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 Chang^{2}
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 IR^{n}_{+} and the second-order
cone (also called Lorentz cone) IK^{n} := {(r, x^{0}) ∈ IR × IR^{n−1} | r ≥ kx^{0}k}. As discussed in
[14], this Lorentz cone IK^{n} can be rewritten as

IK^{n}:=

(

x ∈ IR^{n}

hx, ei ≥ 1

√2kxk

)

with e = (1, 0) ∈ IR × IR^{n−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) =

(

x^{0}+ λe ∈ H

x^{0} ∈ hei^{⊥} and λ ≥ r

√1 − r^{2} kx^{0}k

)

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

2) coincides with IK^{n}. In view of this,
we shall call K(e,^{√}^{1}_{2}) 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(e_{1}) and IK(e_{2}) associated with dif-
ferent unit elements e_{1} and e_{2} in H are isometric. This means there exists a bijective
isometry P which maps IK(e_{1}) onto IK(e_{2}) such that kP xk = kxk for any x ∈ IK(e_{1}). For

example, let e_{1} = (1, 0, 0) and e_{2} = (0, 0, 1). Then, for any x ∈ IK(e_{1}) and y ∈ IK(e_{2}), 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 = x^{0}+ λe with x^{0} ∈ 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 = x^{0}+ λe ∈ H,
we can decompose x as

x = α_{1}(x) · v_{x}^{(1)}+ α_{2}(x) · v_{x}^{(2)}, (1)
where α_{1}(x), α_{2}(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)^{i}kx^{0}k , (2)

v_{x}^{(i)} =

1

2 e + (−1)^{i} x^{0}
kx^{0}k

!

, x^{0} 6= 0
1

2

e + (−1)^{i}w^{}, x^{0} = 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 (α_{1}(x))v_{x}^{(1)}+ f (α_{2}(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^{H}so 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 = x^{0}+ λe ∈ H and y = y^{0}+ µe ∈ H, we define the Jordan product of x and y
by

x ◦ y := (µx^{0}+ λy^{0}) + hx, yie, (6)
and write x^{2} = x ◦ x. Clearly, when H = R^{n} and e = (1, 0) ∈ R × R^{n−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 α_{1}(x), α_{2}(x) be the spectral values of x ∈ H and α_{1}(y), α_{2}(y) be the
spectral values of y ∈ H. Then we have

|α_{1}(x) − α_{1}(y)|^{2}+ |α_{2}(x) − α_{2}(y)|^{2} ≤ 2kx − yk^{2}, (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 = x^{0}+ λe ∈ H and y = y^{0}+ µe ∈ H.

(a) If x^{0} 6= 0 and y^{0} 6= 0, then we have

v^{(i)}_{x} − v^{(i)}_{y} ^{
}^{
}_{
} ≤ 1

kx^{0}kkx − yk ∀i = 1, 2, (8)

where v_{x}^{(i)}, v_{y}^{(i)} are the spectral vectors of x and y, respectively.

(b) If either x^{0} = 0 or y^{0} = 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 g^{0}(x)h = hx, hi
kxk .
(b) If g(x) = x

kxk, we have g^{0}(x)h = h

kxk − hx, hi
kxk^{3} 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 −^{q}hx, xi

qhx, xi ·^{q}hx + h, x + hi · x

= h

kx + hk− 2hx, hi + hh, hi

qhx, xi ·^{q}hx + h, x + hi^{q}hx + h, x + hi +^{q}hx, xi^{} · x

= h

kxk− hx, hi

kxk^{3} x + o(khk).

From the above, it is clear that g^{0}(x)h = h

kxk− hx, hi

kxk^{3} 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 : IR^{n} → 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 ∂_{B}F (x), where

∂_{B}F (x) :=

lim

x^{j}→x∇F (x^{j})|F is differentiable at x^{j} ∈ IR^{n}

.

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 : IR^{n} → 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
D_{0} ⊂ 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 D_{0}.

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

∂_{S}F (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 ∂_{S}F (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 ∈ ∂_{S}F (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

∂_{S}F (x) and ∂_{B}F (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

^{H}

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 = x^{0}+ λe ∈ H with spectral values α_{1}(x), α_{2}(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 α_{1}(x), α_{2}(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 = x^{0}+ 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 α_{1}(x),
α_{2}(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 α_{1}(x), α_{2}(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{δ_{1}, δ_{2}} and for any y, z ∈ B(x, δ), we have
f^{H}(y) − f^{H}(z)

= ^{}f (α_{1}(y))v_{y}^{(1)}+ f (α_{2}(y))v^{(2)}_{y} ^{}−^{}f (α_{1}(z))v^{(1)}_{z} + f (α_{2}(z))v_{z}^{(2)}^{} (14)

= f (α_{1}(y))^{}v_{y}^{(1)}− v_{z}^{(1)}^{}+

f (α_{1}(y)) − f (α_{1}(z))

v_{z}^{(1)}
+f (α2(y))^{}v_{y}^{(2)}− v_{z}^{(2)}^{}+

f (α2(y)) − f (α2(z))

v_{z}^{(2)}

where y = α_{1}(y)v^{(1)}_{y} + α_{2}(y)v_{y}^{(2)} and z = α_{1}(z)v_{z}^{(1)}+ α_{2}(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

^{H}

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 = x^{0}+ λe ∈ H with spectral values α_{1}(x), α_{2}(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 α_{1}(x), α_{2}(x). Fix x = x^{0}+λe ∈ H
and any direction h = h^{0}+ le ∈ H, we discuss two cases as below.

Case (i): If x^{0} 6= 0, then we have f^{H}(x) = f (α_{1}(x))v_{x}^{(1)} + f (α_{2}(x))v_{x}^{(2)} where α_{i}(x) =
λ + (−1)^{i}kx^{0}k and v_{x}^{(i)}= ^{1}_{2}^{}e + (−1)^{i x}_{kx}^{0}0k

for i = 1, 2. Now x + th = (x^{0}+ th^{0}) + (λ + tl)e
with spectral values α_{i}(x + th) = λ + tl + (−1)^{i}kx^{0}+ th^{0}k and spectral vectors v^{(i)}_{x+th} =

1 2

e + (−1)^{i x}_{kx}^{0}0^{+th}+th^{0}^{0}k

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 (α_{1}(x + th))^{}v_{x+th}^{(1)} − v_{x}^{(1)}^{}+

f (α_{1}(x + th)) − f (α_{1}(x))

v_{x}^{(1)} (15)
+f (α_{2}(x + th))^{}v^{(2)}_{x+th}− v^{(2)}_{x} ^{}+

f (α_{2}(x + th)) − f (α_{2}(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))
α_{2}(x) − α_{1}(x) ,

˜b = δ^{+}f (α_{2}(x); k_{2}) + δ^{+}f (α_{1}(x); k_{1})

2 , (16)

˜

c = δ^{+}f (α_{2}(x); k_{2}) − δ^{+}f (α_{1}(x); k_{1})

2 ,

where k_{i} = hh, ei + (−1)^{i hx}_{kx}^{0}^{,hi}0k 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 − hx^{0}, hi
kx^{0}k^{2} x^{0}

!

+ ˜be + ˜c x^{0}

kx^{0}k. (17)

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

Subcase (a). If h^{0} 6= 0, from the spectral decomposition, we choose v^{(i)}_{x} = ^{1}_{2}^{}e + (−1)^{i h}_{kh}^{0}0k

for i = 1, 2 such that

f^{H}(x + th) = f (λ + th_{1})v^{(1)}_{x} + f (λ + th_{2})v_{x}^{(2)}
f^{H}(x) = f (λ)v^{(1)}_{x} + f (λ)v^{(2)}_{x}

where h_{i} = l + (−1)^{i}kh^{0}k for i = 1, 2. Now, we compute

t→0lim^{+}

f^{H}(x + th) − f^{H}(x)
t

= lim

t→0^{+}

f (λ + th_{1}) − f (λ)

t v^{(1)}_{x} + lim

t→0^{+}

f (λ + th_{2}) − f (λ)

t v_{x}^{(2)} (18)

= δ^{+}f (λ; l − kh^{0}k)v^{(1)}_{x} + δ^{+}f (λ; l + kh^{0}k)v_{x}^{(2)}.
This shows that δ^{+}f^{H}(x; h) exists under this subcase.

Subcase (b). If h^{0} = 0, we choose v^{(i)}_{x} = ^{1}_{2}(e + (−1)^{i}w) 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
x^{0} = 0 and its directional derivative δ^{+}f^{H}(x; h) is either in form of (18) or (19).

(⇒) Suppose f^{H}is directionally differentiable at x ∈ H, we will prove that f is direction-
ally differentiable at α_{1}, α_{2}. For α_{1} ∈ R and any direction d1 ∈ R, let h = d1v^{(1)}_{x} + 0v_{x}^{(2)}
where x = α_{1}v_{x}^{(1)}+ α_{2}v_{x}^{(2)}. Then, x + th = (α_{1}+ td_{1})v_{x}^{(1)}+ α_{2}v_{x}^{(2)} and

f^{H}(x + th) − f^{H}(x)

t = f (α_{1}+ td_{1}) − f (α_{1})
t v_{x}^{(1)}.

Since f^{H} is directionally differentiable at x, the above equation implies that
δ^{+}f (α_{1}; d_{1}) = lim

t→0^{+}

f (α_{1}+ td_{1}) − f (α_{1})

t exists.

This means f is directionally differentiable at α_{1}. Similarly, it can be verified that f is
also directionally differentiable at α_{2}. 2

Proposition 4.2 Suppose x = x^{0}+ λe ∈ H with spectral values α_{1}(x), α_{2}(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 α_{1}(x), α_{2}(x).

Proof. (⇐) Suppose f is differentiable at α_{1}, α_{2}. Fix x = x^{0}+λe ∈ H and h = h^{0}+le ∈ H,
we discuss two cases as below.

Case (i): If x^{0} 6= 0, then we have f^{H}(x) = f (α_{1})v^{(1)}_{x} + f (α_{2})v_{x}^{(2)} where α_{i} = λ + (−1)^{i}kx^{0}k
and v^{(i)}_{x} = ^{1}_{2}^{}e + (−1)^{i x}_{kx}^{0}0k

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 (α_{2}) − f (α_{1})
α2− α1

, b = f^{0}(α_{2}) + f^{0}(α_{1})

2 , c = f^{0}(α_{2}) − f^{0}(α_{1})

2 . (20)

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

kx^{0}k^{2} x^{0}

!

+ c

kx^{0}k(hx^{0}, hie + hh, eix^{0}). (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 x^{0} = 0, then

(f^{H})^{0}(x)h = f^{0}(λ)h. (22)

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

Proposition 4.3 Suppose x = x^{0}+ λe ∈ H with spectral values α_{1}(x), α_{2}(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 α_{1}(x), α_{2}(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 = x^{0}+ λe ∈ H with spectral values α_{1}(x), α_{2}(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 α_{1}(x), α_{2}(x).

Proof. (⇐) If f is B-differentiable at α_{1}(x), α_{2}(x), f is directionally differentiable at
α_{1}(x), α_{2}(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 = x^{0}+ λe and h = h^{0}+ le ∈ H. Again, two cases will be discussed.

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

f^{H}(x + h) − f^{H}(x)

= f (α_{1}(x + h))^{}v_{x+h}^{(1)} − v_{x}^{(1)}^{}+ (f (α_{1}(x + h)) − f (α_{1}(x))) v^{(1)}_{x} (23)
+f (α_{2}(x + h))^{}v^{(2)}_{x+h}− v_{x}^{(2)}^{}+ (f (α_{2}(x + h)) − f (α_{2}(x))) v_{x}^{(2)}.

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

f (α_{1}(x + h))^{}v^{(1)}_{x+h}− v_{x}^{(1)}^{}+ f (α_{2}(x + h))^{}v^{(2)}_{x+h}− v_{x}^{(2)}^{}

= (f (α2(x + h)) − f (α1(x + h))) ·1

2 · x^{0}+ h^{0}

kx^{0}+ h^{0}k− x^{0}
kx^{0}k

!

= f (α_{2}(x + h)) − f (α_{1}(x + h))

2kx^{0}k h^{0}−hx^{0}, h^{0}i

kx^{0}k^{2} x^{0}+ o(kh^{0}k)

!

(24)

= f (α_{2}(x + h)) − f (α_{1}(x + h))

α2(x) − α1(x) h − hh, eie − hx^{0}, hi

kx^{0}k^{2} x^{0}+ o(kh^{0}k)

!

,

where the second equality is due to Lemma 2.3(b) and the last equality uses the fact that
α_{2}(x) − α_{1}(x) = 2kx^{0}k. From (17), we know that

δ^{+}f^{H}(x; h)

= f (α_{2}(x)) − f (α_{1}(x))

α_{2}(x) − α_{1}(x) h − hh, eie − hx^{0}, hi
kx^{0}k^{2} x^{0}

!

(25)
+δ^{+}f (α_{1}(x); k_{1})v_{x}^{(1)}+ δ^{+}f (α_{2}(x); k_{2})v_{x}^{(2)}

where k_{i} = hh, ei + (−1)^{i hx}_{kx}^{0}^{,hi}0k for i = 1, 2. Since lim

h→0

h − hh, eie − ^{hx}_{kx}^{0}0^{,hi}k^{2}x^{0}^{}= 0, following
almost the same arguments as in Proposition 4.1 gives

α_{i}(x + h) − α_{i}(x) = l + (−1)^{i}(kx^{0}+ h^{0}k − kx^{0}k)

= hh, ei + (−1)^{i} hx^{0}, h^{0}i

kx^{0}k + o(kh^{0}k)

!

= k_{i}+ (−1)^{i}o(kh^{0}k) ∀i = 1, 2.

Let T_{i} := k_{i} + (−1)^{i}o(kh^{0}k) = 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)^{i}o(kh^{0}k)
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}_{i}k ≤ 2 and k^{e}_{i} can be viewed as a directional vector
here. By the above discussion, we have

h→0lim 1 khk

f (α_{1}(x + h))^{}v^{(1)}_{x+h}− v_{x}^{(1)}^{}+ f (α_{2}(x + h))^{}v^{(2)}_{x+h}− v_{x}^{(2)}^{}

− f (α2(x)) − f (α1(x))

α_{2}(x) − α_{1}(x) h − hh, eie − hx^{0}, hi
kx^{0}k^{2} x^{0}

!!

= lim

h→0

(f (α_{2}(x + h)) − f (α_{2}(x))) − (f (α_{1}(x + h)) − f (α_{1}(x)))

khk · (α_{2}(x) − α_{1}(x)) · h − hh, eie − hx^{0}, hi
kx^{0}k^{2} x^{0}

!

+ lim

h→0

f (α_{2}(x + h)) − f (α_{1}(x + h))

α_{2}(x) − α_{1}(x) · o(kh^{0}k)
khk

= δ^{+}f (α_{2}(x);^{f}k_{2}) − δ^{+}f (α_{1}(x);^{f}k_{1})

α_{2}(x) − α_{1}(x) · 0 + 0 (26)

= 0.

By assumption, f is B-differentiable at α_{1}(x), α_{2}(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_{i}k · kki+ (−1)^{i}o(kh^{0}k)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}_{i}k + lim

h→0δ^{+}f (α_{i}(x); (−1)^{i}o(kh^{0}k)
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 x^{0} = 0, we need to further consider the following two subcases:

Subcase (a): If h^{0} 6= 0, we choose v^{(i)}_{x} = ^{1}_{2}^{}e + (−1)^{i h}_{kh}^{0}0k

for i = 1, 2 such that v_{x+h}^{(i)} =
v_{x}^{(i)}. Then,

f^{H}(x + h) − f^{H}(x)

= (f (α_{1}(x + h)) − f (α_{1}(x))) v_{x}^{(1)}+ (f (α_{2}(x + h)) − f (α_{2}(x))) v^{(2)}_{x} ,
and from Case (ii)(a) of Proposition 4.1, we have

δ^{+}f^{H}(x; h) = δ^{+}f (λ; l − kh^{0}k)v_{x}^{(1)}+ δ^{+}f (λ; l + kh^{0}k)v^{(2)}_{x} ,

where λ = α_{1}(x) = α_{2}(x). Again by the B-differentiability of f at α_{1}(x) and α_{2}(x), we
have

h→0lim

f^{H}(x + h) − f^{H}(x) − δ^{+}f^{H}(x; h)
khk

= lim

h→0

f (α_{1}(x + h)) − f (α_{1}(x)) − δ^{+}f (α_{1}(x); l − kh^{0}k)

khk · v^{(1)}_{x}

+ lim

h→0

f (α_{2}(x + h)) − f (α_{2}(x)) − δ^{+}f (α_{2}(x); l + kh^{0}k)

khk · v_{x}^{(2)}

= 0,

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

Subcase (b): If h^{0} = 0, we choose v_{x}^{(i)} = ^{1}_{2}(e + (−1)^{i}w) 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^{H}is B-differentiable at x, then f^{H}is directionally differentiable at x by definition.

Then, f is also directionally differentiable at α_{i}(x), α_{2}(x) by Proposition 4.1. In order
to prove the B-differentiability of f at α_{i}(x), α_{2}(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 = x^{0} + λe and h = h^{0} + le ∈ H and discuss two cases.

Case (i): If x^{0} 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 (α_{2}(x + h)) − f (α_{2}(x))) − (f (α_{1}(x + h)) − f (α_{1}(x)))

khk · (α_{2}(x) − α_{1}(x)) · h − hh, eie − hx^{0}, hi
kx^{0}k^{2} x^{0}

!

+f (α_{2}(x + h)) − f (α_{1}(x + h))

α_{2}(x) − α_{1}(x) · o(kh^{0}k)
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}^{,hi}0k for i = 1, 2.

Because lim

h→0

h − hh, eie − ^{hx}_{kx}^{0}0^{,hi}k^{2}x^{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 x^{0} = 0, we consider the two subcases of h^{0} = 0 or h^{0} 6= 0. The proof is routine
check as earlier verifications, so we omit it. 2

### 5 S-semismooth properties of f

^{H}

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 ∂_{S}F 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 ∈ ∂_{S}F (x + h). (30)
(b) F is B-differentiable at x, and

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

F (x + h) − F (x) − V h = o(khk) ∀ V ∈ ∂_{S}F (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 ∂_{S}F (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 ∂_{S}F 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_{i}k ≥ δ. (36)

By assumption, F is s-semismooth at x, then for each i ≥ 1 there exist V_{i} ∈ ∂_{S}F (x + h_{i})
and yi ∈ Nx+h such that

kV_{i}− f^{◦}(y_{i})k ≤ kh_{i}k, ky_{i}− (x + h_{i})k ≤ kh_{i}k^{2} (37)
and kF (x + h_{i}) − F (x) − V_{i}h_{i}k

kh_{i}k → 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_{i}k such that
kf^{◦}(x + h_{i}+ t_{i}h_{i})h_{i}− δ^{+}F (x + h_{i}; h_{i})k ≤ kh_{i}k^{2}. (39)
Now we compute

f^{◦}(yi)hi − f^{◦}(x + hi+ tihi)hi

= (F (x + h_{i}+ t_{i}h_{i}) − F (x) − f^{◦}(x + h_{i}+ t_{i}h_{i})(h_{i}+ t_{i}h_{i}))

−(F (y_{i}) − F (x) − f^{◦}(y_{i})(y_{i}− x))
+(F (y_{i}) − F (x + h_{i}+ t_{i}h_{i}))
+f^{◦}(yi)(x + hi− yi)

+f^{◦}(x + h_{i}+ t_{i}h_{i})(t_{i}h_{i}).

(40)

Because F is slantly differentiable at x, the first and second term of (40) implies
F (x + h_{i}+ t_{i}h_{i}) − F (x) − f^{◦}(x + h_{i}+ t_{i}h_{i})(h_{i}+ t_{i}h_{i})

kh_{i}+ t_{i}h_{i}k → 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_{i}h_{i}) − F (x) − f^{◦}(x + h_{i}+ t_{i}h_{i})(h_{i}+ t_{i}h_{i})
kh_{i}k

= F (x + h_{i}+ t_{i}h_{i}) − F (x) − f^{◦}(x + h_{i}+ t_{i}h_{i})(h_{i}+ t_{i}h_{i})

kh_{i}+ t_{i}h_{i}k ·kh_{i}+ t_{i}h_{i}k
kh_{i}k

→ 0 as i → ∞

(41)

and F (y_{i}) − F (x) − f^{◦}(y_{i})(y_{i}− x)
kh_{i}k

= F (y_{i}) − F (x) − f^{◦}(y_{i})(y_{i}− x)

ky_{i}− xk · ky_{i}− xk
kh_{i}k

→ 0 as i → ∞.

(42)

Here we use that fact that kh_{i}+ t_{i}h_{i}k = (1 + t_{i})kh_{i}k and ky_{i}− xk = ky_{i}− x − h_{i}+ h_{i}k ≤
ky_{i}− x − h_{i}k + kh_{i}k ≤ kh_{i}k^{2}+ kh_{i}k. 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_{i}h_{i})k ≤ M ky_{i}− (x + h_{i}+ t_{i}h_{i})k

≤ M (kh_{i}k^{2} + t_{i}kh_{i}k),
kf^{◦}(y_{i})(x + h_{i}− y_{i})k ≤ M kx + h_{i}− y_{i}k ≤ M kh_{i}k^{2}
and

kf^{◦}(x + h_{i}+ t_{i}h_{i})(t_{i}h_{i})k ≤ M kt_{i}h_{i}k ≤ M kh_{i}k^{2}
which implies

kF (y_{i}) − F (x + h_{i}+ t_{i}h_{i})k

kh_{i}k → 0 as i → ∞, (43)

kf^{◦}(y_{i})(x + h_{i}− y_{i})k

kh_{i}k → 0 as i → ∞, (44)

kf^{◦}(x + h_{i}+ t_{i}h_{i})(t_{i}h_{i})k

kh_{i}k → 0 as i → ∞. (45)

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

kf^{◦}(y_{i})h_{i}− f^{◦}(x + h_{i}+ t_{i}h_{i})h_{i}k

kh_{i}k → 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_{i}h_{i}]
+[V_{i}h_{i}− f^{◦}(y_{i})h_{i}]

+[f^{◦}(y_{i})h_{i}− f^{◦}(x + h_{i}+ t_{i}h_{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_{i}k → 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 ∈ ∂_{S}F (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} ∈ ∂_{S}F (x + h_{i}) and y_{i} ∈ N_{x+h}_{i} such
that ky_{i}− (x + h_{i})k ≤ kh_{i}k^{2}, kV_{i}− f^{◦}(y_{i})k ≤ kh_{i}k and

kF (x + h_{i}) − F (x) − V_{i}h_{i}k

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_{i}k → 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_{i}h_{i}]

+[f^{◦}(x + h_{i}+ t_{i}h_{i})h_{i}− f^{◦}(y_{i})h_{i}]

+[δ^{+}F (x + h_{i}; h_{i}) − f^{◦}(x + h_{i}+ t_{i}h_{i})h_{i}].

With similar argument and choice of t_{i} in part (a), we have
kF (x + h_{i}) − F (x) − V_{i}h_{i}k

khik → 0 as i → ∞.

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