On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints

This paper focuses on a class of mathematical programs with symmetric cone complementarity constraints (SCMPCC). The explicit expression of C-stationary condition and SCMPCC-linear independence constraint qualification (denoted by SCMPCC-LICQ) for SCMPCC are first presented. We analyze a parametric smoothing approach for solving this program in which SCMPCC is replaced by a smoothing problem \begin{document}$P_{\varepsilon}$\end{document} depending on a (small) parameter \begin{document}$\varepsilon$\end{document} . We are interested in the convergence behavior of the feasible set, stationary points, solution mapping and optimal value function of problem \begin{document}$P_{\varepsilon}$\end{document} when \begin{document}$\varepsilon \to 0$\end{document} under SCMPCC-LICQ. In particular, it is shown that the convergence rate of Hausdorff distance between feasible sets \begin{document}$\mathcal{F}_{\varepsilon}$\end{document} and \begin{document}$\mathcal{F}$\end{document} is of order \begin{document}$\mbox{O}(|\varepsilon|)$\end{document} and the solution mapping and optimal value of \begin{document}$P_{\varepsilon}$\end{document} are outer semicontinuous and locally Lipschitz continuous at \begin{document}$\varepsilon=0$\end{document} respectively. Moreover, any accumulation point of stationary points of \begin{document}$P_{\varepsilon}$\end{document} is a C-stationary point of SCMPCC under SCMPCC-LICQ.

In Section 4, we construct a smoothing approximation of SCMPCC and discuss the convergence behavior of the feasible set, stationary points, solution mapping and optimal value of the smoothing approximation problem.
The following notations are used throughout the paper. Let X and Y be two finite dimensional real Euclidean spaces. For a given set S ⊆ X , convS denotes the convex hull of S. Let d(·, S) be min{ x − y | y ∈ S} for x ∈ X , where · is the Euclidean norm. Denote B δ (x) := {x| x −x ≤ δ}. Let I be the identity operator, i.e., Ix = x for all x ∈ X . For an operator A, A * denotes the adjoint operator of A. We say that the operator A is onto if equation A * x = 0 implies x = 0 and say A is nonsingular if the equation Ax = 0 has a unique solution x = 0. For a differentiable mapping H : X → Y and z ∈ X , we denote by J H(z) the Jacobian of H at z and ∇H(z) := J H(z) * .

Preliminaries.
2.1. Background in nonsmooth analysis and variational analysis. In this subsection, we briefly review some concepts and results in nonsmooth analysis and variational analysis. First we present the definitions of Bouligand-subdifferential and generalized Jacobian of a Lipschitz continuous function.
The generalized Jacobian in the sense of Clarke [4] is the convex hull of ∂ B F (x), i.e., ∂F (x) = conv{∂ B F (x)}.
Next, we introduce concepts of semicontinuity and continuity of set-valued mappings in [15,Definition 5.4], epi-continuity of function-valued mappings in [15,Definition 7.1] and Pompeiu-Hausdorff distance between two closed nonempty sets in [15,Example 4.13].   A sequence {C k } is said to converge with respect to Pompeiu-Hausdorff distance to C when lim k→∞ H(C k , C) = 0.
The following implicit function theorem for locally Lipschitz continuous function in [19,Lemma 1] is crucial for the convergence analysis in Section 4.
Lemma 2.5. Suppose H : X × Y → X is a locally Lipschitz continuous function in an open neighborhood of (x,ȳ) ∈ X × Y with H(x,ȳ) = 0. If every element in Π x ∂H(x,ȳ), the projection of ∂H(x,ȳ) onto the space X , is nonsingular, then there exist a neighborhood U (ȳ) ofȳ and a unique locally Lipschitz continuous function x(·) : U (ȳ) → X satisfying x(ȳ) =x such that for every y ∈ U (ȳ), H(x(y), y) = 0.

Euclidean Jordan algebras and the Jacobian of Löwner operators.
We give a brief introduction to Euclidean Jordan algebras. Details on Euclidean Jordan algebras can be found in Koecher's lecture notes [11] and the monograph by Faraut and Korányi [7].
A Euclidean Jordan algebra is a triple (V, ·, · , •) := A, where (V, ·, · , •) is a real n-dimensional inner product space and (x, y) → x • y : V × V → V is a bilinear mapping which satisfies the following conditions: We call x • y the Jordan product of x and y. In general, the Jordan product is not associative; i.e., (x • y) • z = x • (y • z) for all x, y, z ∈ V. In addition, we assume that there exists an elements e (called the identity element) such that x • e = e • x = x for all x ∈ V. Given a Euclidean Jordan algebra A, define the set of squares as K := {x 2 | x ∈ V}. From Theorem III.2.1 in [7], K is a symmetric cone in A. In other words, K is self-dual closed convex cone, and for any two elements x, y ∈ intK, there exists an invertible linear transformation Γ : V → V such that Γ(K) = K and Γ(x) = y. For x ∈ V, let l := l(x) be the smallest positive integer such that the set {e, x, x 2 , · · · , x l } is linearly dependent. Then l is said to be the degree of x, which is denoted by deg(x). The rank of A denoted by rk(A) is defined as rk( An idempotent element is primitive if it cannot be written as a sum of two idempotents. A complete system of orthogonal idempotents in A is a finite set {c 1 , c 2 , · · · , c k } of idempotents where c i • c j = 0 for all i = j, and c 1 + c 2 + · · · + c k = e. A Jordan frame in A is a complete system of orthogonal primitive idempotents. The number of elements of any Jordan frame equals the positive integer rk(A).
Note that the Jordan frame {c 1 , c 2 , · · · , c r } in (1) depends on x. We do not write this dependence explicitly sometimes for simplicity of notation. Let C(x) be the set consisting of all such Jordan frames at x, then by [18,Proposition 3.2], C(·) is outer semicontinuous at x.
Next, we recall the Peirce decomposition theorem on the space V, where a Jordan frame {c 1 , c 2 , · · · , c r } is fixed beforehand. In this case, define the following subspaces Theorem 2.7. (Theorem IV.2.1, [7]) Let {c 1 , c 2 , · · · , c r } be a given Jordan frame in a Euclidean Jordan algebra A of rank r. Then V is the orthogonal direct sum of For each x ∈ V, we define the Lyapunov transformation L(x) : V → V by L(x)y = x • y for all y ∈ V, which is a symmetric operator in the sense that L(x)y, z = y, L(x)z for all y, z ∈ V. Meanwhile, the operator Q(x) := 2L 2 (x) − L(x 2 ) is called the quadratic representation of x. We say two elements x, y ∈ V operator commute if L(x)L(y) = L(y)L(x). By Lemma X.2.2 in [7], for a given Jordan frame {c 1 , c 2 , · · · , c r }, it is easy to see that c i , c j operator commute and L(c i )L(c j ) = L(c j )L(c i ) for any i, j ∈ {1, 2, · · · , r}.
We introduce Löwner operators on a Euclidean space in Sun and Sun [18] below.
A direct implication of Theorem 3.2 in [18] is the following property of the Jacobian of Löwner operator P (·).
Then, P (·) is (continuously) differentiable at x if and only if for each i ∈ {1, 2, · · · , r}, p is (continuously) differentiable at λ i (x). In this case, the Jacobian J P (x) is given by Furthermore, J P (x) is a linear and symmetric operator from V into itself.
Define the three index sets of positive, zero, and negative eigenvalues ofx ∈ V, respectively, by Combining Proposition 2.5 with Proposition 2.6 in [21], it yields the following result on the Bouligand subdifferential of Π K (·) atx ∈ V.
if and only if there exist Ω ββ ∈ U |β| and a system of orthogonal idempotents {ĉ i } i∈β such that i∈βĉ i = i∈βc i , {c i } i∈α∪γ ∪ {ĉ i } i∈β form a Jordan frame of A atx and 3. C-stationary condition and SCMPCC-LICQ. In this section, we consider the C(larke)-stationary condition by reformulating SCMPCC as a nonsmooth problem: From [10, Proposition 6], we know that the reformulation NS-SCMPCC is equivalent to SCMPCC. The same with MPCC case, the C-stationary condition introduced below is the nonsmooth KKT condition of NS-SCMPCC by using the Clarke subdifferential. Let F denote the feasible set of SCMPCC.
We say thatz is a C-stationary point of SCMPCC if there exist multiplier (λ, µ, σ G , σ H ) ∈ R q × R p × V × V and V ∈ ∂Π K (x) such that The C-stationary point defined in Definition 3.1 is proposed in the form of Clarke subdifferential of Π K (·). To develop explicit expressions of C-stationary condition, we should derive explicit expressions of equation (5) in Definition 3.1. First, we need some notations. Letc := {c 1 ,c 2 , · · · ,c r } be a given Jordan frame in a Euclidean Jordan algebra A of rank r. Then for any t ∈ R, 1 ≤ i ≤ r, tc i •c i = tc 2 i = tc i , namely, for any t ∈ R, tc i ∈ V ii . Therefore it follows from Theorem 2.7, any element σ ∈ V can be expressed by σ = r i=1 (σc) ici + 1≤k<l≤r (σc) kl , where (σc) i ∈ R, i = 1, 2, · · · , r, and (σc) kl ∈ V kl , 1 ≤ k < l ≤ r. Denote

Theorem 2.7 also implies
and Thus it is easily verified that In virtue of Definition 2.1, for any Then it follows from Theorem 2.10, On the other hand, Then (5) implies that Combining this with (6)-(9) and 0 ≤ Ω k ij ≤ 1, one has for any σ ∈ V, Employing above discussions, we get the explicit expression of C-stationary point of SCMPCC below.
j∈α l∈γ Now we show that the C-stationary condition for SCMPCC could be reduced to the C-stationary condition for MPCC, SOCMPCC or SDCMPCC when K is the nonnegative orthant, a second-order cone, or a positive semidefinite cone.
Example 1. To see that the C-stationary condition for SCMPCC coincides with the C-stationary condition in the MPCC case, see [2,6,9,12,13,14,16,17], we consider the case V = R with the inner product and Jordan product defined by x, y = x • y = xy for x, y ∈ R. Then (R, ·, · , •) forms an Euclidean Jordan algebra, and R + is its cone of squares. The setc := {1} is the unique Jordan frame. Then the SCMPCC is reduced to the MPCC case Example 2. Now we show that the C-stationary condition for SCMPCC coincides with the C-stationary condition in the SOCMPCC case when we consider the Euclidean Jordan algebra A = (R m , ·, · , •), m ≥ 2. x ∈ R m is written as The inner product is x, y = x T y and the Jordan product is defined by The cone of squares of A is the second order cone defined by Letz be a feasible point of SOCMPCC, and It follows from Theorem 2.7, any σ ∈ R m can be expressed by (11), (12) and (13) where I is the identity matrix. (14) that Therefore, by (14), Combining the six cases above with Definition 4.4 in [23],z is a C-stationary point of SOCMPCC.
Example 3. We show that the C-stationary condition for SCMPCC coincides with the C-stationary condition in SDCMPCC case when we consider the Euclidean Jordan algebra A = (S m , ·, · , •), where S m denotes the set of all m × m real symmetric matrices with the inner product and Jordan product defined, respectively, by X, Y = Trace(XY ) and X • Y := XY + Y X 2 . Its cone of squares S m + is the set of all positive semidefinite symmetric matrices. The identity element is the identity matrix I. Then SCMPCC is reduced to the SDCMPCC case LetΓ := P T ΓP . LetΓ J1J2 denote the |J 1 | × |J 2 | sub-matrix ofΓ obtained by removing all the rows ofΓ not in J 1 and all the columns ofΓ not in J 2 . By (11), In the same way, by (12) and (13), we obtain ( Γ H ) γγ = 0, ( Γ H ) βγ = 0, ( Γ H ) γβ = 0 and for any i ∈ α and j ∈ γ, Finally, it follows from (14) that which implies thatz is a C-stationary of SDCMPCC in [5].
SCMPCC-LICQ is the analogue of the well-known MPCC-LICQ in [16] and could be reduced to the corresponding constraint qualification in the cases of SDCMPCC, SOCMPCC or MPCC by [5,16,23]. We now give the following result on the stability of SCMPCC-LICQ. Lemma 3.4. If SCMPCC-LICQ holds atz ∈ F, then there exists a neighborhood U (z) ofz such that SCMPCC-LICQ holds at any z ∈ F ∩ U (z).

Smoothing approximation approach.
For any x, y ∈ K, we have from [8,Theorem 5.13] that x, y ≥ 0. Similar to Proposition 4.1 in [5], Robinson's constraint qualification fails to hold at each feasible point of SCMPCC, hence SCMPCC is a difficult class of optimization problems. To avoid this difficulty, we intend to employ a smoothing approximation approach to solve SCMPCC. As described in the beginning of Section 3, SCMPCC is equivalent to the nonsmooth problem NS-SCMPCC. We construct a smoothing approximation of NS-SCMPCC and focus on discussing the convergence properties of the smoothing approximation problem. For any ε ∈ R, define ψ ε : R → R by Then the corresponding Löwner's operator Ψ ε : V → V takes the following form which can be treated as a smoothing approximation to the "absolute value" function |x| := √ x 2 . Thus 1 2 x + x 2 + ε 2 e is the smoothing approximation to the projection function Π K (x). A well known NCP function is the minimum function In this section, we consider the following parametric smoothing problem with the parameter ε = 0 as an approximation of SCMPCC, where Φ ε (z) = G(z)+H(z)− (G(z) − H(z)) 2 + ε 2 e is the corresponding Löwner's operator of the parameterized NCP function φ ε . When ε = 0, problem P 0 is equivalent to the nonsmooth problem SCMPCC. Let F ε denote the feasible set of P ε . Obviously, F 0 = F, and we will not distinguish the feasible set F 0 from F in the rest of this section. For the smoothing approximation problem P ε , we stress that most of the times we view ε as a parameter, and this explains the notation adopted, where ε is a subscript. However, in some cases, especially in the proofs, we shall view ε as a dependent variable. Hence, from this point of view, the function Φ depends on the two variables (ε, z). Actually, for any z ∈ R n , Φ ε (z) is Lipschitz continuous with respect to ε.
holds for any z ∈ R n with a := e .
Before discussing the convergence behavior of the smoothing method, we give the following assumption on P ε first.
Under this assumption, global solutions of SCMPCC and P ε exist. Actually, this assumption does not mean a restriction since sometimes it is treatable to add some box constraints, such as |z i | ≤ M , i = 1, 2, · · · , n for some large M > 0, to the constraints g(z) ≤ 0.

4.1.
Convergence rate of F ε . In this subsection, we consider the convergence behavior of the feasible set F ε from local and global viewpoints when ε → 0, and quantify the convergence rate for F ε . The following lemma will view ε as a dependent variable and illustrate the nonempty of F ε under SCMPCC-LICQ.
A conclusion about the convergence of F ε at ε = 0 will be given in the following theorem.
By making use of the implicit function theorem, we are now able to show the local convergence rate for F ε .
The convergence result (33) could be obtained in a similar way.
The convergence rate between the feasible sets of MPCC and its perturbed problem P τ in [2] is of order O( √ τ ). Although we use a different smoothing function for the SCMPCC problem in this paper, we have a similar conclusion below on the convergence rate between the feasible sets F ε and F with respect to the Pompeiu-Haudorff distance.

4.2.
Convergence behavior of the optimal value function and solution mapping for P ε . By the compactness assumption, a global minimizer of P ε always exists, assuming F ε = ∅. Now we consider how close the optimal solution set and optimal value of problem P ε are from that of SCMPCC when ε → 0. Let The relationship between the convergence of a set-valued mapping and the epiconvergence of the indicator function of the set-valued mapping is discussed. Lemma 4.6. For a set-valued mapping T : R → R n , the following equivalence holds Proof. Noting that we obtain the equivalence.
Proof. The conclusion is obvious by the definition of epi-convergence. The main convergence results on optimal value function and solution mapping of P ε are obtained below. Theorem 4.9. Let Assumption 4.1 hold and SCMPCC-LICQ hold at each point z ∈ F. Then the optimal value function κ(ε) is locally Lipschitz continuous and the solution mapping S(ε) is outer semicontinuous at ε = 0.
Theorem 4.9 tells us that the optimal solution mapping S(ε) is outer semicontinuous at ε = 0, and the convergence rate for the optimal value function κ(ε) is of order O(|ε|).

4.3.
Local convergence to C-stationary point. Based on the explicit expressions of C-stationary point and SCMPCC-LICQ proposed in Section 3, we consider the limiting behavior of the stationary points of problem P ε when ε → 0.
The following lemma states that SCMPCC-LICQ carries to the standard LICQ for the smoothing approximation problem P ε if |ε| > 0 is sufficiently small. The idea of this lemma is the same with which in [24, Lemma 4.1], we omit the proof here.

5.
Conclusion. The explicit expressions of C-stationary points and SCMPCC-LICQ of SCMPCC are introduced and a parametric smoothing scheme for SCMPCC is studied. We discusse the convergence behavior of feasible set F ε , stationary points, solution mapping and optimal value function of P ε respectively. This paper takes ideas from the convergence analysis of MPCCs in [2,9,17], but it is not a simple extension of these papers. The convergence of the feasible set F ε is discussed by the tool of implicit function theorem for Lipschitz function and variational analysis in [15] and the convergence behavior of solution mapping and optimal value of P ε is based on the epi-continuity of function-valued mappings in [15].
The proposed smoothing approach is only conceptual since it assumes the symmetric cone programs P ε to be solved in each iteration. In our future work, we plan to exploit some efficient algorithms for SCMPCC based on the convergence results of this paper. Also, only the explicit expressions of C-stationary condition are introduced, what about the M-and S-stationary conditions?