Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

to appear in Nonlinear Analysis, Theory, Methods and Applications, 2011

Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

Ching-Yu Yang ¹ Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

Yu-Lin Chang² Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

Jein-Shan Chen ³ Department of Mathematics National Taiwan Normal University

Taipei, Taiwan 11677

July 10, 2010

(1st revised on March 1, 2011) (2nd revised on May 12, 2011)

Abstract Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x ∈ H, a spectral decomposition is introduced, and for any func- tion f : IR → IR, we define a corresponding vector-valued function f^H(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x ∈ H with respect to K. We show that this vector-valued function inherits from f the properties of con- tinuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness.

These results can be helpful for designing and analyzing solutions methods for solving infinite-dimensional second-order cone programs and complementarity problems.

1E-mail: yangcy@math.ntnu.edu.tw

2E-mail: ylchang@math.ntnu.edu.tw

3Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.

The author’s work is partially supported by National Science Council of Taiwan. E-mail:

jschen@math.ntnu.edu.tw.

Key words. Hilbert space, infinite-dimensional second-order cone, strong semismooth- ness.

1 Introduction

Let H be a real Hilbert space endowed with an inner product h·, ·i, and we write the norm induced by h·, ·i as k · k. For any given closed convex cone K ⊆ H,

K^∗ := {x ∈ H | hx, yi ≥ 0, ∀y ∈ K}

is the dual cone of K. A closed convex cone K in H is called self-dual if K coincides with its dual cone K^∗; for example, the non-negative orthant cone IRⁿ₊ and the second-order cone (also called Lorentz cone) IKⁿ := {(r, x⁰) ∈ IR × IRⁿ⁻¹ | r ≥ kx⁰k}. As discussed in [14], this Lorentz cone IKⁿ can be rewritten as

IKⁿ:=

(

x ∈ IRⁿ

 hx, ei ≥ 1

√2kxk

)

with e = (1, 0) ∈ IR × IRⁿ⁻¹.

This motivates us to consider the following closed convex cone in the Hilbert space H:

K(e, r) := {x ∈ H | hx, ei ≥ rkxk}

where e ∈ H with kek = 1 and r ∈ IR with 0 < r < 1. It can be seen that K(e, r) is pointed, i.e., K(e, r) ∩ (−K(e, r)) = {0}. Moreover, by denoting

hei^⊥ := {x ∈ H | hx, ei = 0}, we may express the closed convex cone K(e, r) as

K(e, r) =

(

x⁰+ λe ∈ H

x⁰ ∈ hei^⊥ and λ ≥ r

√1 − r² kx⁰k

)

. When H = IRⁿ and e = (1, 0) ∈ IR × IRⁿ⁻¹, K(e,^√1

2) coincides with IKⁿ. In view of this, we shall call K(e,^√1₂) the infinite-dimensional second-order cone (or infinite-dimensional Lorentz cone) in H determined by e. In the rest of this paper, we shall only consider any fixed unit vector e ∈ H, and denote

IK = K e, 1

√2

!

since two infinite-dimensional second-order cones IK(e₁) and IK(e₂) associated with dif- ferent unit elements e₁ and e₂ in H are isometric. This means there exists a bijective isometry P which maps IK(e₁) onto IK(e₂) such that kP xk = kxk for any x ∈ IK(e₁). For

example, let e₁ = (1, 0, 0) and e₂ = (0, 0, 1). Then, for any x ∈ IK(e₁) and y ∈ IK(e₂), we have the following relation:

y = P x =







0 0 1 0 1 0 1 0 0





x.

Moreover, this mapping preserves the Jordan algebra structure, i.e., P (x ◦ y) = P x ◦ P y.

In the infinite-dimensional Hilbert space, P is indeed a unitary operator. In light of this fact, we can consider the infinite-dimensional second-order cone associated with a fixed arbitrary unit element in H.

Unless specifically stated otherwise, we shall alternatively write any point x ∈ H as x = x⁰+ λe with x⁰ ∈ hei^⊥ and λ = hx, ei. In addition, for any x, y ∈ H, we shall write x _IK y (respectively, x _IK y) if x − y ∈ intIK (respectively, x − y ∈ IK). Now, we introduce the spectral decomposition for any element x ∈ H. For any x = x⁰+ λe ∈ H, we can decompose x as

x = α₁(x) · v_x⁽¹⁾+ α₂(x) · v_x⁽²⁾, (1) where α₁(x), α₂(x) and v_x⁽¹⁾, v_x⁽²⁾are the spectral values and the associated spectral vectors of x, with respect to K, given by

α_i(x) = λ + (−1)ⁱkx⁰k , (2)

v_x⁽ⁱ⁾ =











1

2 e + (−1)ⁱ x⁰ kx⁰k

!

, x⁰ 6= 0 1

2

e + (−1)ⁱw^, x⁰ = 0

(3)

for i = 1, 2 with w being any vector in H satisfying kwk = 1. With this spectral decom- position, for any function f : R → R, the following vector-valued function associated with K is defined:

f^H(x) = f (α₁(x))v_x⁽¹⁾+ f (α₂(x))v⁽²⁾_x ∀x ∈ H. (4) The above definition is analogous to the one in finite-dimensional second-order cone case [22, 6].

The motivation of studying f^H defined as in (4) is from concerning with the comple- mentarity problem associated with infinite-dimensional second-order cone IK, i.e., to find an x ∈ H such that

x ∈ IK, T (x) ∈ IK and hx, T (x)i = 0, (5) where T is a mapping from H to H. We denote this problem (5) as CP(IK, T ). More specifically, when dealing with such complementarity problem by nonsmooth function ap- proach, i.e., recasting it as a nonsmooth system of equations, we need to check what kind

of properties of f can be inherited by f^Hso that we can know to what extent the conver- gence analysis of solutions methods based on such nonsmooth system can be obtained.

Indeed, the format of the aforementioned complementarity problem CP(IK, T ) indeed follows the direction of complementarity problems associated with symmetric cones in Euclidean Jordan algebra. Recently, nonlinear symmetric cone optimization and comple- mentarity problems in finite-dimensional spaces such as semidefinite cone optimization and complementarity problems, second-order cone optimization and complementarity problems, and general symmetric cone optimization and complementarity problems, be- come an active research field of mathematical programming. Taking second-order cone optimization and complementarity problems for example, there have proposed many ef- fective solution methods, including the interior point methods [1, 2, 3, 4], the smoothing Newton methods [5, 6, 7], the semismooth Newton methods [8, 9], and the merit function method [10, 11]. However, there are very limited works about nonlinear symmetric cone optimization and complementarity problems in infinite-dimensional spaces, for instance [12], in which with the JB algebras of finite rank primal-dual interior-point methods are presented for some special type of infinite-dimensional cone optimization problems.

It is our intention to extend the above methods for infinite-dimensional complemen- tarity problem CP(IK, T ), in which the vector-valued function f^H will play a key role.

In this paper, we study the continuity and differential properties of the vector-valued function f^H in general. In particular, we show that the properties of continuity, strict continuity (locally Lipschitz continuity), Lipschitz continuity, directional differentiabil- ity, differentiability, continuous differentiability, and s-semismoothness are each inherited by f^H from f . These results can give some concept in designing solutions methods for solving infinite-dimensional second-order cone programs and infinite-dimensional second- order cone complementarity problems.

2 Preliminaries

For any x = x⁰+ λe ∈ H and y = y⁰+ µe ∈ H, we define the Jordan product of x and y by

x ◦ y := (µx⁰+ λy⁰) + hx, yie, (6) and write x² = x ◦ x. Clearly, when H = Rⁿ and e = (1, 0) ∈ R × Rⁿ⁻¹, this definition coincides with the one given in [13, Chatper II] which is the case of finite-dimensional second-order cone associated with Euclidean Jordan algebra. The following technical lemmas will be frequently used in the subsequent analysis.

Lemma 2.1 Let α₁(x), α₂(x) be the spectral values of x ∈ H and α₁(y), α₂(y) be the spectral values of y ∈ H. Then we have

|α₁(x) − α₁(y)|²+ |α₂(x) − α₂(y)|² ≤ 2kx − yk², (7)

and hence, |α_i(x) − α_i(y)| ≤√

2kx − yk, ∀i = 1, 2.

Proof. The proof can be obtained by direct computation like in [22, Lemma 2]. 2

Lemma 2.2 Let x = x⁰+ λe ∈ H and y = y⁰+ µe ∈ H.

(a) If x⁰ 6= 0 and y⁰ 6= 0, then we have

v⁽ⁱ⁾_x − v⁽ⁱ⁾_y ≤ 1

kx⁰kkx − yk ∀i = 1, 2, (8)

where v_x⁽ⁱ⁾, v_y⁽ⁱ⁾ are the spectral vectors of x and y, respectively.

(b) If either x⁰ = 0 or y⁰ = 0, then we can choose v_x⁽ⁱ⁾, v⁽ⁱ⁾_y such that the left hand side of inequality (8) is zero.

Proof. The proof is similar to [23, Lemma 3.2], so we omit it here. 2

Lemma 2.3 For any x 6= 0 ∈ H, the following hold.

(a) If g(x) = kxk, we have g⁰(x)h = hx, hi kxk . (b) If g(x) = x

kxk, we have g⁰(x)h = h

kxk − hx, hi kxk³ x.

Proof. (a) See Example 3.1 (V) of [14].

(b) First, we compute that g(x + h) − g(x)

= x + h

kx + hk− x kxk

= h

kx + hk− 1

kxk− 1

kx + hk

!

· x

= h

kx + hk−

qhx + h, x + hi −^qhx, xi

qhx, xi ·^qhx + h, x + hi · x

= h

kx + hk− 2hx, hi + hh, hi

qhx, xi ·^qhx + h, x + hi^qhx + h, x + hi +^qhx, xi^ · x

= h

kxk− hx, hi

kxk³ x + o(khk).

From the above, it is clear that g⁰(x)h = h

kxk− hx, hi

kxk³ x. 2

Semismooth function, as introduced by Mifflin [15] for functionals and further ex- tended by Qi and Sun [16] for vector-valued functions, is of particular interest due to the central role it plays in the superlinear convergence analysis of certain generalized Newton methods, see [16, 17] and references therein. Given a mapping F : IRⁿ → IR^m, it is well-known that if F is strictly continuous (locally Lipschitz continuous), then F is almost everywhere differentiable by Rademacher’s Theorem–see [18] and [19, Sec. 9J]. In this case, the generalized Jacobian ∂F (x) of F at x (in the Clarke sense) can be defined as the convex hull of the B-subdifferential ∂_BF (x), where

∂_BF (x) :=



lim

x^j→x∇F (x^j)|F is differentiable at x^j ∈ IRⁿ



.

The notation ∂_B is adopted from [17]. In [19, Chap. 9], the case of m = 1 is considered and the notations “ ¯∇” and “ ¯∂” are used instead of, respectively, “∂_B” and “∂”. Assume F : IRⁿ → IR^m is strictly continuous, then F is said to be semismooth at x if F is directionally differentiable at x and, for any V ∈ ∂F (x + h) and h → 0, we have

F (x + h) − F (x) − V h = o(khk). (9)

Moreover, F is called ρ-order semismooth at x (0 < ρ < ∞) if F is semismooth at x and, for any V ∈ ∂F (x + h) and h → 0, we have

F (x + h) − F (x) − V h = O(khk^1+ρ).

The Rademacher theorem does not hold in function spaces, see [20]. Hence, the aforementioned definitions of generalized Jacobian and semismoothness cannot be used in infinite-dimensional spaces. To overcome this difficulty, in the paper [20], so-called slanting functions and slant differentiability of operators in Banach spaces are proposed and used to formulate a concept of semismoothness in infinite-dimensional spaces. We shall introduce them as below. Let X, Y ⊂ H. A function F : X → Y is said to be directionally differentiable at x if the limit

δ⁺F (x; h) := lim

t→0⁺

F (x + th) − F (x)

t (10)

exists, where δ⁺F (x; h) is called the directional derivative of F at x with respect to the direction h. A function F : X → Y is said to be B-differentiable at x if it is directionally differentiable at x and

h→0lim

F (x + h) − F (x) − δ⁺F (x; h)

khk = 0 (11)

in which we call δ⁺F (x; ·) the B-derivative of F at x. In finite dimensional Euclidean spaces, Shapiro [21] shows that a locally Lipschitz continuous function F is B-differentiable at x if and only if it is directionally differentiable at x. From (9) and (11) (also see [16]), it can be seen that F is semismooth at x if and only if F is B-differentiable (hence directionally differentiable) at x and, for each V ∈ ∂F (x + h), there has

δ⁺F (x; h) − V h = o(khk).

As mentioned earlier, these results do not hold in infinite-dimensional spaces. Therefore, the slant differentiability is introduced to circumvent this hurdle. In what follows, we state its definition.

Definition 2.1 Let D be an open domain in X and L(X, Y ) denote the set of all bounded linear operators from X onto Y .

(a) A function F : D ⊂ X → Y is said to be slantly differentiable at x ∈ D if there exists a mapping f^◦ : D → L(X, Y ) such that the family {f^◦(x + h)} of bounded linear operators is uniformly bounded in the operator norm for h sufficiently small and

h→0lim

F (x + h) − F (x) − f^◦(x + h)h

khk = 0. (12)

The function f^◦ is called a slanting function for F at x.

(b) A function F : D ⊂ X → Y is said to be slantly differentiable in an open domain D₀ ⊂ D if there exists a mapping f^◦ : D → L(X, Y ) such that f^◦ is a slanting function for F at every x ∈ D0. In this case, f^◦ is called a slanting function for F in D₀.

Definition 2.2 Suppose that f^◦ : D → L(X, Y ) is a slanting function for F at x ∈ D We denote the set

∂_SF (x) :=



xlimk→xf^◦(x_k)



(13) and call it the slant derivative of F associated with f^◦ at x ∈ D. Note that f^◦(x) ∈ ∂SF (x) which says ∂_SF (x) is always nonempty.

A function F may be slantly differentiable at all points of D, but there is no common slanting function of F at all points of D. Moreover, a slantly differentiable function F at x can have infinitely many slanting functions at x. A slanting function f^◦ for F at x is a single-valued function, but not continuous in general. In addition, a continuous function is not necessarily slantly differentiable. For more details about slanting functions and slantly differentiability, please refer to [20].

Definition 2.3 A mapping F : X → Y is said to be s-semismooth at x if there is a slanting function f^◦ for F in a neighborhood N_x of x such that f^◦ and the associated slant derivative satisfy the following two conditions.

(a) lim

t→0⁺f^◦(x + th)h exists for every h ∈ X and

khk→0lim lim

t→0⁺f^◦(x + th)h − f^◦(x + h)h

khk = 0.

(b) f^◦(x + h)h − V h = o(khk) for all V ∈ ∂_SF (x + h).

We point it out that the function F defined in Definition 2.3 was called semismooth in [20]. However, we here rename it as “s-semismooth” because when X, Y are both finite-dimensional spaces it does not reduce to the original definition introduced by Qi and Sun [16] in finite-dimensional spaces. The main key causing this is the limits in

∂_SF (x) and ∂_BF (x) are approached by different ways. In order to distinguish such difference, we hence use the term “s-semismooth” to convey concept of semismoothness in infinite-dimensional spaces.

3 Continuous properties of f

In this section, we show properties of continuity and (local) Lipschitz continuity of f^H. The arguments are straightforward by checking their definitions.

Proposition 3.1 Suppose x = x⁰+ λe ∈ H with spectral values α₁(x), α₂(x) and spectral vectors v_x⁽¹⁾, v_x⁽²⁾. Let f^H be defined as in (4). Then, f^H is continuous at x ∈ H if and only if f is continuous at α₁(x), α₂(x).

Proof. (⇒) This part of proof is similar to the argument of [22, Proposition 2(a)].

(⇐) This direction of proof is also similar to [23, Proposition 2.2(a)], we omit it. 2

Proposition 3.2 Suppose x = x⁰+ λe ∈ H with spectral values α1(x), α2(x) and spectral vectors v_x⁽¹⁾, v_x⁽²⁾. Let f^H be defined as in (4). Then, the following hold.

(a) f^H is strictly continuous at x ∈ H if and only if f is strictly continuous at α₁(x), α₂(x).

(b) f^H is Lipschitz continuous (with respect to k · k) if and only if f is Lipschitz contin- uous.

Proof. (a) (⇐) Suppose f is strictly continuous at α₁(x), α₂(x). Then, there exist κ_i > 0 and δ_i > 0 for i = 1, 2, such that

|f (ξ) − f (ζ)| ≤ κi|ξ − ζ| ∀ξ, ζ ∈ [αi(x) − δi, αi(x) + δi] i = 1, 2.

Let δ = ^√1

2min{δ₁, δ₂} and for any y, z ∈ B(x, δ), we have f^H(y) − f^H(z)

= ^f (α₁(y))v_y⁽¹⁾+ f (α₂(y))v⁽²⁾_y ^−^f (α₁(z))v⁽¹⁾_z + f (α₂(z))v_z^(2) (14)

= f (α₁(y))^v_y⁽¹⁾− v_z^(1)+



f (α₁(y)) − f (α₁(z))



v_z⁽¹⁾ +f (α2(y))^v_y⁽²⁾− v_z^(2)+



f (α2(y)) − f (α2(z))



v_z⁽²⁾

where y = α₁(y)v⁽¹⁾_y + α₂(y)v_y⁽²⁾ and z = α₁(z)v_z⁽¹⁾+ α₂(z)v⁽²⁾_z . By Lemma 2.1, Lemma 2.2 and the similar argument in [22, Proposition 6(a)], the proof can be obtained.

(⇒) This part of proof is quite simple and similar to [22, Proposition 6(a)], we omit it here.

(b) The argument of proof is similar to [22, Proposition 6(c)]. 2

4 Differential properties of f

In this section, we show properties of directional differentiability, differentiability, con- tinuous differentiability and B-differentiability of f^H. For simplicity, in the arguments we sometimes abbreviate α_i(x) as α_i when there is no ambiguity in the context. Note that, unlike in finite-dimensional second-order cone case [22], Prop. 4.1 and Prop. 4.2 are proved by different approaches since the chain rule for directional differentiability in infinite-dimensional space does not hold in general, see [21].

Proposition 4.1 Suppose x = x⁰+ λe ∈ H with spectral values α₁(x), α₂(x) and spectral vectors v_x⁽¹⁾, v_x⁽²⁾. Let f^H be defined as in (4). Then, f^H is directionally differentiable at x ∈ H if and only if f is directionally differentiable at α1(x), α2(x).

Proof. (⇐) Suppose f is directionally differentiable at α₁(x), α₂(x). Fix x = x⁰+λe ∈ H and any direction h = h⁰+ le ∈ H, we discuss two cases as below.

Case (i): If x⁰ 6= 0, then we have f^H(x) = f (α₁(x))v_x⁽¹⁾ + f (α₂(x))v_x⁽²⁾ where α_i(x) = λ + (−1)ⁱkx⁰k and v_x⁽ⁱ⁾= ¹₂^e + (−1)^{i x}_kx⁰0k

for i = 1, 2. Now x + th = (x⁰+ th⁰) + (λ + tl)e with spectral values α_i(x + th) = λ + tl + (−1)ⁱkx⁰+ th⁰k and spectral vectors v⁽ⁱ⁾_x+th =

1 2

e + (−1)^{i x}_kx⁰0^+th+th⁰⁰k

 for i = 1, 2. We consider equation (14) again in which replacing y with x + th, then we have

f^H(x + th) − f^H(x)

= f (α₁(x + th))^v_x+th⁽¹⁾ − v_x^(1)+



f (α₁(x + th)) − f (α₁(x))



v_x⁽¹⁾ (15) +f (α₂(x + th))^v⁽²⁾_x+th− v⁽²⁾_x ^+



f (α₂(x + th)) − f (α₂(x))



v⁽²⁾_x .

Because the process of checking argument is similar to [22, Proposition 3], we only present the result here.

By denoting

˜

a = f (α2(x)) − f (α1(x)) α₂(x) − α₁(x) ,

˜b = δ⁺f (α₂(x); k₂) + δ⁺f (α₁(x); k₁)

2 , (16)

˜

c = δ⁺f (α₂(x); k₂) − δ⁺f (α₁(x); k₁)

2 ,

where k_i = hh, ei + (−1)^{i hx}_kx^0,hi0k for i = 1, 2, we can write the expression of δ⁺f^H(x; h) as

δ⁺f^H(x; h) = ˜a h − hh, eie − hx⁰, hi kx⁰k² x⁰

!

+ ˜be + ˜c x⁰

kx⁰k. (17)

Case (ii): If x⁰ = 0, we compute the directional derivative δ⁺f^H(x; h) at x ∈ H for any direction h by definition. Let h = h⁰ + le ∈ H with h⁰ ∈ hei^⊥ and l ∈ R. We discuss two subcases.

Subcase (a). If h⁰ 6= 0, from the spectral decomposition, we choose v⁽ⁱ⁾_x = ¹₂^e + (−1)^{i h}_kh⁰0k



for i = 1, 2 such that

f^H(x + th) = f (λ + th₁)v⁽¹⁾_x + f (λ + th₂)v_x⁽²⁾ f^H(x) = f (λ)v⁽¹⁾_x + f (λ)v⁽²⁾_x

where h_i = l + (−1)ⁱkh⁰k for i = 1, 2. Now, we compute

t→0lim⁺

f^H(x + th) − f^H(x) t

= lim

t→0⁺

f (λ + th₁) − f (λ)

t v⁽¹⁾_x + lim

t→0⁺

f (λ + th₂) − f (λ)

t v_x⁽²⁾ (18)

= δ⁺f (λ; l − kh⁰k)v⁽¹⁾_x + δ⁺f (λ; l + kh⁰k)v_x⁽²⁾. This shows that δ⁺f^H(x; h) exists under this subcase.

Subcase (b). If h⁰ = 0, we choose v⁽ⁱ⁾_x = ¹₂(e + (−1)ⁱw) for any w ∈ H with kwk = 1.

Analogous to (18), we have

t→0lim⁺

f^H(x + th) − f^H(x) t

= lim

t→0⁺

f (λ + tl) − f (λ)

t v⁽¹⁾_x + lim

t→0⁺

f (λ + tl) − f (λ)

t v_x⁽²⁾ (19)

= δ⁺f (λ; l)v_x⁽¹⁾+ δ⁺f (λ; l)v⁽²⁾_x . Hence, δ⁺f^H(x; h) exists under this subcase.

From all the above, we have proved that f^H is directionally differentiable at x ∈ H when x⁰ = 0 and its directional derivative δ⁺f^H(x; h) is either in form of (18) or (19).

(⇒) Suppose f^His directionally differentiable at x ∈ H, we will prove that f is direction- ally differentiable at α₁, α₂. For α₁ ∈ R and any direction d1 ∈ R, let h = d1v⁽¹⁾_x + 0v_x⁽²⁾ where x = α₁v_x⁽¹⁾+ α₂v_x⁽²⁾. Then, x + th = (α₁+ td₁)v_x⁽¹⁾+ α₂v_x⁽²⁾ and

f^H(x + th) − f^H(x)

t = f (α₁+ td₁) − f (α₁) t v_x⁽¹⁾.

Since f^H is directionally differentiable at x, the above equation implies that δ⁺f (α₁; d₁) = lim

t→0⁺

f (α₁+ td₁) − f (α₁)

t exists.

This means f is directionally differentiable at α₁. Similarly, it can be verified that f is also directionally differentiable at α₂. 2

Proposition 4.2 Suppose x = x⁰+ λe ∈ H with spectral values α₁(x), α₂(x) and spectral vectors v_x⁽¹⁾, v⁽²⁾_x . Let f^H be defined as in (4). Then, f^H is differentiable at x ∈ H if and only if f is differentiable at α₁(x), α₂(x).

Proof. (⇐) Suppose f is differentiable at α₁, α₂. Fix x = x⁰+λe ∈ H and h = h⁰+le ∈ H, we discuss two cases as below.

Case (i): If x⁰ 6= 0, then we have f^H(x) = f (α₁)v⁽¹⁾_x + f (α₂)v_x⁽²⁾ where α_i = λ + (−1)ⁱkx⁰k and v⁽ⁱ⁾_x = ¹₂^e + (−1)^{i x}_kx⁰0k

 for i = 1, 2. By using Lemma 2.3 and the chain rule and product rule for differentiation, the argument is similar to [22, Proposition 4] so we omit the process and present the result as following. Denoting

a = f (α₂) − f (α₁) α2− α1

, b = f⁰(α₂) + f⁰(α₁)

2 , c = f⁰(α₂) − f⁰(α₁)

2 . (20)

We can write the expression of (f^H)⁰(x)h as (f^H)⁰(x)h = ah + (b − a) hh, eie +hx⁰, hi

kx⁰k² x⁰

!

+ c

kx⁰k(hx⁰, hie + hh, eix⁰). (21)

Case (ii): The proof is identical to that of Case(ii) in Proposition 4.1, but with th replaced by h. We omit it and only present the formula of (f^H)⁰(x)h as below. If x⁰ = 0, then

(f^H)⁰(x)h = f⁰(λ)h. (22)

(⇒) This part of proof is similar to [23, Proposition 2.2(c)]. 2

Proposition 4.3 Suppose x = x⁰+ λe ∈ H with spectral values α₁(x), α₂(x) and spectral vectors v⁽¹⁾_x , v⁽²⁾_x . Let f^H be defined as in (4). Then, f^H is continuously differentiable (smooth) at x ∈ H if and only if f is continuously differentiable at α₁(x), α₂(x).

Proof. (⇐) This part of proof is similar to [23, Proposition 2.2(d)], so we omit it.

(⇒) This direction of proof is some variant of argument in [22, Proposition 5], we also skip it here. 2

Proposition 4.4 Suppose x = x⁰+ λe ∈ H with spectral values α₁(x), α₂(x) and spectral vectors v⁽¹⁾_x , v_x⁽²⁾. Let f^H be defined as in (4). Then, f^H is B-differentiable at x ∈ H if and only if f is B-differentiable at α₁(x), α₂(x).

Proof. (⇐) If f is B-differentiable at α₁(x), α₂(x), f is directionally differentiable at α₁(x), α₂(x). By Proposition 4.1, f^H is directionally differentiable at x. It remains to verify that

h→0lim

f^H(x + h) − f^H(x) − δ⁺f^H(x; h)

khk = 0.

We write x = x⁰+ λe and h = h⁰+ le ∈ H. Again, two cases will be discussed.

Case (i): If x⁰ 6= 0, considering equation (15) in which we replace x + th with x + h, it yields

f^H(x + h) − f^H(x)

= f (α₁(x + h))^v_x+h⁽¹⁾ − v_x^(1)+ (f (α₁(x + h)) − f (α₁(x))) v⁽¹⁾_x (23) +f (α₂(x + h))^v⁽²⁾_x+h− v_x^(2)+ (f (α₂(x + h)) − f (α₂(x))) v_x⁽²⁾.

Indeed, sum of the first and third can be simplified as

f (α₁(x + h))^v⁽¹⁾_x+h− v_x^(1)+ f (α₂(x + h))^v⁽²⁾_x+h− v_x^(2)

= (f (α2(x + h)) − f (α1(x + h))) ·1

2 · x⁰+ h⁰

kx⁰+ h⁰k− x⁰ kx⁰k

!

= f (α₂(x + h)) − f (α₁(x + h))

2kx⁰k h⁰−hx⁰, h⁰i

kx⁰k² x⁰+ o(kh⁰k)

!

(24)

= f (α₂(x + h)) − f (α₁(x + h))

α2(x) − α1(x) h − hh, eie − hx⁰, hi

kx⁰k² x⁰+ o(kh⁰k)

!

,

where the second equality is due to Lemma 2.3(b) and the last equality uses the fact that α₂(x) − α₁(x) = 2kx⁰k. From (17), we know that

δ⁺f^H(x; h)

= f (α₂(x)) − f (α₁(x))

α₂(x) − α₁(x) h − hh, eie − hx⁰, hi kx⁰k² x⁰

!

(25) +δ⁺f (α₁(x); k₁)v_x⁽¹⁾+ δ⁺f (α₂(x); k₂)v_x⁽²⁾

where k_i = hh, ei + (−1)^{i hx}_kx^0,hi0k for i = 1, 2. Since lim

h→0

h − hh, eie − ^hx_kx⁰0^,hik²x^0= 0, following almost the same arguments as in Proposition 4.1 gives

α_i(x + h) − α_i(x) = l + (−1)ⁱ(kx⁰+ h⁰k − kx⁰k)

= hh, ei + (−1)ⁱ hx⁰, h⁰i

kx⁰k + o(kh⁰k)

!

= k_i+ (−1)ⁱo(kh⁰k) ∀i = 1, 2.

Let T_i := k_i + (−1)ⁱo(kh⁰k) = α_i(x + h) − α_i(x) for i = 1, 2, we obtain

h→0lim

f (α_i(x + h)) − f (α_i(x)) khk

= lim

h→0

f (α_i(x) + T_i· 1) − f (α_i(x))

T_i · k_i+ (−1)ⁱo(kh⁰k) khk

= δ⁺f (α_i(x); 1) ·k^e_i

= δ⁺f (α_i(x);k^e_i),

where the last equality uses the positive homogeneity property of δ⁺f (α_i(x); ·) again and ke_i := lim

h→0 ki

khk. We notice that 0 < kk^e_ik ≤ 2 and k^e_i can be viewed as a directional vector here. By the above discussion, we have

h→0lim 1 khk

f (α₁(x + h))^v⁽¹⁾_x+h− v_x^(1)+ f (α₂(x + h))^v⁽²⁾_x+h− v_x^(2)

− f (α2(x)) − f (α1(x))

α₂(x) − α₁(x) h − hh, eie − hx⁰, hi kx⁰k² x⁰

!!

= lim

h→0

(f (α₂(x + h)) − f (α₂(x))) − (f (α₁(x + h)) − f (α₁(x)))

khk · (α₂(x) − α₁(x)) · h − hh, eie − hx⁰, hi kx⁰k² x⁰

!

+ lim

h→0

f (α₂(x + h)) − f (α₁(x + h))

α₂(x) − α₁(x) · o(kh⁰k) khk

= δ⁺f (α₂(x);^fk₂) − δ⁺f (α₁(x);^fk₁)

α₂(x) − α₁(x) · 0 + 0 (26)

= 0.

By assumption, f is B-differentiable at α₁(x), α₂(x) and employ almost the same argu- ments, we compute

h→0lim

f (α_i(x + h)) − f (α_i(x)) − δ⁺f (α_i(x); k_i)

khk · v_x⁽ⁱ⁾

= lim

h→0

f (αi(x) + Ti) − f (αi(x)) − δ⁺f (αi(x); Ti)

kT_ik · kki+ (−1)ⁱo(kh⁰k)k khk

+ δ⁺f (α_i(x); T_i) − δ⁺f (α_i(x); k_i) khk

!

· v⁽ⁱ⁾_x (27)

= 0 · lim

h→0kk^e_ik + lim

h→0δ⁺f (α_i(x); (−1)ⁱo(kh⁰k) khk )

!

· v_x⁽ⁱ⁾ = 0 ∀i = 1, 2.

Now from equations (23), (25), (26) and (27), we see that

h→0lim

f^H(x + h) − f^H(x) − δ⁺f^H(x; h)

khk = 0

which says that f^H is B-differentiable at x.

Case (ii): If x⁰ = 0, we need to further consider the following two subcases:

Subcase (a): If h⁰ 6= 0, we choose v⁽ⁱ⁾_x = ¹₂^e + (−1)^{i h}_kh⁰0k

 for i = 1, 2 such that v_x+h⁽ⁱ⁾ = v_x⁽ⁱ⁾. Then,

f^H(x + h) − f^H(x)

= (f (α₁(x + h)) − f (α₁(x))) v_x⁽¹⁾+ (f (α₂(x + h)) − f (α₂(x))) v⁽²⁾_x , and from Case (ii)(a) of Proposition 4.1, we have

δ⁺f^H(x; h) = δ⁺f (λ; l − kh⁰k)v_x⁽¹⁾+ δ⁺f (λ; l + kh⁰k)v⁽²⁾_x ,

where λ = α₁(x) = α₂(x). Again by the B-differentiability of f at α₁(x) and α₂(x), we have

h→0lim

f^H(x + h) − f^H(x) − δ⁺f^H(x; h) khk

= lim

h→0

f (α₁(x + h)) − f (α₁(x)) − δ⁺f (α₁(x); l − kh⁰k)

khk · v⁽¹⁾_x

+ lim

h→0

f (α₂(x + h)) − f (α₂(x)) − δ⁺f (α₂(x); l + kh⁰k)

khk · v_x⁽²⁾

= 0,

which implies the B-differentiability of f^H at x.

Subcase (b): If h⁰ = 0, we choose v_x⁽ⁱ⁾ = ¹₂(e + (−1)ⁱw) with any w ∈ H with kwk = 1. With almost the same arguments as in Case (ii)-(b) of Proposition 4.1, the B- differentiability of f^H can be verified, we omit the detail here.

(⇒) If f^His B-differentiable at x, then f^His directionally differentiable at x by definition.

Then, f is also directionally differentiable at α_i(x), α₂(x) by Proposition 4.1. In order to prove the B-differentiability of f at α_i(x), α₂(x), all we have to do is proving the following condition:

limt→0

f (α_i(x) + t) − f (α_i(x)) − δ⁺f (α_i(x); t)

|t| = 0 ∀i = 1, 2.

Since f^H is B-differentiable at x, the following condition is true:

h→0lim

f^H(x + h) − f^H(x) − δ⁺f^H(x; h)

khk = 0.

Again, we write x = x⁰ + λe and h = h⁰ + le ∈ H and discuss two cases.

Case (i): If x⁰ 6= 0, from the proof of first part, we know

h→0lim

f^H(x + h) − f^H(x) − δ⁺f^H(x; h) khk

= lim

h→0

"

(f (α₂(x + h)) − f (α₂(x))) − (f (α₁(x + h)) − f (α₁(x)))

khk · (α₂(x) − α₁(x)) · h − hh, eie − hx⁰, hi kx⁰k² x⁰

!

+f (α₂(x + h)) − f (α₁(x + h))

α₂(x) − α₁(x) · o(kh⁰k) khk +

2

X

i=1

f (α_i(x + h)) − f (α_i(x)) − δ⁺f (α_i(x); k_i)

khk · v⁽ⁱ⁾_x

#

where k_i = hh, ei + (−1)^{i hx}_kx^0,hi0k for i = 1, 2.

Because lim

h→0

h − hh, eie − ^hx_kx⁰0^,hik²x^0= 0 and v_x⁽¹⁾ ⊥ v_x⁽²⁾, we must have

h→0lim

f (α_i(x + h)) − f (α_i(x)) − δ⁺f (α_i(x); k_i)

khk = 0 ∀i = 1, 2. (28)

Note that h ∈ H is arbitrary, we can choose h = te where t ∈ R is also arbitrary. Then, we have

ki = αi(x + h) − αi(x) = t ∀i = 1, 2.

This together with the fact that t → 0 as h → 0 gives limt→0

f (α_i(x) + t) − f (α_i(x)) − δ⁺f (α_i(x); t)

|t| = 0 ∀i = 1, 2,

which means that f is B-differentiable at α_i(x) for i = 1, 2.

Case (ii): If x⁰ = 0, we consider the two subcases of h⁰ = 0 or h⁰ 6= 0. The proof is routine check as earlier verifications, so we omit it. 2

5 S-semismooth properties of f

In this section, we show s-semismooth properties of f^H. To this end, we first present some equivalent criteria for s-semismooth functions in infinite-dimensional spaces. In fact, we immediate obtain the following criteria from the very basic definition and combining some known results in [20].

Proposition 5.1 Suppose that F : X → Y is slantly differentiable on a neighborhood N_x of x. Let f^◦ be a slanting function for F in N_x and ∂_SF be the slant derivative associated with f^◦ in Nx. Then, F is s-semismooth at x if and only if one of the following holds:

(a) lim

t→0⁺f^◦(x + th)h exists for every h ∈ X,

khk→0lim lim

t→0⁺f^◦(x + th)h − f^◦(x + h)h

khk = 0, (29)

and

f^◦(x + h)h − V h = o(khk) ∀ V ∈ ∂_SF (x + h). (30) (b) F is B-differentiable at x, and

δ⁺F (x; h) − V h = o(khk) ∀ V ∈ ∂_SF (x + h). (31) (c) F is B-differentiable at x, and

F (x + h) − F (x) − V h = o(khk) ∀ V ∈ ∂_SF (x + h). (32) Proof. (a) This is clear from the original definition of s-semismooth function given as in Definition 2.3.

(b) This is result of [20, Theorem 3.3].

(c) Using part(a) and [20, Theorem 2.9] yield F being B-differentiable at x, and

δ⁺F (x; h) − f^◦(x + h)h = o(khk). (33) Then, by definition of F being B-differentiable, condition (31) holds. 2

The conditions in Proposition 5.1 are indeed hard to be verified since it is difficult to write out the set ∂_SF (x + h). Hence, we further establish some equivalent conditions which are useful in subsequent analysis regarding s-semismooth property which is the main contribution of this paper. We also want to point out the following observation.

Suppose that F : X → Y is slantly differentiable on a neighborhood N_x of x. Let f^◦ be a slanting function for F with uniform bound kf^◦k ≤ L in N_x. It is easy to derive that kF (y) − F (z)k ≤ 2Lky − zk for any y, z ∈ Nx. However, we have no idea whether it is true or not for the opposite direction.

Proposition 5.2 Suppose that F : X → Y is slantly differentiable on a neighborhood N_x of x. Let f^◦ be a slanting function for F in N_x and ∂_SF be the slant derivative associated with f^◦ in N_x. Then, the following hold.

(a) If F is s-semismooth at x, then F is B-differentiable at x, and

F (x + h) − F (x) − δ⁺F (x + h; h) = o(khk) (34) for all x + h at which F is B-differentiable.

(b) If F is B-differentiable on a neighborhood N_x of x and (34) holds for all x + h at which F is B-differentiable, then F is s-semismooth at x.

Proof. (a) The B-differentiability of F at x is clear by Proposition 5.1. It remains to claim that when F is B-differentiable at x + h, there has

kF (x + h) − F (x) − δ⁺F (x + h; h)k

khk → 0 as h → 0. (35)

If not, there exist a δ > 0 and a sequence hi → 0 such that F is B-differentiable at x + hi

for each i = 1, 2, . . . , and

kF (x + h_i) − F (x) − δ⁺F (x + h_i; h_i)k

kh_ik ≥ δ. (36)

By assumption, F is s-semismooth at x, then for each i ≥ 1 there exist V_i ∈ ∂_SF (x + h_i) and yi ∈ Nx+h such that

kV_i− f^◦(y_i)k ≤ kh_ik, ky_i− (x + h_i)k ≤ kh_ik² (37) and kF (x + h_i) − F (x) − V_ih_ik

kh_ik → 0 as h_i → 0. (38)

By [20, Proposition 2.8], for each h_i there exist t_i > 0 with 0 < t_i ≤ kh_ik such that kf^◦(x + h_i+ t_ih_i)h_i− δ⁺F (x + h_i; h_i)k ≤ kh_ik². (39) Now we compute

f^◦(yi)hi − f^◦(x + hi+ tihi)hi

= (F (x + h_i+ t_ih_i) − F (x) − f^◦(x + h_i+ t_ih_i)(h_i+ t_ih_i))

−(F (y_i) − F (x) − f^◦(y_i)(y_i− x)) +(F (y_i) − F (x + h_i+ t_ih_i)) +f^◦(yi)(x + hi− yi)

+f^◦(x + h_i+ t_ih_i)(t_ih_i).

(40)

Because F is slantly differentiable at x, the first and second term of (40) implies F (x + h_i+ t_ih_i) − F (x) − f^◦(x + h_i+ t_ih_i)(h_i+ t_ih_i)

kh_i+ t_ih_ik → 0 as i → ∞

and F (y_i) − F (x) − f^◦(y_i)(y_i− x)

ky_i− xk → 0 as i → ∞

which lead to

F (x + h_i+ t_ih_i) − F (x) − f^◦(x + h_i+ t_ih_i)(h_i+ t_ih_i) kh_ik

= F (x + h_i+ t_ih_i) − F (x) − f^◦(x + h_i+ t_ih_i)(h_i+ t_ih_i)

kh_i+ t_ih_ik ·kh_i+ t_ih_ik kh_ik

→ 0 as i → ∞

(41)

and F (y_i) − F (x) − f^◦(y_i)(y_i− x) kh_ik

= F (y_i) − F (x) − f^◦(y_i)(y_i− x)

ky_i− xk · ky_i− xk kh_ik

→ 0 as i → ∞.

(42)

Here we use that fact that kh_i+ t_ih_ik = (1 + t_i)kh_ik and ky_i− xk = ky_i− x − h_i+ h_ik ≤ ky_i− x − h_ik + kh_ik ≤ kh_ik²+ kh_ik. Besides, for the third, fourth and fifth term of (40), since F is slantly differentiable in a neighborhood N_x of x, kf^◦(x)k is uniformly bounded in N_x, say kf^◦(x)k ≤ M in N_x. Hence we have

kF (y_i) − F (x + h_i+ t_ih_i)k ≤ M ky_i− (x + h_i+ t_ih_i)k

≤ M (kh_ik² + t_ikh_ik), kf^◦(y_i)(x + h_i− y_i)k ≤ M kx + h_i− y_ik ≤ M kh_ik² and

kf^◦(x + h_i+ t_ih_i)(t_ih_i)k ≤ M kt_ih_ik ≤ M kh_ik² which implies

kF (y_i) − F (x + h_i+ t_ih_i)k

kh_ik → 0 as i → ∞, (43)

kf^◦(y_i)(x + h_i− y_i)k

kh_ik → 0 as i → ∞, (44)

kf^◦(x + h_i+ t_ih_i)(t_ih_i)k

kh_ik → 0 as i → ∞. (45)

Combining (41)- (45) all together, we have

kf^◦(y_i)h_i− f^◦(x + h_i+ t_ih_i)h_ik

kh_ik → 0 as i → ∞. (46)

Now consider

F (x + h_i) − F (x) − δ⁺F (x + h_i; h_i)

= [F (x + h_i) − F (x) − V_ih_i] +[V_ih_i− f^◦(y_i)h_i]

+[f^◦(y_i)h_i− f^◦(x + h_i+ t_ih_i)h_i]

+[f^◦(x + hi+ tihi)hi− δ⁺F (x + hi; hi)].

From (38), (37), (46) and (39), we have

kF (x + hi) − F (x) − δ⁺F (x + hi; hi)k

kh_ik → 0 as h_i → 0

This is a contradiction to equation (36), hence (35) holds for all x + h at which F is B-differentiable.

(b) By Proposition 5.1(c), it suffice to show that for each V ∈ ∂_SF (x + h), there has kF (x + h) − F (x) − V hk

khk → 0 as khk → 0.

If not, there exist δ > 0 and a sequence h_i → 0, V_i ∈ ∂_SF (x + h_i) and y_i ∈ N_x+hi such that ky_i− (x + h_i)k ≤ kh_ik², kV_i− f^◦(y_i)k ≤ kh_ik and

kF (x + h_i) − F (x) − V_ih_ik

khik ≥ δ.

By assumption, F is B-differentiable in a neighborhood of x and satisfies (34) which yields

kF (x + h_i) − F (x) − δ⁺F (x + h_i; h_i)k

khik → 0 as kh_ik → 0.

Then, we consider

F (x + hi) − F (x) − Vihi

= [F (x + h_i) − F (x) − δ⁺F (x + h_i; h_i)]

+[f^◦(y_i)h_i− V_ih_i]

+[f^◦(x + h_i+ t_ih_i)h_i− f^◦(y_i)h_i]

+[δ⁺F (x + h_i; h_i) − f^◦(x + h_i+ t_ih_i)h_i].

With similar argument and choice of t_i in part (a), we have kF (x + h_i) − F (x) − V_ih_ik

khik → 0 as i → ∞.

This leads to a contradiction. Thus, the proof is complete. 2