Clarke Generalized Jacobian of the Projection onto Symmetric Cones and Its Applications

Transcription

1 Clarke Generalized Jacobian of the Projection onto Symmetric Cones and Its Applications Lingchen Kong, Levent Tunçel 2, and Naihua Xiu 3 May 27, 2008 (revised: April 4, 2009) C & O Research Report CORR Abstract In this paper, we give an exact expression for Clarke generalized Jacobian of the projection onto symmetric cones, which is linked to rank- matrices As an application, we employ the projection operator to design a semismooth Newton algorithm for solving nonlinear symmetric cone programs The algorithm is proved to be locally quadratically convergent without assuming strict complementarity of the solution Keywords: Clarke generalized Jacobian, Projection, Symmetric cones, Euclidean Jordan algebra, Semismooth Newton algorithm MSC2000 Subject Classification: Primary: 65K05, 90C30; Secondary: 26B05 Abbreviated Title: Clarke Generalized Jacobian of the Projection Introduction This paper focuses on Clarke generalized Jacobian of the projection onto symmetric cones First, we recall some basic concepts Let : Ω X Y be a locally Lipschitz function on an open set Ω, where X and Y are two finite dimensional inner product spaces over the field R Let (x) denote the derivative of at x if is differentiable at x The Clarke generalized Jacobian of at x is defined by (x) := conv{ B (x)}, where B (x) := {lim x x, x D ( x)} is the B-subdifferential of at x, and D is the set of points of Ω where is differentiable We assume that the reader is familiar with the concepts of (strong) semismoothness, and refer to [5, 6, 20, 2, 23] for details It is well-known that the scalar-valued function g : R R + with g(t) = t + := max{0,t} is strongly semismooth and its Clarke generalized Jacobian is specified by t + /t if t 0 and the interval [0,] if t = 0 Note that the projection onto an arbitrary closed convex set is The work was partly supported by a Discovery Grant from NSERC, and the National Natural Science Foundation of China (06700, ) Lingchen Kong, Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G, Canada; Department of Applied Mathematics, Beijing Jiaotong University, Beijing 00044, P R China ( konglchen@26com) 2 Levent Tunçel, Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G, Canada ( ltuncel@mathuwaterlooca) 3 Naihua Xiu, Department of Applied Mathematics, Beijing Jiaotong University, Beijing 00044, P R China ( nhxiu@bjtueducn)

2 Lipschitz continuous In the case of second-order cone, we know that the projection operator is strongly semismooth, see, eg, [3, 4, 2] In particular, Hayashi, Yamashita and Fukushima [2] gave an explicit representation for Clarke generalized Jacobian of the projection onto secondorder cones Malick and Sendov [8] studied the differentiability properties of the projection onto cone of positive semi-definite matrices and worked out its Clarke generalized Jacobian Recently, Sun and Sun [26] showed that the projection onto symmetric cones, which include the nonnegative orthant, the cone of positive semi-definite matrices and the second-order cone as special cases, is strongly semismooth everywhere Kong, Sun and Xiu [6] studied the Clarke generalized Jacobian of the projection onto symmetric cones and gave its upper bound and lower bound Meng, Sun and Zhao [9] studied the Clarke generalized Jacobian of the projection onto nonempty closed convex sets and showed that any element V in its Clarke generalized Jacobian is self-adjoint and V V 2 (Proposition of [9] for details) One of the areas where projection operator is widely applied is the area of complementarity problems, see, eg, [6] Also, projection operator is a fundamental ingredient of many algorithms for solving convex optimization problems (see for instance the survey by Bauschke and Borwein [] and the references therein) Our work is in the setting of symmetric cones which unifies and generalizes the special cases of second-order cones, cones of Hermitian positive semi-definite matrices over reals, complex numbers, quaternions as well as the direct sums of any subset of these cones by On another front, consider the nonlinear symmetric cone program (NSCP), which is defined min f(x) st Ax = q, x K, where f : J R is a twice continuously differentiable function, A : J R m is a linear operator, q R m, J is a n-dimensional inner product space over real field R and V := (J,,, ) is a Euclidean Jordan algebra (see Section 2), K is the symmetric cone in V When f(x) is linear, NSCP becomes a linear symmetric cone program (SCP) NSCP provides a simple, natural, and unified framework for various optimization programs, such as nonlinear second-order cone programs (NSOCP) and nonlinear semidefinite programs (NSDP) The research activity in NSCP has been increasing recently, see, eg, [9, 0,, 7, 26, 27, 28] These papers deal with either certain theoretical properties of some functions on Euclidean Jordan algebras, or interior-point methods and smoothing methods However, to the best of our knowledge, nonsmooth algorithms for NSCP have not enjoyed the same level of activity except for some special cases For instance, in the case of NSOCP, Kanzow, Ferenczi and Fukushima [3] reformulated its optimality conditions as a nonsmooth system of equations by applying a projection operator, and established nonsmooth Newton methods which are locally quadratically convergent under some suitable conditions Motivated by all of the work cited above, we aim to give the exact expression of the Clarke generalized Jacobian of the projection onto symmetric cones x + (or Π K (x), see Section 2), and to establish nonsmooth Newton methods for NSCP Our result generalizes corresponding results of [2] and [8] (from second-order cones and positive semi-definite cones respectively) to symmetric cones Interesting enough, the expression of the Clarke generalized Jacobian of x + is linked to rank one matrices This allows us to obtain the formulae of operators x and x in a similar manner As an application to NSCP, we design semismooth Newton methods by employing the projection operator () This paper is organized as follows In the next section, we establish the preliminaries and 2

3 study the relationship among all the Jordan frames of an element We also introduce the matrix representation of the Jacobian operator of x + Moreover, based on the partition of the index set, we give a decomposition of Euclidean Jordan algebra (see Theorem 28) In Section 3, we present the exact expression of the Clarke generalized Jacobian of the projection operator x + by studying its B-subdifferential In Section 4, we investigate the relationships among B- subdifferentials of x +, x and x In Section 5, we design semismooth Newton methods for NSCP, and show that the proposed algorithm is locally quadratically convergent under some conditions which do not require strict complementarity of the solution 2 Preliminaries 2 Euclidean Jordan algebras We give a brief introduction to Euclidean Jordan algebras More comprehensive introduction to the area can be found in Koecher s lecture notes [5] and in the monograph by Faraut and Korányi [7] A Euclidean Jordan algebra is a triple (J,,, ) (V for short), where (J,, ) is a n- dimensional inner product space over R and (x, y) x y : J J J is a bilinear mapping which satisfies the following conditions: (i) x y = y x for all x,y J, (ii) x (x 2 y) = x 2 (x y) for all x,y J where x 2 := x x, (iii) x y,w = x,y w for all x,y,w J We call x y the Jordan product of x and y In general, (x y) w x (y w) for all x,y,w J We assume that there exists an element e (called the identity element) such that x e = e x = x for all x J Given a Euclidean Jordan algebra V, define the set of squares as K := {x 2 : x J } By Theorem III 2 in [7], K is the symmetric cone, ie, K is a closed, convex, homogeneous and self-dual cone For x J, the degree of x denoted by deg(x) is the smallest positive integer k such that the set {e,x,x 2,,x k } is linearly dependent The rank of V is defined as max{deg(x) : x J } In this paper, r will denote the rank of the underlying Euclidean Jordan algebra An element c J is an idempotent if c 2 = c 0, which is also primitive if it cannot be written as a sum of two idempotents A complete system of orthogonal idempotents is a finite set {c,c 2,,c k } of idempotents where c i c j = 0 for all i j, and c + c c k = e A Jordan frame is a complete system of orthogonal primitive idempotents in V We now review two spectral decomposition theorems for the elements in a Euclidean Jordan algebra Theorem 2 (Spectral Decomposition Type I (Theorem III, [7])) Let V be a Euclidean Jordan algebra Then for x J there exist unique real numbers µ (x),µ 2 (x),,µ r (x), all distinct, and a unique complete system of orthogonal idempotents {b,b 2,,b r } such that x = µ (x)b + + µ r (x)b r Theorem 22 (Spectral Decomposition Type II (Theorem III2, [7])) Let V be a Euclidean Jordan algebra with rank r Then for x J there exist a Jordan frame {c,c 2,,c r } and real numbers λ (x),λ 2 (x),,λ r (x) such that x = λ (x)c + λ 2 (x)c λ r (x)c r (2) 3

4 The numbers λ i (x) (i =,2,,r) are the eigenvalues of x We call (2) the spectral decomposition (or the spectral expansion) of x Note that the above {b,b 2,,b r } and {c,c 2,,c r } depend on x We do not write this dependence explicitly for the sake of simplicity in notation Let C(x) be the set consisting of all Jordan frames in the spectral decomposition of x Let the spectrum σ(x) be the set of all eigenvalues of x Then σ(x) = {µ (x),µ 2 (x),,µ r (x)} and for each µ i (x) σ(x), denoting N i (x) := {j : λ j (x) = µ i (x)} we have b i = j N i (x) c j Let g : R R be a real-valued function Define the vector-valued function G : J J as G(x) := g(λ j (x))c j (x) = g(λ (x))c (x) + g(λ 2 (x))c 2 (x) + + g(λ r (x))c r (x), (22) j= which is a Löwner operator In particular, letting t + := max{0,t}, t := min{0,t} and noting t = t + t (t R), respectively, we define Π K (x) := x + := (λ i (x)) + c i (x), x := (λ i (x)) c i (x) and x := λ i (x) c i (x) Note that z K (z int(k)) if and only if λ i (z) 0 (λ i (z) > 0) ( i {,2,,r}), where int(k) denotes the interior of K It is easy to verify that x + K, x = x + + x, and x = x + x In other words, x + is the projection of x onto K For each x J, we define the Lyapunov transformation L(x) : J J by L(x)y = x y for all y J, which is a symmetric self-adjoint operator in the sense that L(x)y,w = y, L(x)w for all y,w J The operator Q(x) := 2L 2 (x) L(x 2 ) is called the quadratic representation of x We say two elements x, y J operator commute if L(x)L(y) = L(y)L(x) By Lemma X22 in [7], two elements x,y operator commute if and only if they share a common Jordan frame Next, we recall the Peirce decomposition on the space J Let c J be a nonzero idempotent Then J is the orthogonal direct sum of J(c,0),J(c, 2 ) and J(c,), where J(c,ε) := {z J : c z = εz}, ε {0, 2 }, This is called the Peirce decomposition of J with respect to the nonzero idempotent c Fix a Jordan frame {c,c 2,,c r } Defining the following subspaces for i,j {,2,,r}, { J ii := {x J : x c i = x} and J ij := x J : x c i = } 2 x = x c j, i j, we have the Peirce decomposition theorem as follows For more detail, see [7] Theorem 23 (Theorem IV2, [7]) Let {c,c 2,,c r } be a given Jordan frame in a Euclidean Jordan algebra V of rank r Then J is the orthogonal direct sum of spaces J ij (i j) Furthermore, (i) J ij J ij J ii + J jj ; (ii) J ij J jk J ik, if i k; (iii) J ij J kl = {0}, if {i,j} {k,l} = Ø 4

5 For any i j {,2,,r} and k l {,2,,r}, by Corollary IV26 of [7], we have dim(j ij ) = dim(j kl ) =: n Then n = r + n r(r ) (23) 2 For a given Jordan frame {c,c 2,,c r } and i,j {,2,,r}, let C ij (x) be the orthogonal projection operator onto subspace J ij Then, by Theorem 23, we have C jj (x) = Q(c j ) and C ij (x) = 4L(c i )L(c j ) = 4L(c j )L(c i ) = C ji (x), i,j {,2,,r}, (24) and the orthogonal projection operators {C ij (x) : i,j {,2,,r}} satisfy and C ij (x) = C ij(x), C ij (x) 2 = C ij (x), C ij (x)c kl (x) = 0 if {i,j} {k,l},i,j,k,l {,2,,r} i j r C ij (x) = I, where C ij (x) is the adjoint operator of C ij(x) and I is the identity operator Furthermore, we have the following spectral decomposition theorem for L(x), L(x 2 ) and Q(x) For a more detailed exposition, see [5, 26] Theorem 24 (Theorem 3, [26]) Let x J and r j= λ j(x)c j (x) denote its spectral decomposition with λ (x) λ 2 (x) λ r (x) Then, the symmetric operators L(x), L(x 2 ) and Q(x) have the spectral decompositions: L(x) = L(x 2 ) = Q(x) = λ j (x)c jj (x) + j= λ 2 j (x)c jj(x) + j= λ 2 j(x)c jj (x) + j= j<l r j<l r j<l r 2 (λ j(x) + λ l (x))c jl (x), 2 (λ2 j (x) + λ2 l (x))c jl(x), λ j (x)λ l (x)c jl (x) Moreover, the spectra σ(l(x)), σ(l(x 2 )) and σ(q(x)), respectively, consist of all distinct numbers 2 (λ j(x) + λ l (x)), 2 (λ2 j (x) + λ2 l (x)), and λ j(x)λ l (x) for all j,l {,2,,r} In the end of this subsection, we characterize a cone K I (see (25) below) with respect to x, which plays a key role in establishing the connection between B Π K (x) and B Π KI (0) (see Theorem 33) Let x = r j= λ j(x)c j = r µ i(x)b i with λ (x) λ 2 (x) λ r (x) and µ (x) > µ 2 (x) > > µ r (x) In what follows, let ℵ(x) := {i : λ i (x) < 0}, I(x) := {i : λ i (x) = 0} and (x) := {i : λ i (x) > 0} For the sake of simplicity, let ℵ := ℵ(x), I := I(x) and := (x) Thus ℵ I = {,2,,r} Set b I := j Ic j By Theorem 2, it follows that b I is uniquely defined by x In other words, for any Jordan frame { c,, c r } C(x), j I c j = b I Therefore, we can define a subspace J I := J(b I,) := {w J : w b I = w} 5

6 By Lemma 20 in [9], it follows that J I = span { c +,c +2,,c + I } + + j<k + I Then we can verify by direct calculation that J I is also a Euclidean Jordan algebra with its identity element b I This leads us to define the cone of its squares as J jk K I := {w 2 : w J I } (25) Therefore, it is well-defined that Π KI : J I J I is the projection onto the symmetric cone K I We can also define a cone by the finite set {c j : j I} Thus, we arrive at another cone associated with x, denoted by K(x,0) := Cone{c j : j I} (26) {c,,c r} C(x) An interesting question occurs: what is the difference between the cones K I and K(x,0)? Clearly, when x = 0, K I = K(0,0)(= K) The following proposition states that they always equal to each other Proposition 25 Let x = r j= λ j(x)c j with λ (x) λ 2 (x) λ r (x) and b I = j I c j Let K I and K(x,0) be defined by (25) and (26), respectively Then we have K I = K(x,0) = Cone{c j : j I} (27) {c,,c r} C(x) and b I is the identity element in the above symmetric cone Proof It is clear that K(x,0) K I and b I is the identity element in K I We now prove K I K(x,0) Let y K I Then there exists a Jordan frame in J I(x), say, {e,e 2,,e I }, such that y = y e + y 2 e y I e I with y i 0 Clearly, elements e,e 2,,e I belong to K and the sum of them is b I We next show that, for any Jordan frame {c i : i {,,r}} C(x), we can replace {c +,,c + I } by {e,,e I } to get an element in C(x); Ie, In fact, by the above definitions, we have {c,,c,e,,e I,c + I +,,c r } C(x) (28) c k b I = 0, for all k {,,, + I +,,r} Then c k,b I = 0 By b I = e + + e I, it follows that c k,b I = I c k,e i = 0 Since c k,e i K, c k,e i 0 It then follows c k,e i = 0 By Proposition 6 in [9], we have c k e i = 0, for all k {,,, + I +,,r}, i {,, I } This proves (28) Then Cone{e j : j {,, I }} K(x,0), and therefore y K(x,0) Remark From the above analysis, for x = r j= λ j(x)c j = r µ i(x)b i, we can also obtain that for any Jordan frame {c,,c r } C(x), the set {c j : λ j (x) = µ i (x)} is a Jordan frame in J(b i,), and a Jordan frame in each J(b i,) (i {,, r}), say, C bi, is a subset of some Jordan frame in C(x) and the union of C b,,c b r is a Jordan frame in C(x) 6

7 22 Matrix Representation of G(x) Let G(x) be given by (22) Suppose that g is differentiable at τ i,i {,2,,r} Define the first divided difference g [] of g at τ := (τ,τ 2,,τ r ) T R r as the r r symmetric matrix with the ij-th entry (g [] (τ)) ij (i,j {,2,,r}) given by [τ i,τ j ] g := g(τ i ) g(τ j ) τ i τ j if τ i τ j, g (τ i ) if τ i = τ j Based on Theorem 32 of [26], we can directly derive the following Jacobian properties of the Löwner operator G( ) Theorem 26 Let x = r j= λ j(x)c j = r µ i(x)b i (x) Then, G( ) is (continuously) differentiable at x if and only if for each j {,2,,r}, g is (continuously) differentiable at λ j (x) In this case, the Jacobian G(x) is given by G(x) = 2 [λ i (x),λ j (x)] g L(c i )L(c j ) g (λ i (x))l(c i ), (29) or equivalently G(x) = 2 i j,i,j= i,j= [µ i (x),µ j (x)] g L(b i (x))l(b j (x)) + Furthermore, G(x) is a symmetric linear operator from J into itself g (µ i (x))q(b i (x)) (20) Suppose that Löwner operator G( ) is (continuously) differentiable at x = r j= λ j(x)c j with λ (x) λ 2 (x) λ r (x) For a given Jordan frame {c,c 2,,c r }, let L := L(c ) L(c r ) be an operator vector Denote L := (L (c ),, L (c r )), where L (c i ) is the adjoint operator of L(c i ) Since L(c i ) is self-adjoint, we have L = (L (c ),, L (c r )) = (L(c ),, L(c r )) For the real r r matrix Λ(x) := ([λ i (x),λ j (x)] g ) r r, we define L Λ(x)L := [λ i (x),λ j (x)] g L(c i )L(c j ) i,j= Similarly, for the r-vector d(x) = (g (λ (x)),,g (λ r (x))) T, define d (x)l := g (λ i (x))l(c i ) Thus, by (29), we can give a matrix representation of the Jacobian G(x) as follows: G(x) = 2L Λ(x)L d (x)l (2) Using the matrix expression above and Theorem 24, we can obtain the eigenvalues of G(x) explicitly 7

8 Lemma 27 Let x = r j= λ j(x)c j = r µ i(x)b i (x) Suppose that G( ) is differentiable at x Let Λ(x) be defined by (2) using G(x) Then the spectrum of G(x) consists of all the elements of matrix Λ(x) Moreover, the operator G(x) has r single eigenvalues [λ i (x),λ i (x)] g (i {,2,,r}) as well as the eigenvalue [λ i (x),λ j (x)] g (i < j {,2,,r}) with multiplicity d (unless some elements of Λ(x) are equal, in which case the multiplicities change in an obvious way) Proof By direct calculation, we obtain G(x) = 2 = = [λ i (x),λ j (x)] g L(c i )L(c j ) i,j= g (λ i (x))q(c i ) + 4 g (λ i (x))c ii (x) + i<j r i<j r g (λ i (x))l(c i ) [λ i (x),λ j (x)] g L(c i )L(c j ) [λ i (x),λ j (x)] g C ij (x) where the second equality holds by the definition of Q(x) and the third by (24) By Theorem 24, the spectrum of G(x) consists of all scalars [λ i (x),λ j (x)] g (i,j {,2,,r}) or the elements of matrix Λ(x) Then the property about the multiplicities follows from (23) In particular, consider the metric projection Π K ( ) at x = r j= λ j(x)c j Note that t = 0 is the unique point where g = t + is not smooth but strongly semismooth Thus if I = Ø, ie, x is nonsingular, Π K ( ) is continuously differentiable at x In this case, from (2), we write Π K (x) as Π K (x) = 2L Λ(x)L d (x)l where Λ(x) is a symmetric matrix of the form ( E Γ Λ(x) = Γ T 0 ℵ ℵ ( with E being the all ones matrix, the ℵ matrix Γ = and the vector d(x) = matrix ) λ i (x) λ i (x) λ j (x) ) (22) ℵ, ( T,0T ℵ ) T Throughout this paper, En n 2 denotes the n n 2 all ones 23 A Decomposition Theorem of J Note that, for the operator Π K (x), any element of Λ(x) as in (22) belongs to the interval [0,] By (23) and Lemma 27, we can reset the elements of Λ(x) in nonincreasing order, accounting for multiplicities as a (x) a 2 (x) a n (x) Thus, we can define a partition of {,2,,n} as {,2,,n} = α β γ where α := {i : a i (x) = }, β := {i : 0 < a i (x) < }, γ := {i : a i (x) = 0} (23) Observing that a i (x) is the eigenvalue of some corresponding orthogonal projection operator in {C jl (x) : j < l,j,l {,2,,r}}, by (23) and Lemma 27, we rewrite Π K (x) as Π K (x) = n a i (x)p i (x), 8

9 { where P i (x) is the corresponding orthogonal element in C jj, d } C jl (x) : j < l,j,l {,2,,r} In view of the definitions of the index sets α,β,γ, we define three subspaces J α, J β and J γ by J α := i α P i (x)(j ), J β := i β P i (x)(j ), J γ := i γ P i (x)(j ) (24) Observe that J α = i j J ij, J β = i <j J ij, J γ = Clearly, we have the following decomposition of the space J j>i> Theorem 28 Let the index sets α,β,γ be a partition of {,2,,n} given by (23), and let J α, J β and J γ be given by (24) Then J is the orthogonal direct sum of J α, J β and J γ In other words, every y J is of the form where y α J α, y β J β and y γ J γ J ij y = y α + y β + y γ, (25) Likewise, we define the corresponding operators from Π K (x) as follows: [ Π K (x)] α := i α a i (x)p i (x), [ Π K (x)] β := i β a i (x)p i (x), [ Π K (x)] γ := i γ a i (x)p i (x) = 0 Note that [ Π K (x)] β i β a i (x) P i(x) y β = i β a i (x) P i(x) [ Π K (x)] β y β = y β, y β J β (26) Using the above definitions (26), we directly obtain the following fact Lemma 29 Let the index sets α,β,γ be a partition of {,2,,n} given by (23) Let the orthogonal operators [ Π K (x)] α, [ Π K (x)] β and [ Π K (x)] γ be defined by (26) For any y J with y = y α + y β + y γ as in (25), [ Π K (x)] α y = y α, [ Π K (x)] β y = ȳ β, and [ Π K (x)] γ y = 0, where ȳ β is a vector in J β Furthermore, [ Π K (x)] β is a one-to-one mapping from J β to J β and therefore it has an inverse [ Π K (x)] β on J β as follows: [ Π K (x)] β = a i (x) P i(x) i β 3 Clarke Generalized Jacobian of Π K This section deals with the Clarke generalized Jacobian of Π K This is influenced by the recent work [3, 2, 8] in the special settings of second-order cones and positive semi-definite cones, where Kanzow, Ferenczi and Fukushima [3] gave an expression for the B-subdifferential of the projection onto second-order cones, Hayashi, Yamashita and Fukushima [2] gave an explicit 9

10 representation for Clarke generalized Jacobian of the projection onto second-order cones; Malick and Sendov [8] worked out the Clarke generalized Jacobian of the projection onto the cone of symmetric positive semi-definite matrices We generalize the above results to symmetric cones For this purpose, we mainly look at the B-subdifferential of Π K First, we establish our notation For a given Jordan frame {c,c 2,,c r }, we define { {c,,c r} B Π K (x) := lim Π K (x + h) : h span{c,c 2,,c r }, lim Π K (x + h) exists h 0,x+h D ΠK h 0 For a given integer t {0,,, I } we define a r-vector d t (x) by ( ) d t (x) := t 0 I t = +t, (3) 0 r t 0 ℵ and a set of r r matrices Λ t (x) by E E I Γ ℵ ( ) Λ t (x) := E I Λ 0 0 I ℵ E Λ00 : Λ 0 = Γ T Λ ℵ 0 ℵ I 0 T,Λ Λ(t, I ), (32) ℵ ℵ λ where Γ ℵ is a given matrix by the ij-entry Γ ij = i (x) λ i (x) λ j (x) (i,j ℵ), and Λ(t, I ) is a set of t ( I t) matrices (θ ij ) t ( I t) (the rows are indexed by +, + 2,, + t, and the columns by + t +, + t + 2,, + I ) specified by { } Λ(t, I ) := (θ ij ) t ( I t) [0,] t ( I t) : θ ij satisfy (a) and (b) below (a) θ i, +t+ θ i, +t+2 θ i, + I (i { +, + 2,, + t}), θ +,j θ +2,j θ +t,j (j { + t +, + t + 2,, + I }); ( ) (b) θ ij is a matrix of rank at most one t ( I t) Clearly, when x = 0, we have a r-vector and a set of r r matrices Λ t (0) := Λ t := d t (0) := d t := ( t 0 r t ), (33) {( ) } Et t Λ t (r t) Λ T : Λ t (r t) 0 t (r t) Λ(t,r), (34) (r t) (r t) where Λ(t,r) is a set of t (r t) matrices (θ ij ) t (r t) (the rows are indexed by,2,,t, and the columns by t +,t + 2,,r) specified by { } Λ(t,r) := (θ ij ) t (r t) [0,] t (r t) : θ ij satisfy (a ) and (b ) below (a ) θ i,t+ θ i,t+2 θ i,r (i {,2,,t}), θ ( j θ 2j ) θ tj (j {t +,t + 2,,r}); (b ) θ ij is a matrix of rank at most one t (r t) We are now ready to give the formula for the Clarke generalized Jacobian of Π K (x) by its B-subdifferential Our next result generalizes Lemma 26 of [3] (for B-subdifferential), Theorem 37 of [8] and Proposition 48 of [2] (for Clarke generalized Jacobian) 0 }

11 Theorem 3 The B-subdifferential of Π K ( ) at x is given by B Π K (x) = I {c,,c r} C(x) t=0 {2L Λ t (x)l d t(x)l} where d t (x) and Λ t (x) are specified by (3) and (32), respectively Furthermore, the Clarke generalized Jacobian of Π K ( ) at x is I Π K (x) = conv { B Π K (x)} = conv {2L Λ t (x)l d t (x)l} {c,,c r} C(x) t=0 Proof First we prove B Π K (x) I {c,,c r} C(x) t=0 {2L Λ t (x)l d t (x)l} Without loss of generality, fix a Jordan frame {c,,c r } C(x) For any arbitrary but given integer t {0,,, I }, set V := 2L A t (x)l d t (x)l where A t(x) Λ t (x) and d t (x) is given by (3) We need only to show that V B Π K (x) For this purpose, it suffices to demonstrate that there exists a vector h = r λ i(h)c i such that V = lim Π K (x + h) h 0,x+h D ΠK Let A t (x) = (θ ij ) r r We define ς ij := θ ij for i { +,, + t} and j { + t +,, + I }, then B t ( I t) := (ς ij ) t ( I t) can be written as B t ( I t) = π + π +t ( π +t+,,π + I ), where π i [0,+ ], π + π +2 π +t and π +t+ π +t+2 π + I Taking 0 if i, ρ if π i = 0, < i + t, π i ρ 2 if π i 0,+, < i + t, ρ 3 if π i = +, < i + t, λ i (h) = ρ 3 if π i = 0, + t < i + I, π i ρ 2 ρ if π i 0,+, + t < i + I, if π i = +, + t < i + I, 0 if + I < i r, we have x + h = r λ i(x + h)c i = r [λ i(x) + λ i (h)]c i and for sufficiently small ρ (0,), λ (x + h) λ +t (x + h) > 0 > λ +t+ (x + h) λ r (x + h) It is easy to verify that x + h D ΠK and Hence V B Π K (x) lim Π K(x + h) = 2L A t (x)l d t(x)l h 0

12 We next prove that B Π K (x) r {c,,c r} C(x) t=0 {2L Λ t (x)l d t(x)l} Let W B Π K (x) Then there is a vector h := h(w) J such that W = lim h 0,x+h DΠK Π K (x + h) For the above h J, let x + h =: r j= λ j(x + h)c j (x + h) with λ (x + h) λ 2 (x + h) λ r (x + h) In the sense of set convergence (see [24]), we have lim sup h 0,x+h D Πk {(c (x + h),,c r (x + h))} C(x) Let us pick {c,,c r } lim sup{c (x + h),,c r (x + h)} Clearly, lim h 0,x+h D Πk λ(x + h) = λ(x), where λ(x + h) := (λ (x + h),,λ r (x + h)) T and none of λ i (x + h) is zero Suppose that ( T ℵ(x + h), (x + h) are given and t := (x + h) Then d t (x) = T +t r t),0t Set Λ t (x + h) := E E I Γ ℵ (h) E I Λ 0 I ℵ Γ T ℵ (h) 0 ℵ I 0 ℵ ℵ ( ) with Λ Et t Λ2 := Λ T, 2 0 ( ℵ t) ( ℵ t) ( ) λ where Γ ℵ (h) = i (x+h) λ i (x+h) λ j (x+h) (i,j ℵ) and Λ ) 2 = ( Λij with Λ ij = ℵ t ( I t) λ i (x+h) λ i (x+h) λ j (x+h) (i { +,, + t}, j { + t +,, + I }) If lim h 0 Λ 2 exists, then direct calculation yields Θ t (x) := lim h 0 Λ t (x+h) = E E I Γ ℵ E I Λ 0 I ℵ Γ T ℵ 0 ℵ I 0 ℵ ℵ ( Et t Λ with Λ = 2 Λ T 2 0 ( ℵ t) ( ℵ t) where Λ := lim h 0 Λ and λ Λ2 := lim h 0 Λ2 = (θ ij ) t ( I t) with θ ij = lim i (x+h) h 0 λ i (x+h) λ j (x+h) (i { +,, + t}, j { + t +,, + I }) Observing that and λ + (h) λ +2 (h) λ +t (h) > 0 0 < λ +t+ (h) λ +t+2 (h) λ + I (h), we easily show θ ij [0,] and (a) In order to prove (b), let ς ij := lim h 0 λ j (x + h) λ i (x + h) for i { +,, + t}, j { + t +,, + I } Then, θ ij = + ς ij or equivalently ς ij = θ ij (35) Note that θ ij [0,] implies ς ij [0,+ ] and θ ij = 0 ς ij =, θ ij = ς ij = 0 Define B := (ς ij ) t ( I t) Then, it is easy to see that B = lim h 0 λ + (h) λ +t (h) ( λ +t+ (h),, λ + I (h) ) This establishes (b), and hence Θ t (x) Λ t (x) The existence of lim h 0,x+h DΠK Π K (x + h) means that lim h 0 Λ2 (h) exists This proves W = 2L Θ t (x)l d t (x)l {2L Λ t (x)l d t (x)l}, as desired 2 ),

13 From the above proof, we also obtain that for a given Jordan frame {c,c 2,,c r } C(x) {c,,c r} B Π K (x) = I {2L Λ t (x)l d t(x)l} t=0 Note that when x = 0 the above result becomes the following, which provides a formula for B-subdifferential of the projection onto symmetric cone Π K ( ) at the origin Corollary 32 The B-subdifferential of Π K ( ) at 0 is B Π K (0) = {c,,c r} C(0) t=0 r {2L Λ t L d t L} where d t and Λ t are given by (33) and (34), respectively Furthermore, the Clarke generalized Jacobian of Π K ( ) at 0 is r Π K (0) = conv ( B Π K (0)) = conv {2L Λ t L d t L} In general, B Π K (0) Π K (0) {c,,c r} C(0) t=0 Proof We only need show that B Π K (0) Π K (0) in general For this purpose, we look at an example in the case of the second-order cone Λ n + := {(x,x T 2 )T : x x 2,x R,x 2 R n } (n 2) Let {c,c 2 } be a Jordan frame given by c i = 2 ( ( ) i ω ), for i {,2}, with ω being any vector in R n satisfying ω = By direct calculation, we can derive that B Π Λ n + (0) = {0, I, T } and Π Λ n + (0) = conv{0, I, T } where T satisfies T = 4[0,]L(c )L(c 2 ) + Q(c ) + 0 Q(c 2 ) This means that B Π K (0) Π K (0) in general Applying Proposition 25, Theorem 3 and Corollary 32, we immediately obtain the following result, which states the interesting connection between B Π K (x) and B Π KI (0) In the case of S n, it reduces to Proposition 22 in [25] Theorem 33 Let x = r λ i(x)c i with λ (x) λ 2 (x) λ r (x) Let L L = L I L ℵ where L := L(c ) L(c ), L I := L(c + ) L(c + I ), L ℵ := L(c + I + ) L(c r ) 3

14 Then V B Π K (x) if and only if there exists a V I B Π KI (0) such that V = 2L A t (x)l d t(x)l, V I = 2L I A(t, I )L I d t L I where r r matrix A t (x) Λ t (x) and I I matrix A(t, I ) Λ(t, I ) satisfy E E I Γ ℵ A t (x) = E I A(t, I ) 0 I ℵ ; Γ T ℵ 0 ℵ I 0 ℵ ℵ and d t (x) and d t are, respectively, r-vector and I -vector such that d t (x) = ( T,dT t,0 ℵ ) T Proof Let {c,,c r } =: {c,c I,c ℵ } with c := { } c,,c, ci := { } c +,,c + I, and c ℵ := { } c + I +,,c r By Proposition 25, any Jordan frame in JI is a subset of a Jordan frame in C(x), and the part c I of a Jordan frame {c,,c r } in C(x) is a Jordan frame in J I The conclusion follows immediately from Theorem 3 and Corollary 32 4 B-subdifferentials of x and x Employing the same technique as in the proof of Theorem 3, we give below the formulae for the B-subdifferentials of x and x Theorem 4 Let ē be the all ones vector of appropriate size The B-subdifferential of x is given by I B x = {2L (E r r Λ t (x))l (ē d t (x)) L} {c,,c r} C(x) t=0 where d t (x) and Λ t (x) are specified by (3) and (32), respectively Furthermore, Proof Note that for any pair of scalars τ i,τ j, B x = I B Π K (x) (τ i ) (τ j ) τ i τ j + (τ i) + (τ j ) + τ i τ j = or (τ i ) (τ j ) τ i τ j = (τ i) + (τ j ) + τ i τ j In a similar way as in Theorem 3, we obtain the formula for the B-subdifferential of x Before studying the B-subdifferential of at x, we need the following notation For a given integer t {0,,, I } we define a r-vector t (x) by t (x) := t I t ℵ ( = +t r t ), (4) 4

15 and a set of r r matrices Å t (x) by E E I Υ ℵ ( ) Å t (x) := E I Υ 0 E I ℵ Et t Υ : Υ 0 = 00 Υ T Υ ℵ E ℵ I E T,Υ 00 E 00 Υ(t, I ) ( I t) ( I t), ℵ ℵ (42) where Υ ℵ is a given matrix with the ij-entry Υ ij = λ i(x)+λ j (x) λ i (x) λ j (x) (i,j ℵ), and Υ(t, I ) is a set of t ( I t) matrices (ξ ij ) t ( I t) (the rows are indexed by +, +2,, +t, and the columns by + t +, + t + 2,, + I ) specified by { } Υ(t, I ) := (ξ ij ) t ( I t) [,] t ( I t) : ξ ij satisfy (c) and (d) below, (c) ξ i, +t+ ξ i, +t+2 ξ i, + I (i = +, + 2,, + t), ξ +,j ξ +2,j ξ +t,j (j = + t +, + t + 2,, + I ); ( ξij (d) +ξ ij is a matrix of rank at most one )t ( I t) We are in a position to give a formula for the B-subdifferential of x Theorem 42 The B-subdifferential of x is given by B x = I {c,,c r} C(x) t=0 {2L Å t (x)l t(x)l}, where t (x) and Å t (x) are specified by (4) and (42), respectively Proof Note that for any pair of scalars τ i > τ j, τ i τ j τ i τ j = In the case of τ i > 0 > τ j, set ξ ij := τ i+τ j τ i τ j Then if τ i > τ j 0, τ i +τ j τ i τ j if τ i > 0 > τ j, if τ i 0 > τ j ξ ij = τ j τ i + τ, or equivalently τ j j τ τ i i = ξ ij + ξ ij This implies that τ j [0,+ ] ξ ij [,] τ i Based on the above fact and a similar argument as in the proof of Theorem 3, we deduce the claimed identity Note that B x B x + + B x and B x B x + B x, although x = x + + x and x = x + x, respectively For instance, in the case of J = R, it is easy to derive B x + = B x = {0,}, B x = and B x = {,} However, B x + + B x = {0,,2} and B x + B x = {,0,} 5

16 5 Semismooth Newton Methods As an application, in this section we deal with the semismooth Newton methods for NSCP() by using the projection operator We establish sufficient conditions for our methods to be locally quadratically convergent In the setting of second-order cones, Kanzow, Ferenczi and Fukushima [3] reformulated NSOCP as a nonsmooth system of equations and investigated conditions for the local quadratic convergence of nonsmooth Newton methods They also gave the related conditions for the special case of a linear second-order cone program In the setting of positive semi-definite cones, we only know of some similar results for the linear semi-definite programs, see the recent work [2, 8, 4] and the references therein As it is well-known, under certain favorable conditions like convexity of f and a Slater-type constraint qualification (see, eg, [6, 24]), we can reformulate NSCP as a nonsmooth system of equations by the corresponding KKT conditions, which can be written as follows: f(x) A y = 0, Ax q = 0, x K, y K, x,y = 0 By Proposition 6 in [9], the complementarity condition x K, y K, x, y = 0 is the same as x (x y) + = 0 Therefore, the above system is equivalent to the system of equations H(z) = 0, where the mapping H : V R m V V R m V is specified by H(z) := H(x,,y) := f(x) A y Ax q x (x y) + Then we can apply the nonsmooth Newton method (see [2, 22, 23]) (5) z k+ := z k W k H(z k), W k B H(z k ), k {0,,2, }, (52) to the system of equations H(z) = 0, and therefore solve the NSCP under some suitable conditions In order to establish (52), we need to show the semismoothness of H and the nonsingularity of H Theorem 5 The mapping H defined by (5) is semismooth Moreover, if the Hessian 2 f is locally Lipschitz continuous, then the mapping H is strongly semismooth Proof Note that Π K (x) is strongly semismooth by Proposition 33 in [26] The conclusion follows from the operation rules of semismooth functions Lemma 52 Let z = (x,,y ) be a strictly complementary KKT point of NSCP () Then Π K is continuously differentiable at (x y ) Proof By assumption, x K, y K and x,y = 0 Then, by Proposition 6 in [9], x and y operator commute and x y = 0 It follows that there is a Jordan frame {c,c 2,,c r } such that x = λ i (x )c i, y = λ i (y )c i, and λ i (x )λ i (y ) = 0, i {,2,,r} (53) 6

17 Since x, y K, λ i (x ) 0, λ i (y ) 0 for all i {,2,,r} Again, since x + y int(k) by the strict complementarity assumption, λ i (x ) + λ i (y ) > 0, i {,2,,r} This together with (53) yields that λ i (x ) λ i (y ) 0, i {,2,,r} Thus x y = [λ i (x ) λ i (y )]c i, with λ i (x ) λ i (y ) 0, i {,2,,r} The desired result follows immediately from Theorem 26 We are ready to give our main result on the nonsingularity of H at a strictly complementary KKT point of NSCP () In the setting of NSOCP, it reduces to Theorem 35 in [3] by considering the concrete forms of C jl which are related to the decomposition of a symmetric matrix, for details, see [3] and the references therein Theorem 53 Let z = (x,,y ) be a strictly complementary KKT point of NSCP () Let the index sets α,β,γ be defined by (23) with a i (x) = a i (x y ), and let the corresponding subspaces J α,j β,j γ be given by (24) Let V := Π K (x y ), and V β := [ Π K (x y )] β Assume the following conditions hold: (i) The operator 2 f(x ) + V β (I V ) β is positive definite on the subspace S := {s = s α + s β + s γ V : As = 0,s γ = 0}; (ii) The following implication holds: A J γ = 0 Then the Jacobian H exists and is nonsingular In particular, for the linear SCP, the statement holds with condition (i) replaced by the following one: (i ) {s α : As α = 0} J α = {0} Proof The first part is clear by Lemma 52 It remains to show that H is nonsingular It is easy to see that 2 f(x ) A I H(z ) = H(x,,y ) = A 0 0 I V 0 V Observe that V β (I V ) β is positive definite on the subspace J β, ie, w,v β (I V ) βw > 0, w J β, w 0 (54) In fact, since V β = [ Π K (x y )] β, by (26) and Lemma 29, we obtain that 0 < a i (x y ) < for i β, (I V ) β = i β [ a i(x y )]P i (x y ) and V β = [ Π K (x y )] β = i β a i (x y ) P i(x y ) Thus, by (24) V β (I V ) β = i β a i (x y ) a i (x y P i (x y ) ) This proves (54) 7

18 Let z = (x,,y) V R m V be any vector such that H(z )z = 0 Then 2 f(x )x A y = 0, (55) Ax = 0, (56) (I V )x + V y = 0 (57) By Theorem 28, set x := x α + x β + x γ and y := y α + y β + y γ where x α,y α J α, x β,y β J β and x γ,y γ J γ In view of the definitions of α,β and γ, by (26) and (57), we have By Lemma 29, we obtain Therefore, we deduce [(I V ) α x + V α y] + [(I V ) β x + V β y] + [(I V ) γ x + V γ y] = 0 y α = 0, x γ = 0 and (I V ) β x β + V β y β = 0 (58) x,y = x α + x β + x γ,y α + y β + y γ = x α,y α + x β,y β + x γ,y γ = x β,y β = x β, V β (I V ) βx β = x α + x β,v β (I V ) β(x α + x β ), where the second equality holds by the orthogonality property of the spaces J α,j β and J γ, the third by (58) and the fourth by Lemma 29, the last one follows from x α,v β (I V ) βx β = 0 and V β (I V ) βx α = 0 In the meanwhile, by (55), (56) and (58), we have 0 = x, 2 f(x )x A y = x, 2 f(x )x Ax, x,y = x α + x β, 2 f(x )(x α + x β ) + x α + x β,v β (I V ) β(x α + x β ) = x α + x β,[ 2 f(x ) + V β (I V ) β](x α + x β ) These together with assumption (i) yield that x α + x β = 0, and hence x = 0 By (58), this implies y β = 0 Thus, from (55), we have A = y γ J γ By assumption (ii), = 0 and hence y γ = 0 So, we finally obtain that z = 0 The desired conclusion follows In particular, for linear SCP, since 2 f(x ) = 0, from (54) and x,y = x β, V β (I V ) β x β, we deduce that x β = 0 Then, by (56) and assumption (i ), x α = 0 Following the same steps as above, we complete the proof Next, we extend the above theorem to the general case where strict complementarity may not hold Let z = (x,,y ) be a (not necessarily strictly complementary) KKT point of NSCP () Clearly, in that case, Π K is strongly semismooth at (x y ) In order to establish (52), we need to provide suitable conditions which guarantee the nonsingularity of all elements of the B-subdifferential of H at z For this purpose, we give the following notations For any element V B Π K (x) with V = 2L Λ t (x)l d t(x)l as in Theorem 3, from the argument before (23) (see Subsection 23), we obtain the corresponding index sets α(v ), β(v ) and γ(v ) as α(v ) := { i : a V i (x) = }, β(v ) := { i : 0 < a V i (x) < }, γ(v ) := { i : a V i (x) = 0}, (59) 8

19 where the scalars a V i (x) are the elements of Λ t(x), accounting for multiplicities as a V (x) a V 2 (x) av n (x) From the proof of Theorem 53, we can obtain the following result on the nonsingularity of any element of the B-subdifferential of H where the strict complementarity assumption is replaced by some conditions on this element, which is a generalization of Theorem 38 in [3] from NSOCP to NSCP For details, see [3] and the references therein Theorem 54 Let z = (x,,y ) be a (not necessarily strictly complementary) KKT point of NSCP () For any V B Π K (x y ), let the index sets α(v ),β(v ) and γ(v ) be defined by (59) with a V i (x) = av i (x y ), and the corresponding subspaces by J α(v ),J β(v ),J γ(v ) Let V β(v ) = i β(v ) a i(x y )P i (x y ) Assume the following conditions hold: (i) The operator 2 f(x ) + V β(v ) (I V ) β(v ) is positive definite on the subspace S := {s = s α(v ) + s β(v ) + s γ(v ) V : As = 0,s γ(v ) = 0}; (ii) The following implication holds: A J γ(v ) = 0 Then all the elements of B H(z ) are nonsingular In particular, for the linear SCP, the statement holds with condition (i) replaced by the following one: (i ) {s α(v ) : As α(v ) = 0} J α(v ) = {0} When the Hessian 2 f(x ) is positive definite on some subspace (see below Theorem 55), we will establish the following result which also generalizes Theorem 53 to the case where the condition of strict complementarity may not hold For this purpose, using the above notation (59), we define α(x) := α(v ), γ(x) := γ(v ), β(x) := {,,n} \ (α(x) γ(x)) (50) V B Π K (x) V B Π K (x) It is easy to see that the sets α(x), β(x) and γ(x) form a partition of {,2,,n} In view of Theorem 28, we obtain that J is the orthogonal direct sum of J α(x), J β(x) and J γ(x) where J α(x) := i α(x) P i (x)(j ), J β(x) := i β(x) P i (x)(j ), J γ(x) := i γ(x) P i (x)(j ) (5) Theorem 55 Let z = (x,,y ) be a (not necessarily strictly complementary) KKT point of NSCP () Let the index sets α := α(x y ),β := β(x y ),γ := γ(x y ) be specified by (50), and let the corresponding subspaces J α,j β,j γ be given by (5) Assume the following conditions hold: (i) The Hessian 2 f(x ) is positive definite on the subspace S := {s = s α + s β + s γ V : As = 0,s γ = 0}; (ii) Under the decomposition A = (A ) α + (A ) β + (A ) γ, (A ) α = 0 = 0 Then all elements of B H(z ) are nonsingular Proof Choose any W B H(z ), we need to show that W is nonsingular It is easy to see that W has the form as 2 f(x ) A I W = A 0 0, I V 0 V 9

20 where V B Π K (x y ) is the given element related to W Without loss of generality, let V = 2L Λ t (x y )L d t (x y )L as in Theorem 3 and let α(v ),β(v ),γ(v ) be the corresponding index sets specified by (59) with a V i (x y ) being the elements of Λ t (x y ) Suppose that z = (x,,y) V R m V is any vector such that Wz = 0 As in the proof of Theorem 53, letting x := x α(v ) + x β(v ) + x γ(v ) and y := y α(v ) + y β(v ) + y γ(v ) where x α(v ),y α(v ) J α(v ), x β(v ),y β(v ) J β(v ) and x γ(v ),y γ(v ) J γ(v ), we obtain y α(v ) = 0, x γ(v ) = 0 and (I V ) β(v ) x β(v ) + V β(v ) y β(v ) = 0, and x,y = x β(v ),y β(v ) Thus, we have and x β(v ) = V β(v ) x β(v ) V β(v ) y β(v ), (52) 0 = x β(v ),(I V ) β(v ) x β(v ) + V β(v ) y β(v ) = xβ(v ),(I V ) β(v ) x β(v ) + xβ(v ),V β(v ) y β(v ) Noting that (I V ) β(v ) is positive semidefinite on the subspace J β(v ), we obtain xβ(v ),V β(v ) y β(v ) = xβ(v ),(I V ) β(v ) x β(v ) 0 (53) Similarly, By (52), Vβ(V ) y β(v ),y β(v ) 0 (54) xβ(v ),y β(v ) = Vβ(V ) x β(v ),y β(v ) Vβ(V ) y β(v ),y β(v ) This together with (53) and (54) shows that On the other hand, set x,y = x β(v ),y β(v ) 0 (55) S(V ) := {s = s α(v ) + s β(v ) + s γ(v ) V : As = 0,s γ(v ) = 0} By (50), γ(v ) γ, hence, S(V ) S Thus, assumption (i) implies that the Hessian 2 f(x ) is positive definite on the subspace S(V ) Observing that x,y = x, 2 f(x )x A = x, 2 f(x )x Ax, = x α(v ) + x β(v ), 2 f(x )(x α(v ) + x β(v ) ), from (55), we obtain that x α(v ) + x β(v ) = 0 Hence x = 0, and A = y γ(v ) J γ(v ) This implies (A ) α(v ) = 0 Since α α(v ) by (50), (A ) α = 0 By assumption (ii), it follows that = 0 These, along with 2 f(x )x A y = 0 by Wz = 0, yield y = 0 So, we finally obtain that z = 0 The conclusion follows We end this paper by stating the following result about the convergence of the semismooth method (52), which is based on Theorems 5, 54 and 55, and similar techniques as in [22, 23] Note that the sufficient conditions in below Theorem 56 below do not require strict complementarity of the solution 20

21 Theorem 56 Let z = (x,,y ) be a (not necessarily strictly complementary) KKT point of NSCP (), and suppose that the assumptions of Theorems 54 or 55 hold at this KKT point Then the semismooth Newton method (52) applied to the system of equations H(z) = 0 is locally superlinearly convergent If, in addition, f has a locally Lipschitz continuous Hessian, then it is locally quadratically convergent Proof From the assumptions and Theorems 54 (or 55), we know that, at a KKT point z = (x,,y ) of NSCP (), any element of B H(z ) is nonsingular Thus, as in the proof of Lemma 26 in [22] (or Proposition 3 in [23]), we obtain that there are a neighborhood δ(z ) of z and a constant η such that, for any z δ(z ) and W B H(z), W is nonsingular and W η Then, (52) is well-defined in a neighborhood of z for the first step k = 0 Next, in a similar way as in the proof of Theorem 3 in [22] (or Theorem 32 in [23]), we obtain that z k+ z = z k z W k H(zk ) W k [H(zk ) H(z ) H (z ;z k z )] + W k [W k(z k z ) H (z ;z k z )] = o( z k z ) This establishes the locally superlinear convergence of {z k } to z If, in addition, f has a locally Lipschitz continuous Hessian, then from (5) and the strong semismoothness of Π K (x) we obtain that H is strongly semismooth at z Hence, similarly, the semismooth Newton method (52) is proved to be locally quadratically convergent Based on an exact expression for Clarke generalized Jacobian of the projection onto symmetric cones, we designed a semismooth Newton algorithm for solving NSCP, which is regarded as a generalization of the corresponding results of [3] from the setting of NSOCP However, in the case of NSDP, Sun [25] shown that the strong second order sufficient condition, together with constraint nondegeneracy, is equivalent to many conditions, notably the strong regularity of the KKT point and the nonsingularity of Clarke generalized Jacobian of a nonsmooth system at a KKT point Recently, for linear SDP, Chan and Sun [2] established several equivalent links among the primal and dual constraint nondegeneracy conditions, the strong regularity, and the nonsingularity of both the B-subdifferential and Clarke s generalized Jacobian of a nonsmooth system at a KKT point What is the relationship between the conditions in our results and those in the setting of SDP? (This is tied to an open question proposed by Chan and Sun [2], where they asked whether the corresponding results can be extended to linear SCP) We leave it as a future research topic References [] Bauschke, HH, Borwein, JM: On projection algorithms for solving convex feasibility problems SIAM Rev 38, (996) [2] Chan, Z, Sun, D: Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming SIAM J Optim 9, (2008) 2

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