solution of SOCMPCC. We say that SOCMPCC-LICQ holds at z provided that the gradient vectors are linearly independent. g i z i I g z, h i z i =,..., q, G i z i I G z B G z, H i z i I H z B H z In the following theorem we show that under SOCMPCC-LICQ, a local optimal solution of SOCMPCC must be an S-stationary point. Theorem 4. Let z be a local optimal solution of SOCMPCC. If SOCMPCC-LICQ holds at z, then z is an S-stationary point. Proof. Since z is a local optimal solution, it is also a local optimal solution of the problem with the same objective function and with the inactive constraints g i z < i I g z, H i z intk i, i I, G i z intk i, i I 2 deleted from the feasible region, i.e., z is a local optimal solution to the problem: min fz s.t. hz =, g i z, i I g z, G i z =, i I, H i z =, i I 2, K i G i z H i z K i, i I I 2 c, where I := I G z I + H z, I 2 := I + G z I H z. Then fz + N F z, where F := {z F z D} is the feasible region of the above problem with and F z := hz, g Ig z, G I z, H I2 z, G I I 2 cz, H I I 2 cz, D := {} R Ig {}I {} I 2 Ω I I 2 c, Ω I I 2 c := {u i, v i K i u i v i K i, i I I 2 c }. By the SOCMPCC-LICQ, F z has a full column rank. The desired result follows from Propositions 2. and 2.3 by letting λ g i = for i / I gz, λ H i = for i I and λ G i = for i I 2, i.e., letting the multiplies corresponding to the deleted constraints be zero.

12 SOCMPCC 2 5 M-stationary conditions In this section we study the M-stationary condition for SOCMPCC. For this purpose we rewrite the SOCMPCC as an optimization problem with a nonconvex cone constraint: M-SOCMPCC min fz s.t. hz =, gz, G i z, H i z Ω i, i =,..., J, where Ω i := {x, y x K i, y K i, x y}. As in the MPCC case, we will show that the M-stationary condition introduced below is the KKT condition of M-SOCMPCC by using the limiting normal cone. Note that when the dimension m i 2, the condition α iˆλg i + α i λ H i Rξ i for some α i [, ] is redundant and can be omitted. Definition 5. M-stationary point Let z be a feasible solution of SOCMPCC. We say that z is an M-stationary point of SOCMPCC if there exist multipliers λ g, λ h, λ G, λ H such that = fz + gz λ g + hz λ h + J G i z λ G i + J H i z λ H i, λ g, gz T λ g =, λ G i, λh i N Ω i G i z, H i z, i =,..., J, or equivalently such that 5 and the following condition hold: λ G i R m i, λ H i = or λ G i Ĥiz, λ H i RĤiz or λ H i R Ĥ i z, λ G i, Ĥiz if i I G B H, λ G i =, λ H i R m i or λ G i RĜiz, λ H i Ĝiz or λ G i R Ĝ i z, λ H i, Ĝiz if i B G I H, λ G i K i, λ H i K i or λ H i =, λ G i R m i or λ G i =, λ H i R m i or λ G i R ξ i, λ H i ξi or λ H i R ξ i, λ G i ξi or λ G i ξ i, λ H i ˆξ i, α iˆλg i + α i λ H i Rξ i for some α i [, ] and some ξ i C i if i I G I H, where C i := {, w R R m i w = }. Definition 5.2 We say that M-SOCMPCC is Clarke calm at a feasible solution z if there exist positive ε and µ such that, for all r, s, p εb, for all z B ε z F M r, s, p, one has fz fz + µ r, s, p, where F M r, s, p := { z hz + r =, gz + s, G i z, H i z + p i Ω i, i =,..., J }. Theorem 5. Let z be a local optimal solution of SOCMPCC. Suppose that the problem M-SOCMPCC is Clarke calm at z. Then z is an M-stationary point of SOCMPCC.

13 SOCMPCC 3 Proof. By Theorem [6, Theorem 2.2], there exists a multiplier λ g, λ h, λ G, λ H such that = fz + gz λ g + hz λ h + J G i z λ G i + J H i z λ H i, λ g, gz T λ g =, λ G i, λ H i N Ωi G i z, H i z, i =,..., J, and so the desired result follows from using the expression of the limiting normal cone in Proposition 2.4. Definition 5.3 Let z be a feasible solution of SOCMPCC. We say that the constraint system of M-SOCMPCC has a local error bound at z if there exist µ, ε > such that distz, F M,, µ r, s, p, r, s, p εb and z F M r, s, p B ε z. Note that the constraint system of M-SOCMPCC has a local error bound at z if and only if the set-valued mapping F M r, s, p is calm [6] or pseudoupper-lipschitz continuous using the terminology of [24] around,,, z. Hence F M r, s, p being either pseudo-lipschitz continuous [] around,,, z or upper-lipschitz continuous [5] at z implies that the constraint system of M-SOCMPCC has a local error bound at z. The proposition below is an easy consequence of Clarke s exact penalty principle [4, Proposition 2.4.3] and the calmness of the constraint system. See [22, Proposition 4.2] for a proof. Proposition 5. If the objective function is Lipschitz near z and F M r, s, p is calm at,,, z, then the problem M-SOCMPCC is calm at z. Definition 5.4 SOCMPCC-NNAMCQ Let z be a local optimal solution of SOCM- PCC. We say that SOCMPCC-No Nonzero Abnormal Multiplier Constraint Qualification SOCMPCC-NNAMCQ holds at z if there is no nonzero vector λ g, λ h, λ G, λ H such that the following conditions hold: = gz λ g + hz λ h + J G i z λ G i + J H i z λ H i, λ g, gz T λ g =, λ G i, λh i N Ω i G i z, H i z, i =,..., J. Theorem 5.2 Let z be a local optimal solution of SOCMPCC. Then z is an M-stationary point under one of the following constraint qualifications: i The SOCMPCC-NNAMCQ holds at z. ii All mappings h, g, G, H are affine and m i 2 for i =,..., J. Proof. By Theorem 5. and Proposition 5., it suffices to show the calmness of F M. i Similarly as in [22, Theorem 4.4], we can show that under SOCMPCC-NNAMCQ, the constraint system of M-SOCMPCC is pseudo-lipschitz continuous around,,, z and hence has a local error bound at z. ii Since when m i 2, the second-order cone K i is polyhedral and hence the secondorder cone complementarity set Ω i is a union of finitely many polyhedral convex sets. Since all mappings h, g, G, H are affine, the graph of the set-valued mapping F M is a union of polyhedral convex sets and hence F M is a polyhedral set-valued mapping. By [5, Proposition ], F M is upper-lipschitz and hence the local error bound condition holds at z.

14 SOCMPCC 4 6 C-stationary conditions In this section, we consider the C-stationary condition by reformulating SOCMPCC as a nonsmooth problem: C-SOCMPCC min fz s.t. hz =, gz, G i z Π Ki Gi z H i z =, i =,..., J. As in the MPCC case, the C-stationary condition introduced below is the nonsmooth KKT condition of C-SOCMPCC by using the Clarke generalized gradient. Definition 6. C-stationary point Let z be a feasible solution of SOCMPCC. We say that z is a C-stationary point of SOCMPCC if there exists a multiplier λ g, λ h, λ G, λ H such that 5 and the following conditions hold: λ H i RĤiz if i I G B H, λ G i RĜiz if i B G I H, λ G i, λh i for all i =,..., J. We present the first-order optimality condition of SOCMPCC in terms of C-stationary conditions in the following result. Definition 6.2 We say that C-SOCMPCC is Clarke calm at a feasible solution z if there exist positive ε and µ such that, for all r, s, α εb, for all z B ε z F C r, s, α, one has fz fz + µ r, s, α, where F C r, s, α := { z hz + r =, gz + s, G i z Π Ki Gi z H i z + α i =, i =,..., J }. Theorem 6. Let z be a local optimal solution of SOCMPCC. Suppose that the problem C-SOCMPCC is Clarke calm at z. Then z is a C-stationary point of SOCMPCC. Proof. Since the problem is calm, by the Clarke nonsmooth KKT condition [4, Proposition 6.4.4], there exist λ h R q, λ g R p and β i R m i i =,..., J such that c z Lz, λ h, λ g, β, λ g and λ g, gz =, 6 where c z denotes the Clarke generalized gradient with respect to z and Lz, λ h, λ g, β := fz + λ h, hz + λ g, gz + Consider the Clarke generalized gradient of the nonsmooth function J β i, G i z Π Ki Gi z H i z. S i z := β i, Π Ki Gi z H i z.

15 SOCMPCC 5 Applying the Jacobian chain rule [4, Theorem 2.6.6] twice yields c S i z β T i c Π Ki Gi z H i z J G i z J H i z. Therefore, since any element of the Clarke generalized gradient of the metric projection operator to a closed convex set is self-adjoint see e.g., [, Proposition a], we know from 6 that there exists A i c Π Ki Gi z H i z such that fz + hz λ h + gz λ g + J G i z β i J Gi z H i z A i β i =. 7 Define λ G i := β i A i β i and λ H i := A i β i. Then it follows from 6 and 7 that = fz + hz λ h + gz λ g + λ g, λ g, gz =. J G i z λ G i + J H i z λ H i, We now continue to show that 5 holds. Notice that for i I G I + H I+ G I H B G B H, Π Ki is continuously differentiable at G i z H i z. Hence A i = J Π Ki Gi z H i z. Since [ ] λ H i = A i β i = A i I i A i β i + A i β i = A i λ G i + λ H i, where I i denotes the m i -dimensional identity matrix, it follows that λ H i = D Π Ki Gi z H i z λ G i λ H i. Hence λ G i, λh i N Ωi G i z, H i z by [25, Proposition 2.] for i I G I + H I+ G I H B G B H. Using the formula of regular normal cone given in Proposition 2.3 yields λ G i R m i, λ H i = if i I G I + H, λ G i =, λ H i R m i if i I + G I H, λ G i G i z, λ H i H i z, G i z ˆλ G i + H i z λh i RG i z if i B G B H, which implies that 5 holds. Now consider the case where i I G B H. In this case G i z H i z = H i z bdk i \ {} and hence by [4, Lemma iii] we have A i co B Π Ki Gi z H i z { } = co O, 2 H i z 2 Ĥ i z Ĥiz T, which from λ H i = A i β i implies λ H i RĤix. In the case where i B G I H, G i z H i z = G i z bdk i \ {} and hence by [4, Lemma ii] we have A i co B Π Ki Gi z H i z { } = co I, I 2 G i z 2 Ĝ i z Ĝiz T. It follows from λ G i = I i A i β i that λ G i RĜiz. Moreover, from [, Proposition c], we know that A i β i, β i A i β i, which implies λ G i, λ H i for all i =,..., J. The proof of the theorem is complete.

16 SOCMPCC 6 7 Connections between various stationary points In this section, we discuss the relationships among various stationary points and the Clarke calmness conditions for various reformulations given in the previous sections. First, we give the following result. Proposition 7. Let x, y Ω with Ω being the m-dimensional second-order cone complementarity set. Then N K x + R + y, N K y + R + x = R m, if x =, y intk;, R m if x intk, y = ; Rˆx, Rŷ if x, y bdk\{}, x T y = ; K + R + y, R ŷ if x =, y bdk\{}; R ˆx, K + R + x if x bdk\{}, y = ; K, K if x =, y =, 8 and N K x + R + y, N K y + R + x N Ω x, y. 9 Proof. We will prove 8 and 9 simultaneously. Consider the following cases. Let x = and y intk. For any z R m, since y intk, there exists t > such that y tz K. Hence z K+y t K + R + y. Because z R m is arbitrary, we have K + R + y = R m. Thus N K x + R + y, N K y + R + x = K + R + y, = R m, = N Ω x, y. Let x intk and y =. Then similar to the above case we can show that N K x + R + y, N K y + R + x =, K + R + x =, R m = N Ω x, y. Let x, y bdk\{} and x T y =. It follows from Proposition 2.2 that y R +ˆx and x R + ŷ. Note that N K x = R ˆx and N K y = R ŷ. This implies NK x + R + y, N K y + R + x = R ˆx + R +ˆx, R ŷ + R + ŷ = Rˆx, Rŷ, since R = R + R +. For u, v Rˆx, Rŷ, we have u x, v y, and x û + y v Rx due to ŷ R + x. Comparing the formula given in and Proposition 2.3 yields NK x + R + y, N K y + R + x N Ω x, y. Let x = and y bdk\{}, then by Proposition 2.3 we have NK x + R + y, N K y + R + x = K + R + y, R ŷ N Ω x, y, since w + βy, ŷ = w, ŷ for all w K and β R +.

17 SOCMPCC 7 Let x bdk\{} and y =. Similarly as the above case we can show that N K x + R + y, N K y + R + x = R ˆx, K + R + x N Ω x, y. Let x, y =,. Then by Proposition 2.3 we have N K x + R + y, N K y + R + x = K, K = N Ω x, y. Comparing Proposition 7. and 3 leads to the following expression of the K-stationary condition immediately. Corollary 7. A feasible solution z is a K-stationary point of SOCMPCC if and only if there exist a multiplier λ g, λ h, λ G, λ H such that fz + gz λ g + hz λ h + J G i z λ G i + J H i z λ H i =, λ g, gz T λ g =, λ G i, λh i N Ki G i z + R + H i z, N Ki H i z + R + G i z, i =,..., J. In the following proposition we show that 9 becomes an equality when the dimension of K is less or equal to 2. Proposition 7.2 If K is the m-dimensional second-order cone with m 2, then for x, y Ω, N K x + R + y, N K y + R + x = N Ω x, y. Proof. If m =, then the possible cases are x =, y intk or x intk, y = or x = y =. In these three cases, according to 8 and the formula of the regular normal cone given in Proposition 2.3 we have N K x + R + y, N K y + R + x = N Ω x, y. If m = 2, according to the proof of Proposition 7. it only needs to show N K x + R + y, N K y + R + x N Ω x, y for x, y bdk\{} or x =, y bdk\{} or x bdk\{}, y =. Let x, y bdk\{}. Take u, v N Ω x, y. Then it follows from u x and v y that u Rˆx, v Rŷ. Since N K x + R + y, N K y + R + x = Rˆx, Rŷ by, then N K x + R + y, N K y + R + x N Ω x, y. Let x = and y bdk\{}. According to Proposition 2.3 and, it suffices to show that ŷ K + R + y. Let u ŷ, i.e., u y u 2 y 2. Since y = y 2 due to the assumption that y bdk\{}, consider the following two cases. If y = y 2, then u u 2. Let t > be sufficiently large so that ty u 2. Then ty u ty u 2 = ty u 2 = ty 2 u 2. It means ty u K, i.e., u K + R + y. If y = y 2, then u + u 2. Let t > be sufficiently large so that ty + u 2. Then ty u ty + u 2 = ty + u 2 = ty 2 + u 2 = ty 2 u 2. Hence ty u K, i.e., u K + R + y. In both cases, we have shown that ŷ K + R + y.

18 SOCMPCC 8 Let x bdk\{} and y =. The proof is similar to the above case. The following result follows from Propositions 7., 7.2 and Corollary 7.. Corollary 7.2 A K-stationary point is an S-stationary point. Moreover if the dimension of every K i is less or equal to 2, then an S-stationary point is a K-stationary point. It is well known that the KKT conditions and the S-stationary conditions are equivalent for MPCC. However, for SOCMPCC, according to Corollary 7.2 and Example 7. below, this equivalence holds only for the case where all m i 2 but may fail to hold as m i 3 for some i,..., J. Since the S-stationary point is defined in terms of the regular normal cone and the M-stationary point is defined in terms of the limiting normal cone, it is obvious that an S-stationary point must be an M-stationary point. However, unlike MPCC, it is not so easy to see that an M-stationary point must be an C-stationary point. We now verify this implication. Theorem 7. An M-stationary point must be a C-stationary point. Proof. It suffices to show that for every u, v N Ω x, y, one has u, v. The cases where x =, y intk or x intk, y = or x =, y bdk\{} or x bdk\{}, y = are clear. It suffices to prove for the cases where x, y bdk\{} and where x = y =. Let x, y bdk\{}. Then by Proposition 2.4, x û + y v = βx for some β R and u x. Since y = y 2, it follows that v = βx x û y. Hence u, v = y u, βx x û = x y u, û = x y u 2 u 2 2, where the last inequality follows from the fact that u u 2 since u = x T 2 u 2 due to x u. Now consider the case where x = y =. In this case, it only needs to consider the case where there exists α [, ], β R and ξ C such that αû+ αv = βξ, u ξ, v ˆξ. If α =, then v = βξ and hence u v. If α =, then û = βξ, i.e., u = β ˆξ, which in turn implies u v. If α,, then αû + αv = βξ implies that v = βξ αû α. Hence u, v = α α u, βξ αû = u, û = α α α u2 u 2 2, where the last inequality follows from the fact that u ξ and ξ =, w with w =. We can now summarize the relation between various stationary points as follows. K stationary point = S stationary point = M stationary point = C stationary point = W stationary point. The following examples show that the reverse relationships between various stationary points may not hold in general. In all examples, we use the notation K m to denote the mth dimensional second-order cone. Example 7. S-stationary but not K-stationary Consider the following SOCMPCC:

19 SOCMPCC 9 min fz := z 2 + z 2 2 z 3 s.t. gz := z3 2, z + Gz := z 2 + K 3, Hz := Gz Hz. z 3 z z 2 z 3 K 3, It is obvious that the optimal solution is z =,,. The index sets except I g z, B G z, I H z are all empty. Hence the S-stationary condition is fz + gz λ g + Gz λ G + Hz λ H =, λ g, gz T λ g =, λ G R Ĝz, λ H, Ĝz, 2 and the K-stationary condition is fz + gz λ g + Gz λ G + Hz λ H =, λ g, gz T λ g =, λ G R Ĝz, λ H K 3 + R + Gz. 3 Take λ G =,,, λ H =,,. Then the first condition in the S-stationary condition 2 holds: = +λ H + λ g + λ H 2 + λ G + λ H 3 + λ G 2 + λ G 3 Moreover λ G =,, R Ĝz and λ H, Ĝz = and so the third condition in the S-stationary condition 2 holds. However z is not a K-stationary point. In fact, let λ G R Ĝz, i.e., λ G = λ G, λg 2, λg 3 = t,, for some t, then λh = λ H, λh 2, λh 3 = t, t, by the first condition in the K-stationary condition 3. But λ H +ηgz = t, t, +η,, / K 3 for all η, which means λ H / K 3 +R + Gz. Hence z is not a K-stationary point. This example demonstrates that the K-stationary point and S-stationary point may be different when the dimension of one of the second-order cones is more than 2. Example 7.2 M-stationary but not S-stationary Consider the following SOCMPCC:.

20 SOCMPCC 2 min fz := z + z2 2 z + s.t. Gz := K 2, z 2 + z Hz := K 2, z 2 Gz Hz. The optimal solution is z =,. The index sets except B G z, I H z are all empty. Hence the M-stationary condition is fz + Gz λ G + Hz λ H =, λ G =, λ H R 2 or λ G RĜz, λ H Ĝz or λ G R Ĝz, λ H, Ĝz, and the S-stationary condition is { fz + Gz λ G + Hz λ H =, λ G R Ĝz, λ H, Ĝz. Since G, H are affine and m = 2, z must be an M-stationary point by Theorem 5.2. In fact, let λ G = /2, /2 and λ H = /2, /2. Then the first condition in the M-stationary condition 4 holds: = + λ G + λ G 2 + λ H and the second condition in the M-stationary condition 4 holds: + λ H 2 λ G = /2, /2 RĜz and λ H = /2, /2 Ĝz. Hence the M-stationary condition holds. However, z is not an S-stationary point. If λ G, λ H satisfies 5 with λ G R, and λ H,,, then λ G + λh =, λ G 2 + λh 2 =, λg = λg 2 and λh λh 2. So λg = λh λh 2 = + λg 2 = λg. This implies λ G /2, which contradicts with λg. Example 7.3 C-stationary but not M-stationary Consider the following SOCMPCC: 4 5 min fz := z + z 2 z 3 z + s.t. Gz := z + z 2 z 3 K 3, Hz := Gz Hz. z z z z 2 3 z 2 K 3,

21 SOCMPCC 2 We can show that the only feasible solution is,, and hence it is also the unique optimal solution. Indeed, since Hz = z, z z3 2, z 2 K 3, by definition of the second order cone we have z z + z z 2 2 which implies that z and z 2 z + z z 2 2 z + z Hence we obtain z 3 =. Plugging z 3 = in 6, we obtain z 2 =. Similarly from Gz = z +, z + z 2 z 3, z K 3 and z 3 =, we obtain z =. The index sets except B G z, I H z are all empty. Hence the C-stationary condition is { fz + Gz λ G + Hz λ H =, λ G RĜz, λ G, λ H 7, and the M-stationary condition is fz + Gz λ G + Hz λ H =, λ G =, λ H R 3 or λ G RĜz, λ H Ĝz or λ G R Ĝz, λ H, Ĝz. Take λ G =,, and λ H = 2,,. Then the first condition in the C-stationary condition 7 holds: = + λ G + λ G 2 + λ G 3 +λ H + λ H 2 + λ H 3 8 and λ G = Ĝz RĜz, λ G, λ H = λ H λh 2 = >. So z is a C-stationary point. However z is not an M-stationary point. Indeed, from 8, it is clear that λ G must be nonzero. For λ G RĜz = t,, for some t R, then it follows from 8 that t = and λ H λh 2 =. Thus λg R + Ĝz but λ H, Ĝz. So z is not an M-stationary point. Example 7.4 W-stationary but not C-stationary Consider the following SOCMPCC: min fz := z + z2 2 z 3 z + z 2 s.t. Gz := z + z 2 + z 3 K 3, Hz := Gz Hz. z z 3 z z 2 z 2 K 3,

22 SOCMPCC 22 We can show that the feasible region is {z = z, z 2, z 3 z =, z 2 =, z 3 }. Indeed, from Hz K 3, we have z and z 2 z + z z 2 2 z + z 2 2. Hence z = and z 2 =. Then Hz =,, and Gz =, + z 3, z 3 K 3 which implies z 3 [, ]. Hence the optimal solution is z =,,. The index sets except B G z, I H z are all empty. Hence the W-stationary condition is fz + Gz λ G + Hz λ H =, 9 and the C-stationary condition is { fz + Gz λ G + Hz λ H =, λ G RĜz, λ G, λ H. Let λ G =,, and λ H = 2,, 2 or λ G =,, and λ H =,, 3. Then the W-stationary condition 9 holds: = 2 + λ G + λ G 2 + λ G 3 +λ H + λ H 2 + λ H 3. 2 However z is not a C-stationary point. Indeed, for λ G RĜz, i.e., λ G = t,, for some t R, it then follows from 2 that t =, λ H λh 2 = >, and λh 3 = 2. Thus λ G, λ H = tλ H λh 2 = <. Hence z is not a C-stationary point. To study the relationship between the Clarke calmness conditions for the various reformulations we consider the following general optimization problem with cone complementarity constraints. P min fz s.t. K Gz Hz K, where K is a convex symmetric cone of a finite dimensional space X and G, H are continuous. For simplicity we omit the standard inequality and equality constraints. Let t R and α, β X. Consider the following perturbed feasible regions of P. F K t, α, β := {z X Gz, Hz + t, Gz, Hz + α, β K K} F M α, β := {z X Gz, Hz + α, β Ω} F C α := {z X Gz Π K Gz Hz + α = }. Proposition 7.3 Let z be a feasible solution of problem P. a Suppose that there exist positive ε and µ such that, for all t, α, β ε B, for all z B ε z F K t, α, β, one has fz fz + µ t, α, β, 2 then there exist positive ε 2, µ 2 such that for all α, β ε 2 B, for all z B ε2 z F M α, β, one has fz fz + µ 2 α, β. 22

23 SOCMPCC 23 b Suppose that there exist positive ε and µ such that, for all α, β in ε B, for all z B ε z F M α, β, one has fz fz + µ α, β, 23 then for all α ε 2 B, for all z B ε2 z F C α, one has where ε 2 = ε / 2 and µ 2 = 2µ. fz fz + µ 2 α, 24 Proof. a Suppose that z F M α, β, i.e., Gz, Hz + α, β Ω. Then it is easy to verify that Gz, Hz + t M, Gz, Hz + α, β K K with t M := Gz, β + Hz, α + α, β. That is, z F K t M, α, β. Now suppose that there exist positive ε and µ such that, for all t, α, β in ε B and z B ε z F K t, α, β, 2 holds. Then by the continuity of G, H, one can find positive ε 2 < ε and µ 2 such that for all z B ε2 z and α, β in ε 2 B, and t M, α, β = Gz, β + Hz, α + α, β, α, β ε B, µ 2 α, β µ tm, α, β. Combining these and 2 with t = t M ensures that 22 holds. b Suppose that z F C α. Then Gz Π K Gz Hz + α =, which can be rewritten as Gz + α = Π K Gz + α Hz α, i.e., Gz, Hz + α, α Ω. That is, z F M α, α. Now suppose that there exist positive ε and µ such that, for all α, β in ε B, for all z B ε z F M α, β, 23 holds. Let ε 2 = ε / 2 and µ 2 = 2µ. Then for all α ε 2 B and z B ε2 z F C α, we have α, α ε B and hence i.e., 24 holds. fz fz + µ 2 α = fz fz + µ α, α, Definition 7. K-Clarke calmness We say that a feasible solution z of P is K- Clarke calm if there exist positive ε and µ such that, for all t, α, β εb, for all z B ε z F K t, α, β, one has fz fz + µ t, α, β. Definition 7.2 M-Clarke calmness We say that a feasible solution z of P is M- Clarke calm if there exist positive ε and µ such that, for all α, β εb, for all z B ε z F M α, β, one has fz fz + µ α, β. Definition 7.3 C-Clarke calmness We say that a feasible solution z of P is C- Clarke calm if there exist positive ε and µ such that, for all α εb, for all z B ε z F C α, one has fz fz + µ α. According to Proposition 7.3, the following implications hold. Theorem 7.2 K-Clarke calmness = M-Clarke calmness = C-Clarke calmness.

24 SOCMPCC 24 8 New optimality conditions for MPCC via SOCMPCC Consider the vector MPCC: MPCC min fz s.t. hz =, gz, G i z H i z, i =,..., J, where G i z, H i z : R n R. We reformulate MPCC as the associated SOCMPCC: a-socmpcc min fz s.t. hz =, gz, K i G i z H i z K i, i =,..., J, where G i z := G i z,,..., R m i and H i z := G i z,,..., R m i for i =,..., J. Note that the index sets B Gz, B Hz are always empty. For the convenience of the discussion we recall the definition of W-, S-, M- and C-stationary conditions for the problem a-socmpcc. A feasible solution of a-socmpcc is W-stationary point if there exists λ g, λ h, λ G, λ H R p R q R τ R τ such that fz + gz λ g + hz λ h + J G i z λ G i + J H i z λ H i =, λ g, gz T λ g =, λ H i = if i I Gz I + Hz, λ G i = if i I + Gz I Hz. It is an S-, M- and C-stationary point if 25 holds and for all i I Gz I Hz λ G i K i, λ H i K i ; λ G i K i, λ H i K i or λ H i =, λ G i R m i or λ G i =, λ H i R m i or λ G i R ξ i, λ H i ξ i or λ H i R ξ i, λ G i ξ i or λ G i ξ i, λ H i ˆξ i, α iˆ λg i + α i λ H i Rξ i for some α i [, ] and some ξ i C i ; λ G i, λ H i holds respectively. Let us discuss the relationship of the various stationary points between MPCC and its associated SOCMPCC reformulations. Theorem 8. The following statements holds: a If z is an S-stationary point of vector MPCC with λ g, λ h, λ G, λ H R p R q R J R J then z is an S-stationary point of the associated SOCMPCC with λ g, λ h, λ G, λ H R p R q R τ R τ where λ G i = λ G i,,..., Rm i and λ H i = λ H i,,..., Rm i for i =,..., J. Conversely, if z is an S-stationary point of SOCMPCC with λ g, λ h, λ G, λ H R p R q R τ R τ, then z is an S- stationary point of vector MPCC with λ g, λ h, λ G, λ H R p R q R J R J where λ G i = λ G i and λ H i = λ H i for i =,..., J. 25

25 SOCMPCC 25 b If z is an M-,C-stationary point of vector MPCC with λ g, λ h, λ G, λ H R p R q R J R J, then z is an M-,C-stationary point of the associated SOCMPCC with λ g, λ h, λ G, λ H R p R q R τ R τ where λ G i = λ G i,,..., Rm i and λ H i = λ H i,,..., Rm i for i =,..., J. Proof. Part a. Recall that a point z is said to be an S-stationary point of the MPCC if there exists λ g, λ h, λ G, λ H R p R q R J R J such that Note that Let Then fz + gz λ g + hz λ h + J G i z λ G i λ g, gz T λ g =, λ H i = if i I G z I + H z, λ G i = if i I + G z I H z, λ G i, λ H i, if i I G z I H z. + J H i z λ H i =, I Gz = I G z, I + Gz = I + G z, I Hz = I H z, I + Hz = I + H z. λ G = λ G,..., λ G J with λg i = λ G i,,..., R m i, 26 λ H = λ H,..., λ H J with λh i = λ H i,,..., R m i. 27 G i z λ G i = G i z λ G i and H i z λ H i = H i z λ H i. From λ G i, λ H i, we have λ G i K i, λ H i K i. Thus 26 implies that that z is an S-stationary point of a-socmpcc. Conversely, assume that z is an S-stationary point of the SOCMPCC reformulation, i.e., there exists λ g, λ h, λ G, λ H R p R q R τ R τ such that 25-8 hold. Notice that G i z λ G i = λ G i G i z and H i z λ H i = λ H i H i z. In addition, λ G i K i, λh i K i implies λ G i, λ H i. Hence z is an S- stationary point of MPCC with λ g, λ h, λ G, λ H R p R q R J R J satisfying 26 where λ G i = λ G i and λ H i = λ H i for j =,..., J. Part b. Recall that a point z is said to be an M-stationary point of the vector MPCC if there exists λ, µ, u, v R p R q R J R J such that fz + gz λ g + hz λ h + J G i z λ G i λ g, gz T λ g =, λ H i = if i I G z I + H z, λ G i = if i I + G z I H z, λ G i <, λ H i < or λ G i λh i = if i I G z I H z. + J H i z λ H i =,