# 3 Generalized Differentiation of the Projection Operator onto the Circular Cone

## Full text

(1)

VARIATIONAL ANALYSIS OF CIRCULAR CONE PROGRAMS JINCHUAN ZHOU1, JEIN-SHAH CHEN2 and BORIS S. MORDUKHOVICH3

Abstract. This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming important for optimization theory and its applications. First we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/coderivatives of the projection operator associated with the circular cone.

Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs.

Keywords. variational analysis, optimization, generalized differentiation, conic programming, circular cone, second-order cone, projection operator, full and tilt stability

AMS subject classifications. 90C30, 90C31, 49J52

### 1 Introduction

The circular cone [5, 34] is a pointed, closed, convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation. Let its half-aperture angle be θ ∈ (0,π2).

Then the n-dimensional circular cone denoted by Lθcan be expressed as follows (see Figure 1):

Lθ: =x = (x1, x2) ∈ IR × IRn−1

kxk cos θ ≤ x1

=x = (x1, x2) ∈ IR × IRn−1

kx2k ≤ x1tan θ . (1.1) When θ = 45, the circular cone reduces to the well-known second-order cone (SOC for short, also

(a) 0 < θ < 45 (b) θ = 45 (c) 45< θ < 90

Figure 1: The graphs of circular cones.

1Department of Mathematics, School of Sciences, Shandong University of Technology. Zibo 255049, P.R. China (jinchuanzhou@163.com). Research of this author was partly supported by the National Natural Science Foundation of China under grant 11101248, 11271233 and by the Shandong Province Natural Science Foundation under grant ZR2010AQ026, ZR2012AM016.

2Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan (jschen@math.ntnu.edu.tw).

Research of this author was supported by Ministry of Science and Technology, Taiwan.

3Department of Mathematics, Wayne State University, Detroit, MI 48202 and King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (boris@math.wayne.edu). Research of this author was partly supported by the USA National Science Foundation under grant DMS-1007132.

(2)

known as the Lorentz cone and the ice-cream cone) given by Kn: =x = (x1, x2) ∈ IR × IRn−1

kx2k ≤ x1

=x = (x1, x2) ∈ IR × IRn−1

kxk cos 45≤ x1 . (1.2) Concerning SOC, for any vector x = (x1, x2) ∈ IR × IRn−1 we can decompose it as

x = λ1(x)u(1)x + λ2(x)u(2)x , (1.3) where λ1(x), λ2(x) and u(1)x , u(2)x are the spectral values and the associated spectral vectors of x relative to Kn defined by, respectively,

λi(x) : = x1+ (−1)ikx2k,

u(i)x : =





1 2



1, (−1)i x2

kx2k



if x26= 0,

1 2



1, (−1)iw

if x2= 0, i = 1, 2,

with w being any unit vector in IRn−1. If x26= 0, decomposition (1.3) is unique. Using this decomposi- tion, for any f : IR → IR we consider [3, 4] the vector function associated with Kn, n ≥ 1 by

fsoc(x) := f (λ1(x))u(1)x + f (λ2(x))u(2)x , x = (x1, x2) ∈ IR × IRn−1. (1.4) If f is defined only on some subset of IR, then fsoc is defined on the corresponding subset of IRn. Definition (1.4) is unambiguous whether x26= 0 or x2= 0.

Note that circular cone systems described by (1.1) with θ 6= 45 naturally arises in many real-life engineering problems. In particular, we refer the reader to the recent paper  and the bibliographies therein to the important class of optimal grasping manipulation problems for multi-fingered robots in which the grasping force of the i-th finger is subject to a contact friction constraint given by

(ui2, ui3)

≤ µui1, (1.5)

where ui1is the normal force of the i-th finger, ui2 and ui3 are the friction forces of the i-th finger, k · k is the 2-norm, and µ is the friction coefficient; see Figure 2.

Figure 2: The grasping force forms a circular cone where α = tan−1µ < 45.

It is easy to see that (1.5) is a circular cone constraint corresponding to the description ui= (ui1, ui2, ui3) ∈ Lθ in (1.1) with the angle θ = tan−1µ < 45.

(3)

Observe that a possible way to deal with circular cone constraints is to scale Lθas SOC by Lθ= A−1Kn and Kn= ALθ with A =tan θ 0

0 I



, (1.6)

which is justified in [34, Theorem 2.1]. However, this approach may not be acceptable from both theoret- ical and numerical viewpoints. Indeed, the “scaling” step can cause undesirable numerical performance due to round-off errors in computers, which has been confirmed by experiments. Furthermore, we will see in what follows that applying (1.6) does not help to obtain some major results of the paper while being useful in deriving the other ones.

Optimization problems with both SOC and circular cone constraints belong to a broad and important class in modern optimization theory known as conic or cone-constrained programming; see, e.g., [1, 2, 22]

and the references therein. However, the main difference between circular cone constraints and those given by SOC and most of the other constraint systems in conic programming is that the circular cone Lθ is non-self-dual, i.e., nonsymmetric, which makes its study more challenging and rather limited.

In contrast to symmetric conic programming, we are not familiar with a variety of publications de- voted to their nonsymmetric counterparts. Referring the reader to [14, 26, 32, 33] and the bibliographies therein, observe that there is no any unified way to handle nonsymmetric cone constraints, and each study uses certain specific features of the nonsymmetric cones under consideration. The previous papers [34, 35] concerning the circular cone show that some properties holding in the SOC framework can be extended to the circular cone setting. At the same time, some other SOC properties fail to be satisfied for the general nonsymmetric circular cone, where the angle θ 6= 45 plays a crucial role; see .

This paper is mainly devoted to two major interrelated issues of variational analysis and optimization for problems involving circular cone constraints. Our first goal is to calculate, entirely in terms of the initial circular cone data, some generalized differential constructions of variational analysis that have been proven to be important for various aspects of optimization. Namely, we derive explicit formulas to calculate generalized differential constructions for the (metric) projection operator associated with the general circular cone that are known as the B-subdifferential, directional derivative, graphical deriva- tive, regular derivative, regular coderivative, and (limiting) coderivative. Except the B-subdifferential and the (regular and limiting) coderivatives, the results obtained are new even for the symmetric SOC case. The obtained calculations allow us, in particular, to prove the strong semismoothness of the pro- jection operator onto the circular cone, which is important for many applications including those to numerical optimization. Furthermore, we establish new relationships between these generalized differen- tial constructions for the projection operator onto the circular cone and the metric projection onto the orthogonal spaces to the spectral vectors in the circular cone representation.

The second major goal of this paper is to completely characterize the notions of tilt stability and full stability of mathematical programs with circular cone constraints. These fundamental stability concepts were introduced in optimization theory by Rockafellar and his collaborators [12, 28] and then have been intensively studied by many researchers, especially in the recent years, for various classes of optimization problems; see, e.g., [2, 7, 8, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28] and the references therein. The construction of the second-order subdifferential/generalized Hessian in the sense of Mordukhovich 

(i.e., the coderivative of the first-order subgradient mapping) plays a crucial role in the characterization of tilt and full stability obtained in the literature. In this paper we establish, by using the obtained second-order calculations and the recent results of , complete characterizations of full and tilt stability for locally optimal solutions to mathematical programs with circular cone constraints expressed entirely in terms of the initial program data via certain second-order growth and strong sufficient optimality conditions under appropriate constraint qualifications.

The rest of the paper is organized as follows. In Section 2 we recall and briefly discuss the generalized differential constructions of variational analysis employed in deriving the main results of this paper.

Section 3 is devoted to calculating the generalized derivatives listed above for the projection operator onto the circular cone. In Section 4 we represent these generalized differential constructions for the

(4)

aforementioned projection operator via the orthogonal projections generated by the spectral vectors of the circular cone. Finally, Section 5 applies the second-order subdifferential of the indicator function associated with the circular cone and related to the above coderivative calculations to establish complete characterizations of full and tilt stability of mathematical programs with circular cone constraints.

Throughout the paper we use the standard notation and terminology of variational analysis; see, e.g., [16, 31]. Given a set-valued mapping/multifunction F : Rn →→Rm, recall that the constructions

Lim sup

x→¯x

F (x) :=n

y ∈ Rm

∃ sequences xk→ ¯x, yk → y such that yk ∈ F (xk) for all k ∈ IN := {1, 2, . . .}o

,

(1.7)

Lim inf

x→¯x F (x) :=n

y ∈ Rm

for any xk→ ¯x, ∃ yk → y such that

yk ∈ F (xk) for all k ∈ INo (1.8) are known as the (Painlev´e-Kuratowski) outer limit and inner limit of F as x → ¯x, respectively. For a set Ω ⊂ Rn, the symbol x→ ¯ x signifies that x → Ω with x ∈ Ω.

### 2 Tools of Variational Analysis

In this section we briefly review those tools of generalized differentiation in variational analysis, which are widely used in the subsequent sections. We start with geometric notions.

Given a set Ω ⊂ IRn locally closed around x ∈ Ω, the (Bouligand-Severi) tangent/contingent cone to Ω at ¯x ∈ Ω is defined by

T(¯x) := Lim sup

t↓0

Ω − ¯x t =n

d ∈ IRn

∃tk ↓ 0, dk→ d with ¯x + tkdk ∈ Ωo

(2.1) via the outer limit (1.7), while the (Clarke) regular tangent cone to Ω at ¯x ∈ Ω is given by

Tb(¯x) := Lim inf

x→¯x

T(x) (2.2)

via the inner limit (1.8). The (Fr´echet) regular normal cone to Ω at ¯x ∈ Ω is Nb(¯x) :=n

z ∈ IRn

hz, x − ¯xi ≤ o kx − ¯xk

for all x ∈ Ωo

, (2.3)

and the (Mordukhovich, limiting) normal cone to Ω at ¯x ∈ Ω can be equivalently defined by N(¯x) := Lim sup

x→¯x

Nb(x) = Lim sup

x→¯x

nconex − Π(x)o

, (2.4)

where Π denotes the (Euclidean) projection operator onto Ω, and where “cone” stands for the conic (may not be convex) hull of the set in question.

Consider next a set-valued mapping H : IRn⇒ IRmwith its graph and domain given by gph H :=(x, y) ∈ IRn× IRm

y ∈ H(x)

and dom H :=x ∈ IRn

H(x) 6= ∅ , respectively. The graphical derivative of H at (¯x, ¯y) ∈ gph H is defined by

DH(¯x, ¯y)(w) :=z ∈ IRm

(w, z) ∈ TgphH(¯x, ¯y) , w ∈ Rn, (2.5) via the tangent cone (2.1), while the (limiting) coderivative is defined via the normal cone (2.4) by

DH(¯x, ¯y)(y) :=x∈ IRn

(x, −y) ∈ NgphH(¯x, ¯y) , y∈ Rm, (2.6)

(5)

where we drop ¯y in the derivative/coderivative notion if H is single-valued at ¯x. Similarly, the regular derivative and the regular coderivative of H at (¯x, ¯y) is defined via, respectively, (2.2) and (2.3) by

DH(¯b x, ¯y)(w) :=z ∈ IRm

(w, z) ∈ bTgphH(¯x, ¯y) , w ∈ Rn, (2.7) DbH(¯x, ¯y)(y) :=x∈ IRn

(x, −y) ∈ bNgphH(¯x, ¯y) , y∈ Rm. (2.8) Now let f : IRn → R := (−∞, ∞] be an extended-real-valued function finite at ¯x ∈ Rn. To define the second-order subdifferential construction needed in what follows, we proceed in the way of [15, 16] and begin with the first-order (limiting) subdifferential of f at ¯x given by

∂f (¯x) :=v ∈ Rn

(v, −1) ∈ Nepif x, f (¯¯ x)

(2.9) via the normal cone (2.4) of the epigraph {(x, µ) ∈ Rn× R| µ ≥ f(x)} of f. Observe the representation N(¯x) = ∂δ(¯x) the normal cone (2.4) via the subdifferential (2.9) of the indicator function δ(x) of Ω equal to 0 if x ∈ Ω and ∞ otherwise. The second-order subdifferential (or generalized Hessian) of f at

¯

x relative to ¯y ∈ ∂f (¯x) is defined as the coderivative (2.6) of the first-order subdifferential (2.9) by

2f (¯x, ¯y)(u) := (D∂f )(¯x, ¯y)(u), u ∈ IRn. (2.10) Finally in this section, consider a single-valued mapping F : IRn→ IRmlocally Lipschitzian around ¯x and recall that F is almost everywhere differentiable in a neighborhood of ¯x with the derivative ∇F (x) by the classical Rademacher theorem; see . Then the B-subdifferential of F at ¯x is defined by

BF (¯x) :=



xlimk→x∇F (xk)

F is differentiable at xk



. (2.11)

Recall also that F is directionally differentiable at ¯x if the limit F0(x; h) := lim

t→0+

F (x + th) − F (x)

t exists for all h ∈ IRn. (2.12)

Having this, F is said to be semismooth at ¯x if F is locally Lipschitzian around ¯x, directionally differen- tiable at this point, and satisfies the relationship

V h − F0(x; h) = o khk

for any V ∈ co ∂BF (x + h) as h → 0. (2.13) Furthermore, F is ρ-order semismooth at x with 0 < ρ < ∞ if (2.13) is replaced above by

V h − F0(x; h) = O khk1+ρ

for any V ∈ co ∂BF (x + h) as h → 0. (2.14) The case of ρ = 1 in (2.13) corresponds to strongly semismooth mappings.

### 3 Generalized Differentiation of the Projection Operator onto the Circular Cone

In this section we derive precise formulas for calculating the above generalized derivatives of the projec- tion operator onto the circular cone (1.1). First we recall the following spectral decomposition from [34, Theorem 3.1] of any vector x = (x1, x2) ∈ IR × IRn−1relative to the circular cone Lθ:

x = λ1(x)u1x+ λ2(x)u2x, (3.1)

where the spectral values λ1(x) and λ2(x) are defined by

λ1(x) := x1− kx2kctanθ, λ2(x) := x1+ kx2k tan θ, (3.2)

(6)

and where the spectral vectors u1xand u2xare written as

u1x:= 1 1 + ctan2θ

1 0

0 ctanθ

  1

−¯x2



, u2x:= 1 1 + tan2θ

1 0

0 tan θ

  1

¯ x2



(3.3)

with ¯x2:= x2/kx2k if x26= 0 and ¯x2equal to any unit vector w ∈ IRn−1otherwise. Given any f : R → R we construct the vector function

fLθ(x) := f λ1(x)u1x+ f λ2(x)u2x. (3.4) associated with circular cone. It follows from  that the projection ΠLθ(x) of x onto Lθ, which is a single-valued and Lipschitzian operator, corresponds to f (t) := (t)+= max{t, 0} in (3.4), i.e., we have

ΠLθ(x) = x1− kx2kctanθ

+u1x+ x1+ kx2k tan θ

+u2x. (3.5)

Our first result in this section provides a complete calculation of the B-subdifferential (2.11) of the projection operator (3.5) entirely in terms of the initial data of the general circular cone (1.1). This result is widely used in what follows.

Lemma 3.1 (calculating the B-subdifferential of the projection operator). For any x ∈ IRn with the spectral decomposition (3.1), the B-subdifferential of the projection operator ΠLθ is calculated as follows:

(a) If λ1(x)λ2(x) 6= 0, then ΠLθ is differentiable at x and ∂BLθ)(x) = {∇ΠLθ(x)}.

(b) If λ1(x) = 0 and λ2(x) > 0, then

BLθ)(x) =



I, I + 1

tan θ + ctanθ

− tan θ x¯2

¯

x2 −ctanθ¯x2T2



.

(c) If λ1(x) < 0 and λ2(x) = 0, then

BLθ)(x) =



0, 1

tan θ + ctanθ

ctanθ x¯T2

¯

x2 tan θ ¯x2T2



.

(d) If λ1(x) = λ2(x) = 0, then

BLθ)(x)

=

( 1

tan θ + ctanθ

"ctanθ wT

w 

tan θ + ctanθ aI +h

tan θ − a(ctanθ + tan θ)i wwT

#

a ∈ [0, 1]

kwk = 1 )

[ 0, I

 .

Proof. In case (a) the function f (t) = (t)+ is differentiable at λi(x) for i = 1, 2. Hence it follows from [35, Theorem 2.3] that ΠLθ is also differentiable at x. Furthermore, in this case we have by (3.5) that

ΠLθ(x) =

x if λ1(x) > 0 and λ2(x) > 0, 0 if λ1(x) < 0 and λ2(x) < 0, (x1+ kx2k tan θ)u2x if λ1(x) < 0 and λ2(x) > 0.

In particular, kx2k 6= 0 when λ1(x) < 0 and λ2(x) > 0, and thus ∇kx2k = ¯x2. This gives us

BΠLθ(x) =∇ΠLθ(x) ,

(7)

where the derivative of ΠLθ at x is calculated by

∇ΠLθ(x) =









I if λ1(x) > 0 and λ2(x) > 0, 0 if λ1(x) < 0 and λ2(x) < 0, 1

tan θ + ctanθ

ctanθ x¯T2

¯ x2

x1+ kx2k tan θ

kx2k I − x1

kx2kx¯2T2

 if λ1(x) < 0 and λ2(x) > 0.

(3.6) In case (b) we have kx2k 6= 0, and so it follows from [35, Theorem 3.1] that

B fLθ(x) =









 ξ %¯xT2

%¯x2 aI + (η − a)¯x2T2



a = f λ2(x) − f λ1(x) λ2(x) − λ1(x) ξ − %ctanθ ∈ ∂Bf (λ1(x)) ξ + % tan θ ∈ ∂Bf (λ2(x)) η = ξ − %(ctanθ − tan θ)









 .

This implies by the obvious calculation

B(t)+=

1 for t > 0, {0, 1} for t = 0, 0 for t < 0 that the B-subdifferential of the projection operator is represented as

BLθ)(x) =





 ξ %¯xT2

%¯x2 aI + (η − a)¯x2¯xT2



a = 1 ξ − %ctanθ ∈ {0, 1}

ξ + % tan θ = 1 η = ξ − %(ctanθ − tan θ)





. (3.7)

Analyzing (3.7) in the case of ξ − %ctanθ = 1 and ξ + % tan θ = 1 shows that ξ = 1, % = 0, and η = 1.

Hence (3.7) reduces in this case to I. For ξ − %ctanθ = 0 we know that ξ = ctanθ

tan θ + ctanθ, % = 1

tan θ + ctanθ, and η = tan θ tan θ + ctanθ, and so equation (3.7) in this case takes the form of

ctanθ tan θ + ctanθ

1

tan θ + ctanθx¯T2 1

tan θ + ctanθx¯2 I +

 tan θ

tan θ + ctanθ − 1



¯ x2T2

= I + 1

tan θ + ctanθ

− tan θ x¯T2

¯

x2 −ctanθ¯x2T2

 ,

which gives us the B-subdifferential representation

B ΠLθ(x) =



I, I + 1

tan θ + ctanθ

− tan θ x¯2

¯

x2 −ctanθ¯x2T2



.

In case (c) we also have x26= 0. Similarly to case (b), it is not hard to verify that

B ΠLθ(x) =



0, 1

tan θ + ctanθ

ctanθ x¯T2

¯

x2 tan θ ¯x2T2



.

It remains to consider case (d) when x = 0. Then the result of [35, Theorem 3.4] tells us that

B ΠLθ(x)

(8)

=













 ξ %wT

%w aI + (η − a)wwT



either a = ξ ∈ {0, 1}, % = 0 or a ∈ [0, 1]

ξ − %ctanθ = 0 ξ + % tan θ = 1

η = ξ − %(ctanθ − tan θ) kwk = 1













=

( 1

tan θ + ctanθ

"ctanθ wT

w 

tan θ + ctanθ aI +

tan θ − a(ctanθ + tan θ) wwT

#

a ∈ [0, 1]

kwk = 1 )

[ (

0, I )

,

which thus completes the proof of the lemma. 2

Our next goals is to verify the directional differentiability of the projection operator (3.5) and derive formulas for calculating its directional derivative (2.12). Observe to this end that the result of [35, Theorem 2.2] tells us that the vector function fLθ from (3.4) is directionally differentiable at x provided that f is directionally differentiable at λi(x) for i = 1, 2. Moreover, for x2= 0 we have

fLθ0

(x; h) = 1

1 + ctan2θf0

x1; h1− kh2kctanθ

1 0

0 ctanθ

  1

−¯h2



+ 1

1 + tan2θf0

x1; h1+ kh2k tan θ

1 0

0 tan θ

  1

¯h2



(3.8)

= f0

x1; h1− kh2kctanθ

u1h+ f0

x1; h1+ kh2k tan θ u2h. On the other hand, for x26= 0 we denote

Mx2 :=

0 0

0 I − x2xT2 kx2k2

and arrive at the following relationships:

fLθ0 (x; h)

= 1

1 + ctan2θf0



λ1(x); h1−xT2h2

kx2kctanθ

  1 0 0 ctanθ

  1

−¯x2



− ctanθ 1 + ctan2θ

f (λ1(x))

kx2k Mx2h (3.9)

+ 1

1 + tan2θf0



λ2(x); h1+xT2h2

kx2k tan θ

  1 0 0 tan θ

  1

¯ x2



+ tan θ 1 + tan2θ

f (λ2(x)) kx2k Mx2h

= f0



λ1(x); h1−xT2h2

kx2kctanθ

 u1x+ f0



λ2(x); h1+xT2h2

kx2k tan θ



u2x+f (λ2(x)) − f (λ1(x)) λ2(x) − λ1(x) Mx2h.

This leads us to calculate the directional derivative (2.12) of the projection operator (3.5).

Lemma 3.2 (calculating the directional derivative of the projection operator). The projector operator (3.5) is directionally differentiable at any point x ∈ IRn with the spectral decomposition (3.1), and its directional derivative at x in any direction h ∈ Rn is calculated as follows:

(a) If λ1(x)λ2(x) 6= 0, then Π0L

θ(x; h) = ∇ΠLθ(x)h.

(b) If λ1(x) = 0 and λ2(x) > 0, then Π0Lθ(x; h) = h − (1 + ctan2θ) (u1x)Th

u1x.

(9)

(c) If λ1(x) < 0 and λ2(x) = 0, then Π0L

θ(x; h) = (1 + tan2θ) (u2x)Th

+u2x. (d) If λ1(x) = λ2(x) = 0, then Π0L

θ(x; h) = ΠLθ(h).

Proof. The directional differentiability of (3.5) at x follows from the discussions above. Moreover, in case (a), corresponding to f (t) = (t)+ in (3.4), we get the differentiability of ΠLθ at this point, and hence Π0L

θ(x; h) = ∇ΠLθ(x)h for all h ∈ Rn.

In case (b) we have x26= 0. It follows from (3.9) that Π0Lθ(x; h) =



h1−xT2h2 kx2kctanθ



+

u1x+



h1+xT2h2 kx2k tan θ



u2x+ Mx2h

= (1 + ctan2θ) (u1x)Th

+u1x+ h +

tan2θ 1 + tan2θ



−h1+xT2h2 kx2kctanθ



 tan θ 1 + tan2θ



h1+xT2h2 kx2k tan θ



−xT2h2 kx2k



¯ x2

= (1 + ctan2θ) (u1x)Th

+u1x+ h −

tan2θ 1 + tan2θ



h1−xT2h2 kx2kctanθ

 tan2θ

1 + tan2θ



h1− ¯xT2h2ctanθ

− ctanθ¯x2

= (1 + ctan2θ) (u1x)Th

+u1x+ h − tan2θ

1 + tan2θ(1 + ctan2θ) (u1x)Th1 0 0 ctanθ

  1

−¯x2



= (1 + ctan2θ) (u1x)Th

+u1x+ h − (1 + ctan2θ) (u1x)Thu1x

= h − (1 + ctan2θ) (u1x)Th

u1x,

where the representations t = (t)++ (t) for all t ∈ IR is used together with tan θ

1 + tan2θ



h1+xT2h2

kx2k tan θ



−xT2h2

kx2k

= tan θ

1 + tan2θ



h1+xT2h2

kx2k tan θ − 1 + tan2θ tan θ

xT2h2

kx2k



= tan θ

1 + tan2θ



h1−xT2h2

kx2kctanθ

 , and (u1x)Th

+− (u1x)Th = − (u1x)Th

.

In case (c) we employ (3.9) again to get the conclusion claimed. The final case (d) yields x = 0, and hence representation (3.8) gives us the equalities

Π0L

θ(x; h) = (h1− kh2kctanθ)+u1h+ (h1+ kh2k tan θ)+u2h= ΠLθ(h), which therefore complete the proof of the lemma. 2

The following theorem uses the previous considerations to establish the strongly semismoothness property of the projection operator ΠLθ. It has been well recognized the importance of this property of Lipschitzian mappings in many aspects of variational analysis and optimization; in particular, to establish the quadratic rate of convergence of the so-called semismooth Newton method; see [9, 30].

Theorem 3.3 (strong semismoothness of the projection operator). The projection operator ΠLθ

in (3.5) is strongly semismooth over Rn.

Proof. The proof is inspired by [10, Proposition 4.5]. Note first that the directional differentiability of the Lipschitz continuous projection operator ΠLθ from Lemma 3.2, and thus it remains to show that representation (2.14) holds for it with ρ = 1.

(10)

To verify our claim, deduce from the proof of Lemma 3.1 that

ΠLθ(x) =





x if λ1(x) ≥ 0 and λ2(x) ≥ 0,

1

1 + tan2θ(x1+ kx2k tan θ)1 0 0 tan θ

  1

¯ x2



if λ1(x) < 0 and λ2(x) > 0,

0 if λ1(x) ≤ 0 and λ2(x) ≤ 0.

(3.10)

Then we split the subsequent proof into two cases: x26= 0 and x2= 0.

Case 1. When x26= 0, we can easily observe that in all the formulas from (3.10) corresponding to this case, the projection operator ΠLθ is a piecewise C2-smooth mapping whose strong semismoothness is well known in optimization . It verifies the claim in this case.

Case 2. For x2 = 0, suppose first that x1 6= 0. Then λi(x) = x1 6= 0, i = 1, 2. Since λi(y) is Lipschitz continuous by [35, Lemma 2.1], we get from (3.10) that ΠLθ(y) is either 0 or y when y is in a neighborhood of x. Thus ΠLθ is surely strongly semismooth at x in this setting. In the remaining setting of x1= 0 we have x = 0. Note that the projection operator ΠLθis obviously positively homogeneous, i.e., ΠLθ(tz) = tΠLθ(z) for z ∈ Rnand t ≥ 0. This implies that Π0L

θ(h; h) = ΠLθ(h) and Π0L

θ(0; h) = ΠLθ(h).

Hence ∇ΠLθ(h0)(h0) = ΠLθ(h0) as h0 ∈ DΠ. Since DΠ is a dense subset of Rn, for any fixed h 6= 0 and V ∈ ∂BΠLθ(h), there exists h0 ∈ DΠ such that kh0− hk ≤ khk2 and kV − ∇ΠLθ(h0)k ≤ khk.

Hence for h sufficiently close to 0 we have

kV h − Π0Lθ(0; h)k = kV h − ∇ΠLθ(h0)(h0) + ΠLθ(h0) − Π0Lθ(0; h)k

= kV h − ∇ΠLθ(h0)(h) + ∇ΠLθ(h0)(h) − ∇ΠLθ(h0)(h0) + ΠLθ(h0) − ΠLθ(h)k

≤ kV − ∇ΠLθ(h0)kkhk + k∇ΠLθ(h0)kkh − h0k + kh − h0k

≤ (r + 2)khk2,

where r is a bounded from above of k∂BΠLθ(·)k near 0 since ΠLθ is Lipschitz. Thus L := lim sup

h→0

kV h − Π0Lθ(0; h)k

khk2 < ∞, i.e., (3.11)

V h − Π0L

θ(0; h) = O(khk2) for all V ∈ ∂BΠLθ(h).

Now let us show that V h − Π0L

θ(0; h) = O(khk2) for any V ∈ co ∂BΠLθ(h), i.e., for any hk → 0 and Vk ∈ co ∂BΠLθ(hk) we have Vkhk− Π0Lθ(0; hk) = O(khkk2). Since Vk ∈ co ∂BΠLθ(hk), it follows from the Carath´eodory theorem that there are Vki∈ ∂BΠLθ(hk) and λik ≥ 0 for i = 1, . . . , n + 1 such that

Vk=

n+1

X

i=1

λikVki and

n+1

X

i=1

λik = 1.

Since Vki∈ ∂BΠLθ(hk), it follows from (3.11) that lim sup

k→0

kVkihk− Π0L

θ(0; hk)k khkk2 ≤ L.

Due to the boundedness of {λik}, we can assume without loss of generality that {λik} converge to some λ¯i for i = 1, . . . , n + 1. Hence

lim sup

k→0

kVkhk− Π0L

θ(0; hk)k

khkk2 = lim sup

k→0

n+1

P

i=1

λikVkihk− Π0Lθ(0; hk) khkk2

≤ lim sup

k→0 n+1

X

i=1

λikkVkihk− Π0L

θ(0; hk)k khkk2

(11)

n+1

X

i=1

λ¯iL = L.

Thus V h − Π0L

θ(0; h) = O(khk2) for any V ∈ co ∂BΠLθ(h), i.e., ΠLθ is strongly semismooth at 0. 2 The next result, which easily follows from Lemma 3.2, provides the calculation of the graphical derivative (2.5) for the projection operator onto the circular cone.

Proposition 3.4 (calculating the graphical derivative of the projection operator). For any x ∈ Rn with decomposition (3.1), the graphical derivative of ΠLθ(x) is calculated by

Lθ(x)(w) =Π0Lθ(x; w)

for any w ∈ IRn. (3.12)

Proof. It follows from [31, formula 8(14)] that the graphical derivative of any closed-graph operator, and hence of ΠLθ in particular, can be equivalently represented as

Lθ(x)(w) = Lim sup

τ &0 w0 →w

ΠLθ(x + τ w0) − ΠLθ(x)

τ . (3.13)

By Lemma 3.2 the Lipschitzian mapping ΠLθ is directionally differentiable at x. Thus the right-hand side of (3.13) reduces to Π0Lθ(x; w), which justifies (3.12). 2

Based on the calculations provided in Lemma 3.1 and Lemma 3.2, we are now ready to establish precise formulas for computing the regular and limiting coderivatives of the projection operator ΠLθ onto the general circular cone (1.1). We proceed similarly to the proofs of the main results of the paper 

by Outrata and Sun while using our calculations given above as well as in the proofs of the theorems.

Taking into account relationships (1.6) between the circular and second-order cones, it is appealing to reduce deriving coderivative formulas for the projection onto the circular cone to those obtained for the second-order one. However, it does not seem to be possible; see more discussions in Remark 4.7.

Theorem 3.5 (calculating the regular coderivative of the projection operator). For any x ∈ Rn with decomposition (3.1) and any y∈ Rn, the regular coderivative (2.8) of the projection operator ΠLθ(x) onto the circular cone (1.1) is calculated as follows:

(a) If λ1(x)λ2(x) 6= 0, then bDΠLθ(x)(y) =∇ΠLθ(x)y . (b) If λ1(x) = 0 and λ2(x) > 0, then

DbΠLθ(x)(y) =x∈ Rn

y− x∈ IR+u1x, hx, u1xi ≥ 0 . (c) If λ1(x) < 0 and λ2(x) = 0, then

DbΠLθ(x)(y) =x∈ Rn

x∈ IR+u2x, hy− x, u2xi ≥ 0 . (d) If λ1(x) = λ2(x) = 0, then

DbΠLθ(x)(y) =x∈ Rn

x∈ Lθ, y− x∈ Lπ

2−θ .

Proof. Due to the well-known duality between the regular coderivative and the graphical derivative of a mapping (see ) and by the established directional differentiability of the projection operator onto the circular cone, we have the equivalence

x∈ bDΠLθ(x)(y) ⇐⇒ hx, hi ≤y, Π0Lθ(x; h)

for all h ∈ IRn. (3.14)

(12)

Employing (3.14) and the calculation of the directional derivative of ΠLθ in Lemma 3.2 allows us to derive the claimed formulas for the regular coderivative of ΠLθ in all the cases (a)–(d) of the theorem.

In case (a), pick any x∈ bDΠLθ(x)(y) and get by using Lemma 3.2(a) and duality (3.14) that hx, hi ≤ hy, Π0L

θ(x; h)i ⇐⇒ hx, hi ≤ hy, ∇ΠLθ(x)hi

⇐⇒ hx− ∇ΠLθ(x)y, hi ≤ 0,

where the last step comes from the fact that the operator ∇ΠLθ is self-adjoint by (3.6). Hence we have x= ∇ΠLθ(x)y, i.e., bDΠLθ(x)(y) = {∇ΠLθ(x)y}.

In case (b) we employ Lemma 3.2(b), which gives us together with (3.14) that x∈ bDΠLθ(x)(y) ⇐⇒ hx, hi ≤

y, h − (1 + ctan2θ)

(u1x)Th

u1x

⇐⇒ hx− y, hi + (1 + ctan2θ)

y,

(u1x)Th

u1x

≤ 0

⇐⇒

 hx− y, hi ≤ 0, (u1x)Th ≥ 0

hx− y, hi + (1 + ctan2θ)(u1x)Th(y)Tu1x≤ 0, (u1x)Th ≤ 0

⇐⇒ ∃ α ≥ 0 and β ≥ 0 such that y− x= αu1x and x− y+ (1 + ctan2θ) (y)Tu1xu1x= βu1x

⇐⇒ ∃ α ≥ 0 and β ≥ 0 such that y− x= αu1x and (1 + ctan2θ) (y)Tu1xu1x= (α + β)u1x

⇐⇒ ∃ α ≥ 0 such that y− x= αu1x

and (1 + ctan2θ)hy, u1xi ≥ α (3.15)

⇐⇒ ∃ α ≥ 0 such that y− x= αu1x and hx, u1xi ≥ 0. (3.16) The last equivalence above comes from the following arguments: if (3.16) holds, then

(1 + ctan2θ)hy, u1xi = (1 + ctan2θ)hx+ αu1x, u1xi ≥ α(1 + ctan2θ)ku1xk2= α;

conversely, the validity of (3.15) implies that

hx, u1xi = hy, u1xi − αhu1x, u1xi ≥ 1

1 + ctan2θα − 1

1 + ctan2θα = 0.

In case (c) we have the equivalencies by using Lemma 3.2(c) and duality (3.14):

x∈ bDΠLθ(x)(y) ⇐⇒ hx, hi ≤

y, (1 + tan2θ)

(u2x)Th

+

u2x

⇐⇒

( hx, hi ≤ 0, (u2x)Th ≤ 0,

D

x− (1 + tan2θ)

(y)Tu2x u2x, hE

≤ 0, (u2x)Th ≥ 0

⇐⇒ ∃ α ≥ 0 such that x= αu2x and (1 + tan2θ)(y)Tu2x≥ α

⇐⇒ ∃ α ≥ 0 such that x= αu2x and hy− x, u2xi ≥ 0, which readily justify the claimed result in this case.

In case (d) we have x = 0 and then proceed by using Lemma 3.2(d) together with (3.14). This yields x∈ bDΠLθ(0)(y) ⇐⇒ x, h ≤ y, ΠLθ(h)

for all h ∈ IRn

(13)

⇐⇒ x, ΠLθ(h) + ΠLθ(h) ≤ y, ΠLθ(h)

for all h ∈ IRn

⇐⇒ x− y, ΠLθ(h) + x, ΠLθ(h) ≤ 0 for all h ∈ IRn (3.17)

⇐⇒ x∈ Lθ and y− x∈ Lπ

2−θ, (3.18)

where the last equivalence is justified as follows. Relationship (3.18) =⇒ (3.17) is implied by the inclusion x− y ∈ −Lπ2−θ = (Lθ). For the converse implication, observe that the validity of (3.17) gives us hx− y, hi ≤ 0 for all h ∈ Lθ and hx, hi ≤ 0 for all h ∈ (Lθ), which yields in turn the fulfillment of x− y∈ (Lθ)= −Lπ

2−θ and x∈ ((Lθ))= Lθ since Lθ is a closed and convex cone. 2

To calculate next the coderivative (2.6) of the projection operator ΠLθ, for any x, y∈ Rn we define A(x, y) :=x∈ Rn

y− x∈ IR+u1x, hx, u1xi ≥ 0 , (3.19) B(x, y) :=x∈ Rn

x∈ IR+u2x, hy− x, u2xi ≥ 0 . (3.20) Theorem 3.6 (calculating the coderivative of the projection operator). For any x ∈ Rn with decomposition (3.1) and any y ∈ Rn, the coderivative (2.8) of the projection operator ΠLθ(x) onto the circular cone (1.1) is calculated as follows:

(a) If λ1(x)λ2(x) 6= 0, then DΠLθ(x)(y) =∇ΠLθ(x)y . (b) If λ1(x) = 0 and λ2(x) > 0, then

DΠLθ(x)(y) =h

BLθ)(x)yi [ n

x ∈ Rn

y− x∈ IR+u1x, hx, u1xi ≥ 0o .

(c) If λ1(x) < 0 and λ2(x) = 0, then DΠLθ(x)(y) =h

BLθ)(x)yi [ n

x ∈ Rn

x∈ IR+u2x, hy− x, u2xi ≥ 0o .

(d) If λ1(x) = λ2(x) = 0, then DΠLθ(x)(y) =h

BLθ)(x)yi [ h [

ξ∈bd(Lπ 2−θ)/{0}

n

x ∈ IRn

y− x∈ IR+ξ, hx, ξi ≥ 0oi

[ h [

η∈bd(Lθ)/{0}

n x

x∈ IR+η, hy− x, ηi ≥ 0oi [ hn x

x∈ Lθ, y− x∈ Lπ

2−θ

oi ,

where the B-subdifferential of ΠLθ at x is calculated in Lemma 3.1.

Proof. Using the well-known representation of the coderivative (2.6) via the outer limit (1.7) of the regular one (see [16, 31]) as well as the continuity of ΠLθ, we get

DΠLθ(x)(y) = Lim sup

v→x v∗ →y∗

DbΠLθ(v)(v). (3.21) This allows us to calculate DΠLθ by passing to the limit in the relationships of Theorem 3.5.

In case (a) we easily get from (3.21) and Theorem 3.5(a) that DΠLθ(x)(y) = Lim sup

v→x v∗ →y∗

DbΠLθ(v)(v) (3.22)

= Lim sup

v→x v∗ →y∗

∇ΠLθ(v)v = ∇ΠLθ(x)y .

Updating...

## References

Related subjects :