Variational analysis of circular cone programs

Variational analysis of circular cone programs

Jinchuan Zhou^a, Jein-Shah Chen^band Boris S. Mordukhovich^cd∗

aDepartment of Mathematics, School of Sciences, Shandong University of Technology, Zibo, P.R.

China;^bDepartment of Mathematics, National Taiwan Normal University, Taipei, Taiwan;

cDepartment of Mathematics, Wayne State University, Detroit, MI, USA;^dKing Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

(Received 18 January 2014; accepted 12 July 2014)

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/co- derivatives 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 [1,2] 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 beθ ∈ (0,^π₂). Then the n-dimensional circular cone denoted by L_θcan be expressed as follows (see Figure1):

L_θ : =

x= (x1, x2) ∈ IR × IRⁿ⁻¹x cos θ ≤ x1



=

x= (x1, x2) ∈ IR × IRⁿ⁻¹x₂ ≤ x1tanθ

. (1.1)

Whenθ = 45^◦, the circular cone reduces to the well-known second-order cone (SOC for short, also known as the Lorentz cone and the ice-cream cone) given by

Kⁿ: =

x= (x1, x2) ∈ IR × IRⁿ⁻¹x₂ ≤ x1



=

x= (x1, x2) ∈ IR × IRⁿ⁻¹x cos 45^◦≤ x1

. (1.2)

∗Corresponding author. Email: boris@math.wayne.edu

© 2014 Taylor & Francis

Figure 1. The graphs of circular cones.

Concerning SOC, for any vector x = (x1, x2) ∈ IR × IRⁿ⁻¹we can decompose it as x= λ1(x)u⁽¹⁾x + λ2(x)u⁽²⁾x , (1.3) whereλ1(x), λ2(x) and u⁽¹⁾x , u⁽²⁾x are the spectral values and the associated spectral vectors of x relative toKⁿdefined by, respectively,

λi(x) : = x1+ (−1)ⁱx2,

u⁽ⁱ⁾_x : =

⎧⎨

⎩

1 2

1, (−1)ⁱ x₂

x2

if x2= 0,

1 2

1, (−1)ⁱw

if x2= 0, i = 1, 2,

withw being any unit vector in IRⁿ⁻¹. If x2= 0, decomposition (1.3) is unique. Using this decomposition, for any f : IR → IR we consider [3,4] the vector function associated with Kⁿ, n≥ 1 by

f^soc(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 f^socis defined on the corresponding subset of IRⁿ. Definition (1.4) is unambiguous whether x2= 0 or x2= 0.

Note that circular cone systems described by (1.1) withθ = 45^◦ naturally arises in many real-life engineering problems. In particular, we refer the reader to the recent paper [5] 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

ui 2, ui 3) ui 1, (1.5)

where ui 1is the normal force of the i th finger, ui 2and ui 3are the friction forces of the i th finger, ·  is the 2-norm and μ is the friction coefficient; see Figure2.

It is easy to see that (1.5) is a circular cone constraint corresponding to the description ui = (ui 1, ui 2, ui 3) ∈ L_θin (1.1) with the angleθ = tan⁻¹μ < 45^◦.

Observe that a possible way to deal with circular cone constraints is to scaleL_θas SOC by

Lθ = A⁻¹Kⁿ and Kⁿ= ALθ with A=

tanθ 0

0 I

, (1.6)

Figure 2. The grasping force forms a circular cone whereα = tan⁻¹μ < 45^◦.

which is justified in [2, Theorem 2.1]. However, this approach may not be acceptable from both theoretical 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 pro- gramming; see, e.g. [6–8] 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 coneL_θis 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 devoted to their nonsymmetric counterparts. Referring the reader to [9–12]

and the bibliographies therein, observe that there is no unified way to handle nonsymmetric cone constraints, and each study uses certain specific features of the nonsymmetric cones under consideration. The previous papers [2,13] 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 angleθ = 45^◦plays a crucial role; see [14].

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 derivative, 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 projection operator onto the circular cone, which is important for many applications including those to numerical optimization. Furthermore, we establish new relationships between these generalized differential 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 [15,16] and then have been intensively studied by many researchers, especially in the recent years, for various classes of optimization problems; see, e.g.

[7,8,16–27] and the references therein. The construction of the second-order subdifferen- tial/generalized Hessian in the sense of Mordukhovich [28] (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 [25], 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 Section2we recall and briefly discuss the generalized differential constructions of variational analysis employed in deriving the main results of this paper. Section3is devoted to calculating the generalized derivatives listed above for the projection operator onto the circular cone. In Section4we represent these generalized differential constructions for the aforementioned projection operator via the orthogonal projections generated by the spectral vectors of the circular cone. Finally, Section5applies 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. [29,30]. Given a set-valued mapping/multifunction F: Rⁿ→→ R^m, recall that the constructions

Lim sup

x→ ¯x F(x) :=

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

, (1.7)

Lim inf

x→ ¯x F(x) :=

y∈ R^m for any xk → ¯x, ∃ yk→ y such that

y_k∈ F(xk) for all k ∈ IN (1.8) are known as the (Painlevé–Kuratowski) outer limit and inner limit of F as x → ¯x, respectively. For a set ⊂ Rⁿ, the symbol x→ ¯x signifies that x →  with x ∈ .^

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  ⊂ IRⁿ locally closed around x ∈ , the (Bouligand–Severi) tan- gent/contingent cone to at ¯x ∈  is defined by

T_( ¯x) := Lim sup

t↓0

 − ¯x

t =

d ∈ IRⁿ ∃tk↓ 0, dk → d with ¯x + tkd_k∈  (2.1)

via the outer limit (1.7), while the (Clarke) regular tangent cone to at ¯x ∈  is given by T _( ¯x) := Lim inf

x→ ¯x^

T_(x) (2.2)

via the inner limit (1.8). The (Fréchet) regular normal cone to at ¯x ∈  is N _( ¯x) :=

z∈ IRⁿ z, x − ¯x ≤ o

x − ¯x

for all x∈ 

, (2.3)

and the (Mordukhovich, limiting) normal cone to at ¯x ∈  can be equivalently defined by

N_( ¯x) := Lim sup

x→ ¯x^

N_(x) = Lim sup

x→ ¯x

 cone

x− (x)

, (2.4)

where_denotes the (Euclidean) projection operator onto, 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⇒ IR^m with its graph and domain given by

gph H :=

(x, y) ∈ IRⁿ× IR^my∈ H(x)

and dom H :=

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

D H( ¯x, ¯y)(w) :=

z∈ IR^m (w,z) ∈ Tgph H( ¯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

D^∗H( ¯x, ¯y)(y^∗) :=

x^∗∈ IRⁿ (x^∗, −y^∗) ∈ Ngph H( ¯x, ¯y)

, y^∗∈ R^m, (2.6) where we drop ¯y in the derivative/coderivative notion if H is single-valued at ¯x. Simi- larly, the regular derivative and the regular coderivative of H at( ¯x, ¯y) are defined via, respectively, (2.2) and (2.3) by

D H ( ¯x, ¯y)(w) :=

z∈ IR^m (w,z) ∈ T_{gph H}( ¯x, ¯y)

, w ∈ Rⁿ, (2.7)

D ^∗H( ¯x, ¯y)(y^∗) :=

x^∗∈ IRⁿ (x^∗, −y^∗) ∈ N_{gph H}( ¯x, ¯y)

, y^∗∈ R^m. (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 [28,29] and begin with the first-order (limiting) subdifferential of

f at ¯x given by

∂ f ( ¯x) :=

v ∈ Rⁿ (v,−1) ∈ Nepi f

¯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 functionδ(x) of  equal to 0 if x ∈  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 → IR^m locally Lipschitzian around ¯x and recall that F is almost everywhere differentiable in a neigh- bourhood of ¯x with the derivative ∇ F(x) by the classical Rademacher theorem; see [30].

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 F^(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, direc- tionally differentiable at this point, and satisfies the relationship

V h− F^(x; h) = o

h

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

V h− F^(x; h) = O

h^1+ρ

for any V ∈ co ∂BF(x + h) as h → 0. (2.14) The case ofρ = 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 projection operator onto the circular cone (1.1). First we recall the following spectral decomposition from [2, Theorem 3.1] of any vector x = (x1, x2) ∈ IR × IRⁿ⁻¹relative to the circular coneL_θ:

x= λ1(x)u¹_x+ λ2(x)u²_x, (3.1) where the spectral valuesλ1(x) and λ2(x) are defined by

λ1(x) := x1− x2ctanθ, λ2(x) := x1+ x2 tan θ, (3.2) and where the spectral vectors u¹_x and u²_xare written as

u¹_x := 1 1+ ctan²θ

1 0 0 ctanθ

  1

− ¯x2

, u²x := 1 1+ tan²θ

1 0 0 tanθ

  1

¯x2

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

f^Lθ(x) := f λ1(x)

u¹_x+ f λ2(x)

u²_x. (3.4)

associated with circular cone. It follows from [2] that the projection_Lθ(x) of x onto L_θ, which is a single-valued and Lipschitzian operator, corresponds to f(t) := (t)₊ = max{t, 0} in (3.4), i.e. we have

_Lθ(x) =

x₁− x2ctanθ

+u¹_x+

x₁+ x2 tan θ

+u²_x. (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.

Le m m a 3.1 (calculating the B-subdifferential of the projection operator) For any x∈ IRⁿwith the spectral decomposition (3.1), the B-subdifferential of the projection operator

_Lθ is calculated as follows:

(a) Ifλ1(x)λ2(x) = 0, then _Lθis differentiable at x and∂B(_Lθ)(x) = {∇_Lθ(x)}.

(b) Ifλ1(x) = 0 and λ2(x) > 0, then

∂B(_Lθ)(x) =



I, I + 1 tanθ + ctanθ

− tan θ ¯x2

¯x2 −ctanθ ¯x2¯x₂^T 

.

(c) Ifλ1(x) < 0 and λ2(x) = 0, then

∂B(Lθ)(x) =



0, 1

tanθ + ctanθ

ctanθ ¯x₂^T

¯x2 tanθ ¯x2¯x₂^T 

.

(d) Ifλ1(x) = λ2(x) = 0, then

∂B(_Lθ)(x) =

 1

tanθ + ctanθ

×

ctanθ w^T

w 

tanθ + ctanθ a I+

tanθ − a(ctanθ + tan θ) ww^T



×

a∈ [0, 1]

w = 1

 0, I

 .

Proof In case (a) the function f(t) = (t)₊is differentiable atλi(x) for i = 1, 2. Hence, it follows from [13, Theorem 2.3] that_Lθ is also differentiable at x. Furthermore, in this case we have by (3.5) that

Lθ(x) =

⎧⎨

⎩

x if λ1(x) > 0 and λ2(x) > 0, 0 if λ1(x) < 0 and λ2(x) < 0, (x1+ x2 tan θ)u²x if λ1(x) < 0 and λ2(x) > 0.

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

∂B_Lθ(x) =

∇_Lθ(x) , where the derivative of_Lθ at x is calculated by

∇_Lθ(x)

=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

I if λ1(x) > 0 and λ2(x) > 0,

0 if λ1(x) < 0 and λ2(x) < 0,

1 tanθ + ctanθ

⎡

⎣ctanθ ¯x₂^T

¯x2

x₁+ x2 tan θ

x2 I− x₁

x2¯x2¯x₂^T

⎤

⎦ if λ1(x) < 0 and λ2(x) > 0.

(3.6)

In case (b) we havex2 = 0, and so it follows from [13, Theorem 3.1] that

∂B

f^Lθ (x) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

 ξ  ¯x₂^T  ¯x2 a I + (η − a) ¯x2¯x₂^T







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

∂B(Lθ)(x) =

⎧⎪

⎪⎨

⎪⎪

⎩

 ξ  ¯x₂^T  ¯x2 a I+ (η − a) ¯x2¯x₂^T







a= 1

ξ − ctanθ ∈ {0, 1}

ξ +  tan θ = 1

η = ξ − (ctanθ − tan θ)

⎫⎪

⎪⎬

⎪⎪

⎭. (3.7)

Analysing (3.7) in the case ofξ −ctanθ = 1 and ξ + tan θ = 1 shows that ξ = 1,  = 0, andη = 1. Hence (3.7) reduces in this case to I . Forξ − ctanθ = 0 we know that

ξ = ctanθ

tanθ + ctanθ,  = 1

tanθ + ctanθ, and η = tanθ tanθ + ctanθ, and so Equation (3.7) in this case takes the form of

⎡

⎢⎣

ctanθ tanθ + ctanθ

1

tanθ + ctanθ¯x₂^T 1

tanθ + ctanθ¯x2 I+

$ tanθ

tanθ + ctanθ − 1

%

¯x2¯x₂^T

⎤

⎥⎦

= I + 1

tanθ + ctanθ

− tan θ ¯x₂^T

¯x2 −ctanθ ¯x2¯x₂^T

 ,

which gives us the B-subdifferential representation

∂B

Lθ

(x) =



I, I + 1 tanθ + ctanθ

− tan θ ¯x2

¯x2 −ctanθ ¯x2¯x₂^T 

.

In case (c) we also have x2= 0. Similarly to case (b), it is not hard to verify that

∂B

_Lθ (x) =

'

0, 1

tanθ + ctanθ

ctanθ ¯x₂^T

¯x2 tanθ ¯x2¯x₂^T

(

.

It remains to consider case (d) when x = 0. Then the result of [13, Theorem 3.4] tells us that

∂B

_Lθ (x)

=

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

 ξ w^T

w aI + (η − a)ww^T 







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

ξ − ctanθ = 0 ξ +  tan θ = 1

η = ξ − (ctanθ − tan θ)

w = 1

⎫⎪

⎪⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎪

⎭

=

 1

tanθ + ctanθ

ctanθ w^T

w 

tanθ + ctanθ a I +

tanθ − a(ctanθ + tan θ) ww^T



×

a ∈ [0, 1]

w = 1

 0, I

 ,

which thus completes the proof of the lemma. 

Our next goal 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 [13, Theorem 2.2] tells us that the vector function f^Lθ from (3.4) is directionally differentiable at x provided that f is directionally differentiable atλi(x) for i= 1, 2. Moreover, for x2= 0 we have

 f^Lθ

_

(x; h) = 1

1+ ctan²θ f^



x₁; h1− h2ctanθ 1 0 0 ctanθ

  1

− ¯h2

+ 1

1+ tan²θ f^



x₁; h1+ h2 tan θ 1 0 0 tanθ

 1

¯h₂ 

= f^

x₁; h1− h2ctanθ

u¹_h+ f^

x₁; h1+ h2 tan θ

u²_h. (3.8) On the other hand, for x2= 0 we denote

M_x2 :=

⎡

⎣0 0

0 I− x2x₂^T

x2²

⎤

⎦

and arrive at the following relationships:

 f^Lθ

_ (x; h)

= 1

1+ ctan²θ f^ )

λ1(x); h1−x₂^Th₂

x2ctanθ

* 1 0 0 ctanθ

  1

− ¯x2

− ctanθ 1+ ctan²θ

f(λ1(x))

x2 M_x2h

+ 1

1+ tan²θ f^ )

λ2(x); h1+x₂^Th₂

x2 tanθ

* 1 0 0 tanθ

 1

¯x2

+ tanθ 1+ tan²θ

f(λ2(x))

x2 M_x2h

= f^ )

λ1(x); h1− x₂^Th2

x2ctanθ

* u¹_x+ f^

)

λ2(x); h1+x₂^Th2

x2 tanθ

* u²_x +f(λ2(x)) − f (λ1(x))

λ2(x) − λ1(x) Mx₂h. (3.9)

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

Le m m a 3.2 (calculating the directional derivative of the projection operator) The pro- jector operator (3.5) is directionally differentiable at any point x ∈ IRⁿwith the spectral decomposition (3.1), and its directional derivative at x in any direction h∈ Rⁿis calculated as follows:

(a) Ifλ1(x)λ2(x) = 0, then ^_Lθ(x; h) = ∇_Lθ(x)h.

(b) Ifλ1(x) = 0 and λ2(x) > 0, then ^_Lθ(x; h) = h − (1 + ctan²θ) (u¹_x)^Th

−u¹_x. (c) Ifλ1(x) < 0 and λ2(x) = 0, then ^_Lθ(x; h) = (1 + tan²θ)

(u²x)^Th

+u²_x. (d) Ifλ1(x) = λ2(x) = 0, then ^_Lθ(x; h) = Lθ(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

_Lθ at this point, and hence^_Lθ(x; h) = ∇_Lθ(x)h for all h ∈ Rⁿ. In case (b) we have x2= 0. It follows from (3.9) that

^_Lθ(x; h) = )

h₁−x₂^Th₂

x2ctanθ

*

+

u¹_x+ )

h₁+ x₂^Th₂

x2 tanθ

*

u²_x+ Mx₂h

= (1 + ctan²θ) (u¹x)^Th

+u¹_x+ h

+

⎛

⎜⎜

⎜⎜

⎝

tan²θ 1+ tan²θ

)

−h1+x₂^Th₂

x2ctanθ

*

 tanθ 1+ tan²θ

)

h₁+x₂^Th₂

x2 tanθ

*

−x₂^Th₂

x2



¯x2

⎞

⎟⎟

⎟⎟

⎠

= (1 + ctan²θ) (u¹_x)^Th

+u¹_x+ h

−

⎛

⎜⎜

⎜⎝

tan²θ 1+ tan²θ

)

h₁− x₂^Th₂

x2ctanθ

*

tan²θ 1+ tan²θ



h1− ¯x2^Th2ctanθ 

− ctanθ ¯x2



⎞

⎟⎟

⎟⎠

= (1 + ctan²θ) (u¹x)^Th

+u¹_x+ h

− tan²θ

1+ tan²θ(1 + ctan²θ)

(u¹_x)^Th  1 0 0 ctanθ

  1

− ¯x2

= (1 + ctan²θ) (u¹x)^Th

+u¹_x+ h − (1 + ctan²θ) (u¹x)^Th

u¹_x

= h − (1 + ctan²θ) (u¹x)^Th

−u¹_x,

where the representations t = (t)₊+ (t)₋for all t∈ IR are used together with tanθ

1+ tan²θ )

h1+x₂^Th₂

x2 tanθ

*

−x₂^Th₂

x2

= tanθ 1+ tan²θ

)

h₁+x₂^Th2

x2 tanθ −1+ tan²θ tanθ

x₂^Th2

x2

*

= tanθ 1+ tan²θ

)

h₁−x₂^Th₂

x2ctanθ

* , 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

^_Lθ(x; h) = (h1− h2ctanθ)₊u¹_h+ (h1+ h2 tan θ)₊u²_h= _Lθ(h),

which therefore complete the proof of the lemma. 

The following theorem uses the previous considerations to establish the strongly semis- moothness property of the projection operator_Lθ. It has been well recognized the impor- tance 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 [31,32].

Th e o r e m 3.3 (strong semismoothness of the projection operator) The projection oper- ator_Lθ in (3.5) is strongly semismooth overRⁿ.

Proof The proof is inspired by [33, Proposition 4.5]. Note first that the directional differ- entiability of the Lipschitz continuous projection operatorLθ from Lemma3.2, and thus it remains to show that representation (2.14) holds for it withρ = 1.

To verify our claim, deduce from the proof of Lemma3.1that

_Lθ(x) =

⎧⎪

⎪⎨

⎪⎪

⎩

x if λ1(x) ≥ 0 and λ2(x) ≥ 0,

1

1+ tan²θ(x1+ x2 tan θ) 1 0

0 tanθ  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: x2= 0 and x2= 0.

Case 1 When x2 = 0, we can easily observe that in all the formulas from (3.10) corresponding to this case, the projection operator_Lθ is a piecewiseC²-smooth mapping whose strong semismoothness is well known in optimization.[34] It verifies the claim in this case.

Case 2 For x2= 0, suppose first that x1= 0. Then λi(x) = x1= 0, i = 1, 2. Since λi(y) is Lipschitz continuous by [13, Lemma 2.1], we get from (3.10) that_Lθ(y) is either 0 or y when y is in a neighbourhood of x. Thus_Lθ is surely strongly semismooth at x in this setting. In the remaining setting of x1= 0 we have x = 0. Note that the projection operatorLθ is obviously positively homogeneous, i.e.Lθ(tz) = tLθ(z) for z ∈ Rⁿ

and t ≥ 0. This implies that ^_Lθ(h; h) = _Lθ(h) and ^_Lθ(0; h) = _Lθ(h). Hence

∇_Lθ(h^)(h^) = _Lθ(h^) as h^ ∈ D_Lθ. Since D_Lθ is a dense subset ofRⁿ, for any fixed h= 0 and V ∈ ∂B_Lθ(h), there exists h^ ∈ D_Lθ such thath^− h ≤ h²and

V − ∇Lθ(h^) ≤ h. Hence for h sufficiently close to 0 we have

V h − ^_Lθ(0; h) = V h − ∇_Lθ(h^)(h^) + _Lθ(h^) − ^_Lθ(0; h)

= V h − ∇Lθ(h^)(h) + ∇Lθ(h^)(h) − ∇Lθ(h^)(h^) +_Lθ(h^) − _Lθ(h)

≤ V − ∇Lθ(h^)h + ∇Lθ(h^)h − h^ + h − h^

≤ (r + 2)h²,

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

h→0

V h − ^_Lθ(0; h)

h² < ∞, i.e. (3.11)

V h− ^_Lθ(0; h) = O(h²) for all V ∈ ∂B_Lθ(h).

Now let us show that V h− ^_Lθ(0; h) = O(h²) for any V ∈ co ∂BLθ(h), i.e. for any h_k → 0 and Vk ∈ co ∂BLθ(hk) we have Vkh_k − ^_Lθ(0; hk) = O(hk²). Since Vk ∈ co∂BLθ(hk), it follows from the Carathéodory theorem that there are V_kⁱ ∈ ∂BLθ(hk) andλⁱ_k ≥ 0 for i = 1, . . . , n + 1 such that

V_k =

n+1

1

i=1

λⁱkV_kⁱ and

n+1

1

i=1

λⁱk = 1.

Since V_kⁱ ∈ ∂BLθ(hk), it follows from (3.11) that

lim sup

k→0

V_kⁱh_k− ^_Lθ(0; hk)

hk² ≤ L.

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

lim sup

k→0

Vkhk− ^_Lθ(0; hk)

hk² = lim sup

k→0

n2+1

i=1λⁱ_kV_kⁱh_k− ^_Lθ(0; hk)

hk²

≤ lim sup

k→0 n+1

1

i=1

λⁱ_kV_kⁱhk− ^_Lθ(0; hk)

hk²

≤

n+1

1

i=1

¯λⁱL = L.

Thus V h − ^_Lθ(0; h) = O(h²) for any V ∈ co ∂BLθ(h), i.e. Lθ is strongly

semismooth at 0. 

The next result, which easily follows from Lemma3.2, provides the calculation of the graphical derivative (2.5) for the projection operator onto the circular cone.

Pr o p o s it io n 3.4 (calculating the graphical derivative of the projection operator) For any x∈ Rⁿwith decomposition (3.1), the graphical derivative ofLθ(x) is calculated by

DLθ(x)(w) =

^_Lθ(x; w)

for any w ∈ IRⁿ. (3.12) Proof It follows from [30, formula 8(14)] that the graphical derivative of any closed graph operator, and hence ofLθ in particular, can be equivalently represented as

DLθ(x)(w) = Lim sup

w→wτ0

_Lθ(x + τw^) − _Lθ(x)

τ . (3.13)

By Lemma3.2the Lipschitzian mapping_Lθ is directionally differentiable at x. Thus the right-hand side of (3.13) reduces to^_Lθ(x; w), which justifies (3.12).  Based on the calculations provided in Lemmas3.1and3.2, we are now ready to establish precise formulas for computing the regular and limiting coderivatives of the projection operator_Lθonto the general circular cone (1.1). We proceed similarly to the proofs of the main results of the paper [35] 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 Remark4.7.

Th e o r e m 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 operatorLθ(x) onto the circular cone (1.1) is calculated as follows:

(a) Ifλ1(x)λ2(x) = 0, then D^∗_Lθ(x)(y^∗) =

∇_Lθ(x)y^∗ . (b) Ifλ1(x) = 0 and λ2(x) > 0, then

D ^∗_Lθ(x)(y^∗) =

x^∗∈ Rⁿy^∗− x^∗∈ IR₊u¹_x, x^∗, u¹_x ≥ 0 . (c) Ifλ1(x) < 0 and λ2(x) = 0, then

D ^∗Lθ(x)(y^∗) =

x^∗∈ Rⁿx^∗∈ IR+u²_x, y^∗− x^∗, u²x ≥ 0 . (d) Ifλ1(x) = λ2(x) = 0, then

D^∗Lθ(x)(y^∗) =

x^∗∈ Rⁿx^∗∈ Lθ, y^∗− x^∗∈ L^π_2−θ .

Proof Due to the well-known duality between the regular coderivative and the graphical derivative of a mapping (see [30]) and by the established directional differentiability of the projection operator onto the circular cone, we have the equivalence

x^∗∈ D^∗Lθ(x)(y^∗) ⇐⇒ x^∗, h ≤3

y^∗, ^_Lθ(x; h)4

for all h ∈ IRⁿ. (3.14)

Employing (3.14) and the calculation of the directional derivative of_Lθ in Lemma3.2 allows us to derive the claimed formulas for the regular coderivative ofLθ in all the cases (a)–(d) of the theorem.

In case (a), pick any x^∗ ∈ D^∗Lθ(x)(y^∗) and get by using Lemma3.2(a) and duality (3.14) that

x^∗, h ≤ y^∗, ^_Lθ(x; h) ⇐⇒ x^∗, h ≤ y^∗, ∇Lθ(x)h

⇐⇒ x^∗− ∇_Lθ(x)y^∗, h ≤ 0,

where the last step comes from the fact that the operator∇_Lθ is self-adjoint by (3.6).

Hence, we have x^∗= ∇_Lθ(x)y^∗, i.e. D^∗_Lθ(x)(y^∗) = {∇_Lθ(x)y^∗}.

In case (b) we employ Lemma3.2(b), which gives us together with (3.14) that

x^∗∈ D^∗_Lθ(x)(y^∗) ⇐⇒ x^∗, h ≤ 5

y^∗, h − (1 + ctan²θ) (u¹x)^Th

−u¹_x 6

⇐⇒ x^∗− y^∗, h + (1 + ctan²θ) 5

y^∗, (u¹x)^Th

−u¹_x 6

≤ 0

⇐⇒

 x^∗− y^∗, h ≤ 0, (u¹x)^Th≥ 0

x^∗− y^∗, h + (1 + ctan²θ)(u¹_x)^Th(y^∗)^Tu¹_x ≤ 0, (u¹_x)^Th≤ 0

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

(y^∗)^Tu¹_x

u¹_x = βu¹_x

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

(y^∗)^Tu¹_x

u¹_x= (α + β)u¹_x

⇐⇒ ∃ α ≥ 0 such that y^∗− x^∗= αu¹x

and (1 + ctan²θ) y^∗, u¹_x ≥ α (3.15)

⇐⇒ ∃ α ≥ 0 such that y^∗− x^∗= αu¹x and x^∗, u¹x ≥ 0. (3.16) The last equivalence above comes from the following arguments: if (3.16) holds, then

(1 + ctan²θ) y^∗, u¹x = (1 + ctan²θ) x^∗+ αu¹x, u¹x ≥ α(1 + ctan²θ)u¹x²= α;

conversely, the validity of (3.15) implies that x^∗, u¹x = y^∗, u¹x − α u¹x, u¹x ≥ 1

1+ ctan²θα − 1

1+ ctan²θα = 0.

In case (c) we have the equivalencies by using Lemma3.2(c) and duality (3.14):

x^∗∈ D^∗Lθ(x)(y^∗) ⇐⇒ x^∗, h ≤ 5

y^∗, (1 + tan²θ) (u²x)^Th

+u²_x 6

⇐⇒

' x7 ^∗, h ≤ 0, (u²x)^Th≤ 0, x^∗− (1 + tan²θ)

(y^∗)^Tu²_x

u²_x, h8

≤ 0, (u²x)^Th≥ 0

⇐⇒ ∃ α ≥ 0 such that x^∗= αu²x and (1 + tan²θ)(y^∗)^Tu²_x≥ α

⇐⇒ ∃ α ≥ 0 such that x^∗= αu²x and y^∗− x^∗, u²x ≥ 0, which readily justify the claimed result in this case.