Clarke Generalized Jacobian of the Projection onto Symmetric Cones

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1 Clarke Generalized Jacobian of the Projection onto Symmetric Cones Lingchen Kong 1, Levent Tunçel 2, and Naihua Xiu 3 June 26, 2008, revised April 14, 2009 Abstract In this paper, we give an exact expression for Clarke generalized Jacobian of the projection onto symmetric cones, which generalizes and unifies the existing related results on secondorder cones and the cones of symmetric positive semi-definite matrices over the reals Our characterization of the Clarke generalized Jacobian exposes a connection to rank-one matrices Keywords: Clarke generalized Jacobian; Projection; Symmetric cones; Euclidean Jordan algebra MSC2000 Subject Classification: primary: 49R50, 49J52, secondary: 26E25, 47A75 Abbreviated Title: Clarke Generalized Jacobian of the Projection 1 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 [4, 5, 14, 15, 16] for details It is well-known that the scalar-valued function g : R R + with gt = t + := max{0,t} is strongly semismooth and its Clarke generalized Jacobian is specified by t + /t if t 0 and the interval [0,1] if t = 0 Note that the projection onto an arbitrary closed convex set is Lipschitz continuous In the case of second-order cone, we know that the projection operator is strongly semismooth, see, eg, [2, 3, 8] In particular, Hayashi, Yamashita and Fukushima [8] gave an explicit representation for Clarke generalized Jacobian of the projection onto second-order 1 Lingchen Kong, Department of Applied Mathematics, Beijing Jiaotong University, Beijing , P R China; Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada konglchen@126com 2 Levent Tunçel, Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada ltuncel@mathuwaterlooca 3 Naihua Xiu, Department of Applied Mathematics, Beijing Jiaotong University, Beijing , P R China nhxiu@bjtueducn 1

2 cones Malick and Sendov [12] 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 [19] showed that the projection onto symmetric cones, which include the nonnegative orthant, the cone of symmetric positive semi-definite matrices and the second-order cone as special cases, is strongly semismooth everywhere Kong, Sun and Xiu [11] studied the Clarke generalized Jacobian of the projection onto symmetric cones and gave its upper bound and lower bound Meng, Sun and Zhao [13] 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 see Proposition 1 of [13] for details One of the areas where projection operator is widely applied is the area of complementarity problems, see, eg, [5] Also, projection operator is a fundamental ingredient of many algorithms for solving convex optimization problems see for instance the survey by Bauschke and Borwein [1] 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 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 Our result generalizes corresponding results of [8] and [12] 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 This paper is organized as follows In the next section, we establish the preliminaries and study the relationship among all the Jordan frames of an element We also introduce the matrix representation of the Jacobian operator of x + 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 2 Preliminaries 21 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 [10] and in the monograph by Faraut and Korányi [6] 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 21 in [6], 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 degx is the smallest positive 2

3 integer k such that the set {e,x,x 2,,x k } is linearly dependent The rank of V is defined as max{degx : 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 1,c 2,,c k } of idempotents where c i c j = 0 for all i j, and c 1 + 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 21 Spectral Decomposition Type I Theorem III11, [6] Let V be a Euclidean Jordan algebra Then for x J there exist unique real numbers µ 1 x,µ 2 x,,µ r x, all distinct, and a unique complete system of orthogonal idempotents {b 1,b 2,,b r } such that x = µ 1 xb µ r xb r Theorem 22 Spectral Decomposition Type II Theorem III12, [6] Let V be a Euclidean Jordan algebra with rank r Then for x J there exist a Jordan frame {c 1,c 2,,c r } and real numbers λ 1 x,λ 2 x,,λ r x such that x = λ 1 xc 1 + λ 2 xc λ r xc r 21 The numbers λ i x i = 1,2,,r are the eigenvalues of x We call 21 the spectral decomposition or the spectral expansion of x Note that the above {b 1,b 2,,b r } and {c 1,c 2,,c r } depend on x We do not write this dependence explicitly for the sake of simplicity in notation Let Cx be the set consisting of all Jordan frames in the spectral decomposition of x Let the spectrum σx be the set of all eigenvalues of x Then σx = {µ 1 x,µ 2 x,,µ r x} and for each µ i x σx, 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 Gx := r gλ j xc j x = gλ 1 xc 1 x + gλ 2 xc 2 x + + gλ r xc r x, 22 j=1 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 + := r λ i x + c i x, x := i=1 r r λ i x c i x and x := λ i x c i x i=1 i=1 Note that z K z intk if and only if λ i z 0 λ i z > 0 i {1,2,,r}, where intk 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 Lx : J J by Lxy = x y for all y J, which is a symmetric self-adjoint operator in the sense that Lxy,w = y, Lxw for all y,w J The operator Qx := 2L 2 x Lx 2 is called the quadratic representation of 3

4 x We say two elements x, y J operator commute if LxLy = LyLx By Lemma X22 in [6], 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 Jc,0,Jc, 1 2 and Jc,1, where Jc,ε := {z J : c z = εz}, ε {0, 12 },1 This is called the Peirce decomposition of J with respect to the nonzero idempotent c Fix a Jordan frame {c 1,c 2,,c r } Defining the following subspaces for i,j {1,2,,r}, { J ii := {x J : x c i = x} and J ij := x J : x c i = 1 } 2 x = x c j, i j, we have J ii = J c i,1 and J ij = J c i,1/2 J c j,1/2 = J ji We state the Peirce decomposition theorem as follows For more detail, see [6] Theorem 23 Theorem IV21, [6] Let {c 1,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 In a n-dimensional Euclidean Jordan algebra V of rank r, by Corollary IV26 of [6], we have dimj ij = dimj kl =: n for i,j,k,l {1,2,,r} such that i j,k l Then n = r + n rr For a given Jordan frame {c 1,c 2,,c r } and i,j {1,2,,r}, let C ij x be the orthogonal projection operator onto subspace J ij Then, by Theorem 23, we have C jj x = Qc j and C ij x = 4Lc i Lc j = 4Lc j Lc i = C ji x, i,j {1,2,,r}, 24 and the orthogonal projection operators {C ij x : i,j {1,2,,r}} satisfy and C ij x = C ijx, C ij x 2 = C ij x, C ij xc kl x = 0 if {i,j} {k,l},i,j,k,l {1,2,,r} 1 i j r C ij x = I, where C ij x is the adjoint operator of C ijx and I is the identity operator Furthermore, we have the following spectral decomposition theorem for Lx, Lx 2 and Qx For a more detailed exposition, see [10, 19] Theorem 24 Theorem 31, [19] Let x J and r j=1 λ jxc j x denote its spectral decomposition with λ 1 x λ 2 x λ r x Then, the symmetric operators Lx, Lx 2 and 4

5 Qx have the spectral decompositions: Lx = Lx 2 = Qx = r λ j xc jj x + j=1 r λ 2 j xc jjx + j=1 r λ 2 jxc jj x + j=1 1 j<l r 1 j<l r 1 j<l r 1 2 λ jx + λ l xc jl x, 1 2 λ2 j x + λ2 l xc jlx, λ j xλ l xc jl x Moreover, the spectra σlx, σlx 2 and σqx, respectively, consist of all distinct numbers 1 2 λ jx + λ l x, 1 2 λ2 j x + λ2 l x, and λ jxλ l x for all j,l {1,2,,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=1 λ jxc j = r i=1 µ ixb i with λ 1 x λ 2 x λ r x and µ 1 x > µ 2 x > > µ r x In what follows, let ℵx := {i : λ i x < 0}, Ix := {i : λ i x = 0} and x := {i : λ i x > 0} For the sake of simplicity, let ℵ := ℵx, I := Ix and := x Thus ℵ I = {1,2,,r} Set b I := j Ic j By Theorem 21 and the argument after Theorem 22, b I = b i where the subscript i is specified by µ i x = 0 It follows that b I is uniquely defined by x In other words, for any Jordan frame { c 1,, c r } Cx, j I c j = b I Therefore, we can define a subspace By Lemma 20 in [7], it follows that J I := Jb I,1 := {w J : w b I = w} J I = span { c +1,c +2,,c + I } + +1 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 Observe that b I := j I c j and {c j : j I} is a complete system of orthogonal idempotents Thus, {c j : j I} forms a Jordan frame in J I and the rank of J I is 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 Kx,0 := Cone{c j : j I} 26 {c 1,,c r} Cx An interesting question occurs: what is the difference between the cones K I and Kx,0? Clearly, when x = 0, K I = K0,0= K The following proposition states that they are always equal to each other 5

6 Proposition 25 Let x = r j=1 λ jxc j with λ 1 x λ 2 x λ r x and b I = j I c j Let K I and Kx,0 be defined by 25 and 26, respectively Then we have K I = Kx,0 = Cone{c j : j I} 27 {c 1,,c r} Cx and b I is the identity element in the above symmetric cone Proof It is clear that Kx,0 K I and b I is the identity element in K I We now prove K I Kx,0 Let y K I Then, by the argument before 25, there exists a Jordan frame in J I, say, {e 1,e 2,,e I }, such that y = y 1 e 1 +y 2 e 2 + +y I e I with y i 0 Clearly, elements e 1,e 2,,e I belong to K and their sum is b I We next show that, for any Jordan frame {c i : i {1,,r}} Cx, we can replace {c +1,,c + I } by {e 1,,e I } to get an element in Cx; Ie, In fact, by the above definitions, we have {c 1,,c,e 1,,e I,c + I +1,,c r } Cx 28 c k b I = 0, for all k {1,,, + I + 1,,r} Then c k,b I = 0 By b I = e e I, it follows that c k,b I = I i=1 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 [7], we have c k e i = 0, for all k {1,,, + I + 1,,r}, i {1,, I } This proves 28 Then Cone{e j : j {1,, I }} Kx,0, and therefore y Kx,0 The above proof yields the following interesting fact which is a decomposition result on Jordan frames determined by x Proposition 26 Let x = r j=1 λ jxc j = r i=1 µ ixb i Then for any Jordan frame {c 1,,c r } in Cx, the set {c j : λ j x = µ i x} is a Jordan frame in Jb i,1, and a Jordan frame in each Jb i,1 i {1,, r}, say, C bi, is a subset of some Jordan frame in Cx Moreover, the union of C b1,,c b r forms a Jordan frame in Cx 22 Matrix Representation of Gx Let Gx be given by 22 Suppose that g is differentiable at τ i,i {1,2,,r} Define the first divided difference g [1] of g at τ := τ 1,τ 2,,τ r T R r as the r r symmetric matrix with the ij-th entry g [1] τ ij i,j {1,2,,r} given by [τ i,τ j ] g := gτ i gτ j τ i τ j if τ i τ j, g τ i if τ i = τ j Based on Theorem 32 of [19], Kong, Sun and Xiu [11] introduced the following Jacobian properties of the Löwner operator G Theorem 27 Theorem 28 of [11] Let x = r j=1 λ jxc j = r i=1 µ ixb i x Then, G is continuously differentiable at x if and only if for each j {1,2,,r}, g is continuously differentiable at λ j x In this case, the Jacobian Gx is given by Gx = 2 r [λ i x,λ j x] g Lc i Lc j i,j=1 6 r g λ i xlc i, 29 i=1

7 or equivalently Gx = 2 r i j,i,j=1 [µ i x,µ j x] g Lb i xlb j x + Furthermore, Gx is a symmetric linear operator from J into itself r i=1 g µ i xqb i x 210 Suppose that Löwner operator G is continuously differentiable at x = r j=1 λ jxc j with λ 1 x λ 2 x λ r x For a given Jordan frame {c 1,c 2,,c r }, let L := Lc 1 Lc r be an operator vector with respect to {c 1,c 2,,c r } Denote L := L c 1,, L c r, where L c i is the adjoint operator of Lc i Since Lc i is self-adjoint, we have L = L c 1,, L c r = Lc 1,, Lc r For the real r r matrix Λx := [λ i x,λ j x] g r r, we define L ΛxL := r [λ i x,λ j x] g Lc i Lc j i,j=1 Similarly, for the r-vector dx = g λ 1 x,,g λ r x T, define d xl := r g λ i xlc i Thus, by 29, we can give a matrix representation of the Jacobian Gx as follows: i=1 Gx = 2L ΛxL d xl 211 In particular, consider the metric projection Π K at x = r j=1 λ jxc 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 211, we write Π K x as Π K x = 2L ΛxL d xl where Λx is a symmetric matrix of the form E Γ Λx = Γ T 0 ℵ ℵ with E being the all ones matrix, the ℵ matrix Γ = and the vector dx = matrix λ i x λ i x λ j x 212 ℵ, 1 T,0T ℵ T Throughout this paper, En1 n 2 denotes the n 1 n 2 all ones 7

8 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 [8, 9, 12] in the special settings of second-order cones and positive semi-definite cones, where Kanzow, Ferenczi and Fukushima [9] gave an expression for the B-subdifferential of the projection onto second-order cones, Hayashi, Yamashita and Fukushima [8] gave an explicit representation for Clarke generalized Jacobian of the projection onto second-order cones; Malick and Sendov [12] 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 1,c 2,,c r }, we define { {c 1,,c r} B Π K x := lim Π K x + h : h span{c 1,c 2,,c r }, lim Π K x + h exists h 0,x+h D ΠK h 0 For a given integer t {0,1,, I } we define a r-vector d t x by 1 d t x := 1 t 0 I t = 1 +t, 31 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 00 0 ℵ ℵ,Λ 00 Λt, I, 32 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 Λt, I is a set of t I t matrices θ ij t I t the rows are indexed by + 1, + 2,, + t, and the columns by + t + 1, + t + 2,, + I specified by { } Λt, I := θ ij t I t [0,1] t I t : θ ij satisfy a and b below a θ i, +t+1 θ i, +t+2 θ i, + I i { + 1, + 2,, + t}, θ +1,j θ +2,j θ +t,j j { + t + 1, + t + 2,, + I }; b 1 θ ij 1 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 := 1t 0 r t, 33 { } Et t Λ t r t Λ T : Λ t r t 0 t r t Λt,r, 34 r t r t where Λt,r is a set of t r t matrices θ ij t r t the rows are indexed by 1,2,,t, and the columns by t + 1,t + 2,,r specified by { } Λt,r := θ ij t r t [0,1] t r t : θ ij satisfy a and b below 8

9 a θ i,t+1 θ i,t+2 θ i,r i {1,2,,t}, θ 1j θ 2j θ tj j {t + 1,t + 2,,r}; b 1 θ ij 1 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 [9] for B-subdifferential, Theorem 37 of [12] and Proposition 48 of [8] for Clarke generalized Jacobian Theorem 31 The B-subdifferential of Π K at x is given by B Π K x = I {c 1,,c r} Cx t=0 {2L Λ t xl d t xl} where d t x and Λ t x are specified by 31 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 xl d txl} {c 1,,c r} Cx t=0 Proof First we prove B Π K x I {c 1,,c r} Cx t=0 {2L Λ t xl d txl} Without loss of generality, fix a Jordan frame {c 1,,c r } Cx For any arbitrary but given integer t {0,1,, I }, set V := 2L A t xl d txl where A t x Λ t x and d t x is given by 31 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 i=1 λ ihc 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 := 1 θ ij 1 for i { + 1,, + t} and j { + t + 1,, + I }, then B t I t := ς ij t I t can be written as B t I t = π +1 π +t π +t+1,,π + I, where π i [0,+ ], π +1 π +2 π +t and π +t+1 π +t+2 π + I Taking 0 if 1 i, ρ if π i = 0, < i + t, 1 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, 9

10 we have x + h = r i=1 λ ix + hc i = r i=1 [λ ix + λ i h]c i and for sufficiently small ρ 0,1, λ 1 x + h λ +t x + h > 0 > λ +t+1 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 Kx + h = 2L A t xl d t xl h 0 We next prove that B Π K x r {c 1,,c r} Cx t=0 {2L Λ t xl d txl} Let W B Π K x Then there is a vector h := hw J such that W = lim h 0,x+h DΠK Π K x + h For the above h J, let x + h =: r j=1 λ jx + hc j x + h with λ 1 x + h λ 2 x + h λ r x + h In the sense of set convergence see [17], we have lim sup {c 1 x + h,,c r x + h} Cx h 0,x+h D Πk Let us pick {c 1,,c r } lim sup{c 1 x + h,,c r x + h} Clearly, lim h 0,x+h D Πk λx + h = λx, where λx + h := λ 1 x + h,,λ r x + h T and none of λ i x + h is zero Suppose that T ℵx + h, x + h are given and t := x + h Then d t x = 1 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 Λ12 := Λ T, 12 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 12 = Λ ij with Λ ij = ℵ t I t λ i x+h λ i x+h λ j x+h i { + 1,, + t}, j { + t + 1,, + I } If lim h 0 Λ 12 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 = 12 Λ T 12 0 ℵ t ℵ t where Λ := lim h 0 Λ and λ Λ12 := lim h 0 Λ12 = θ ij t I t with θ ij = lim i x+h h 0 λ i x+h λ j x+h i { + 1,, + t}, j { + t + 1,, + I } Observing that and λ +1 h λ +2 h λ +t h > 0 0 < λ +t+1 h λ +t+2 h λ + I h, we easily show θ ij [0,1] and a In order to prove b, let ς ij := lim h 0 λ j x + h λ i x + h, 10

11 for i { + 1,, + t}, j { + t + 1,, + I } Then, θ ij = ς ij or equivalently ς ij = 1 θ ij 1 35 Note that θ ij [0,1] implies ς ij [0,+ ] and θ ij = 0 ς ij =, θ ij = 1 ς ij = 0 Define B := ς ij t I t Then, it is easy to see that B = lim h 0 1 λ +1 h 1 λ +t h λ +t+1 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 Λ12 h exists This proves W = 2L Θ t xl d t xl {2L Λ t xl d t xl}, as desired From the above proof, we also obtain that for a given Jordan frame {c 1,c 2,,c r } Cx {c 1,,c r} B Π K x = I {2L Λ t xl d txl} 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 1,,c r} C0 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 1,,c r} C0 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 1,x T 2 T : x 1 x 2,x 1 R,x 2 R n 1 } n 2 Let {c 1,c 2 } be a Jordan frame given by c i = i 1 ω, for i {1,2}, with ω being any vector in R n 1 satisfying ω = 1 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[0,1]Lc 1 Lc 2 + Qc Qc 2 This means that B Π K 0 Π K 0 in general 11

12 Applying Proposition 25, Theorem 31 and Corollary 32, we immediately obtain the following result, which states the interesting connection between B Π K xrespectively, Π K x and B Π KI 0respectively, Π KI 0 In the case of S n, it reduces to Proposition 22 in [18] Theorem 33 Let x = r i=1 λ ixc i with λ 1 x λ 2 x λ r x Then V B Π K x respectively, V Π K x if and only if there exists a V I B Π KI 0 respectively, V I Π KI 0 such that V = W + V I, where W is independent on the Jordan frame {c 1,,c r } and specified by W = Q c i + 4L c i L c i + 4 λ i x λ i i i x λ j x Lc ilc j i I i,j ℵ Proof Let x = r i=1 λ ixc i = r i=1 µ ixb i with λ 1 x λ 2 x λ r x and µ 1 x > µ 2 x > > µ r x, and set {c 1,,c r } =: {c,c I,c ℵ } with c := { } { } c 1,,c, ci := c +1,,c + I and cℵ := { } c + I +1,,c r By Proposition 25, any Jordan frame in JI is a subset of a Jordan frame in Cx, and the part c I of a Jordan frame {c,,c r } in Cx is a Jordan frame in J I Let L L = L I L ℵ, where L := Lc 1 Lc, L I := Lc +1 Lc + I, L ℵ := Lc + I +1 Lc r It therefore follows immediately from Theorem 31 and Corollary 32 that V B Π K x if and only if there exists a V I B Π KI 0 such that V = 2L A t xl d t xl, V I = 2L I At, I L I d t L I, where r r matrix A t x Λ t x and I I matrix At, I Λt, I satisfy and r-vector d t x and I -vector d t satisfy E E I Γ ℵ A t x = E I At, I 0 I ℵ ; Γ T ℵ 0 ℵ I 0 ℵ ℵ d t x = 1 T,dT t,0 ℵ T By the symmetry of Lx and A t x, direct calculation yields V = 2L A t xl d txl = 2L E L + 4L E I L I + 4L Γ ℵ L ℵ + 2L I At, I L I d txl = 2L E L 1 T L + 4L E I L I + 4L Γ ℵ L ℵ 12

13 = + 2L I At, I d t L I 2 Lc i + 4 Lc i Lc j i,j i i,j I +4 λ i x λ i x λ j x Lc ilc j + V I i,j ℵ = Q c i + 4L c i L c i + 4 i i i I Lc i Lc j i,j ℵ λ i x λ i x λ j x Lc ilc j + V I This implies that V B Π K x if and only if there is a V I B Π KI 0 such that where W is given as W := Q i c i + 4L i c i V = W + V I, L c i + 4 i I i,j ℵ λ i x λ i x λ j x Lc ilc j Observe the following equations c i = b i, i i {i:µ i >0} c i = b i, i I i {i:µ i =0} and i,j ℵ λ i x λ i x λ j x Lc ilc j = i {i:µ i >0},j {i:µ i <0} µ i x µ i x µ j x Lb ilb j From the fact above and the uniqueness of b i by Theorem 21, we obtain that W is unique and independent on the Jordan frame {c 1,,c r } By the definition of Clarke generalized Jacobian, it holds that V Π K x if and only if there is a V I Π KI 0 such that V = W + V I We complete the proof 4 B-subdifferentials of x and x Employing the same technique as in the proof of Theorem 31, we give below the formulae for the B-subdifferentials of x and x Theorem 41 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 xl ē d t x L}, {c 1,,c r} Cx t=0 where d t x and Λ t x are specified by 31 and 32, respectively Furthermore, B x = I B Π K x 13

14 Proof Note that for any pair of scalars τ i,τ j, τ i τ j τ i τ j + τ i + τ j + τ i τ j = 1 or τ i τ j τ i τ j = 1 τ i + τ j + τ i τ j In a similar way as in Theorem 31, 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 {0,1,, I } we define a r-vector t x by 1 t x := 1 t 1 I t = 1 +t, 41 1 r t 1 ℵ and a set of r r matrices Å t x by Å t x := E E I Υ ℵ E I Υ 0 E I ℵ Υ T ℵ E ℵ I E ℵ ℵ : Υ 0 = E1 Υ 00 Υ T 00 E 2,Υ 00 Υt, I, 42 where E 1 = E t t,e 2 = E I t I t, 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 Υt, I is a set of t I t matrices ξ ij t I t the rows are indexed by +1, +2,, +t, and the columns by +t+1, +t+2,, + I specified by Υt, I := { } ξ ij t I t [ 1,1] t I t : ξ ij satisfy c and d below, c ξ i, +t+1 ξ i, +t+2 ξ i, + I i = + 1, + 2,, + t, ξ +1,j ξ +2,j ξ +t,j j = + t + 1, + t + 2,, + I ; 1 ξij d 1+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 1,,c r} Cx t=0 {2L Å t xl t xl}, where t x and Å t x are specified by 41 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 1 if τ i > τ j 0, τ i +τ j τ i τ j if τ i > 0 > τ j, 1 if τ i 0 > τ j ξ ij = 1 τ j τ i 1 + τ, or equivalently τ j j τ τ i i 14 = 1 ξ ij 1 + ξ ij

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