Nonsingularity Conditions for FB System of Reformulating Nonlinear Second-Order Cone Programming

Abstract

This paper is a counterpart of Bi et al., 2011. For a locally optimal solution to the nonlinear second-order cone programming (SOCP), specifically, under Robinson’s constraint qualification, we establish the equivalence among the following three conditions: the nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for the Karush-Kuhn-Tucker conditions, the strong second-order sufficient condition and constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

1. Introduction

The nonlinear second-order cone programming (SOCP) problem can be stated as min_{𝜁∈ℝ𝑛}𝑓(𝜁)(𝜁)𝑔(𝜁)s.t.ℎ=0,∈𝒦,(1) where 𝑓∶ℝ^𝑛→ℝ, ℎ∶ℝ^𝑛→ℝ^𝑚, and 𝑔∶ℝ^𝑛→ℝ^𝑛 are given twice continuously differentiable functions, and 𝒦 is the Cartesian product of some second-order cones, that is, 𝒦∶=𝒦^𝑛1×𝒦^𝑛2×⋯×𝒦^𝑛𝑟,(2) with 𝑛₁+⋯+𝑛_𝑟=𝑛 and 𝒦^𝑛𝑗 being the second-order cone (SOC) in ℝ^𝑛𝑗 defined by 𝒦^𝑛𝑗∶=𝑥_𝑗1,𝑥_𝑗2∈ℝ×ℝ^𝑛𝑗−1∣𝑥_𝑗1≥𝑥_𝑗2.(3) By introducing a slack variable to the second constraint, the SOCP (1) is equivalent to min_{𝜁,𝑥∈ℝ𝑛}𝑓(𝜁)(𝜁)𝑔(𝜁)s.t.ℎ=0,−𝑥=0,𝑥∈𝒦.(4) In this paper, we will concentrate on this equivalent formulation of problem (1).

Let 𝐿∶ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑛×𝒦→ℝ be the Lagrangian function of problem (4) 𝐿𝜁,𝑥,𝜇,𝑠,𝑦∶=𝑓(𝜁)+⟨𝜇,ℎ(𝜁)⟩+⟨𝑔(𝜁)−𝑥,𝑠⟩−⟨𝑥,𝑦⟩,(5) and denote by 𝒩_𝒦(𝑥) the normal cone of 𝒦 at 𝑥 in the sense of convex analysis [1]: 𝒩_𝒦(𝑥)=𝑑∈ℝ^𝑛∶⟨𝑑,𝑧−𝑥⟩≤0∀𝑧∈𝒦,if𝑥∈𝒦,∅,if𝑥∉𝒦.(6) Then the Karush-Kuhn-Tucker (KKT) conditions for (4) take the following form: 𝒥_𝜁,𝑥𝐿(𝜁)𝑔(𝜁)𝜁,𝑥,𝜇,𝑠,𝑦=0,ℎ=0,−𝑥=0,−𝑦∈𝒩_𝒦(𝑥),(7) where 𝒥_𝜁,𝑥𝐿(𝜁,𝑥,𝜇,𝑠,𝑦) is the derivative of 𝐿 at (𝜁,𝑥,𝜇,𝑠,𝑦) with respect to (𝜁,𝑥). Recall that 𝜙^soc is an SOC complementarity function associated with the cone 𝒦 if 𝜙^soc𝑥,𝑦=0⟺𝑥∈𝒦,𝑦∈𝒦,𝑥,𝑦=0⟺−𝑦∈𝒩_𝒦(𝑥).(8) With an SOC complementarity function 𝜙^soc associated with 𝒦, we may reformulate the KKT optimality conditions in (7) as the following nonsmooth system: 𝐸𝒥𝜁,𝑥,𝜇,𝑠,𝑦∶=^𝜁,𝑥𝐿ℎ(𝜁)𝑔(𝜁)𝜙𝜁,𝑥,𝜇,𝑠,𝑦−𝑥_soc𝑥,𝑦=0.(9)

The most popular SOC complementarity functions include the vector-valued natural residual (NR) function and Fischer-Burmeister (FB) function, respectively, defined as 𝜙_NR𝑥,𝑦∶=𝑥−Π_𝒦𝑥−𝑦,∀𝑥,𝑦∈ℝ^𝑛,𝜙_FB𝑥,𝑦∶=𝑥+𝑦−𝑥²+𝑦²,∀𝑥,𝑦∈ℝ^𝑛,(10) where Π_𝒦(⋅) is the projection operator onto the closed convex cone 𝒦, 𝑥²=𝑥∘𝑥 means the Jordan product of 𝑥 and itself, and √𝑥 denotes the unique square root of 𝑥∈𝒦. It turns out that the FB SOC complementarity function 𝜙_FB enjoys almost all favorable properties of the NR SOC complementarity function 𝜙_NR (see [2]). Also, the squared norm of 𝜙_FB induces a continuously differentiable merit function with globally Lipschitz continuous derivative [3, 4]. This greatly facilitates the globalization of the semismooth Newton method [5, 6] for solving the FB nonsmooth system of KKT conditions: 𝐸_FB𝒥𝜁,𝑥,𝜇,𝑠,𝑦∶=^𝜁,𝑥𝐿ℎ(𝜁)𝑔(𝜁)𝜙𝜁,𝑥,𝜇,𝑠,𝑦−𝑥_FB𝑥,𝑦=0.(11)

Recently, with the help of [7, Theorem 30] and [8, Lemma 11], Wang and Zhang [9] gave a characterization for the strong regularity of the KKT point of the SOCP (1) via the nonsingularity study of Clarke's Jacobian of the NR nonsmooth system 𝐸_NR𝒥𝜁,𝑥,𝜇,𝑠,𝑦∶=^𝜁,𝑥𝐿ℎ(𝜁)𝑔(𝜁)𝜙𝜁,𝑥,𝜇,𝑠,𝑦−𝑥_NR𝑥,𝑦=0.(12) They showed that the strong regularity of the KKT point, the nonsingularity of Clarke's Jacobian of 𝐸_NR at the KKT point, and the strong second-order sufficient condition and constraint nondegeneracy [7] are all equivalent. These nonsingularity conditions are better structured than those of [10] for the nonsingularity of the 𝐵-subdifferential of the NR system. Then, it is natural to ask the following: is it possible to obtain a characterization for the strong regularity of the KKT point by studying the nonsingularity of Clarke's Jacobian of 𝐸_FB. Note that up till now one even does not know whether the 𝐵-subdifferential of the FB system is nonsingular or not without the strict complementarity assumption.

In this work, for a locally optimal solution to the nonlinear SOCP (4), under Robinson's constraint qualification, we show that the strong second-order sufficient condition and constraint nondegeneracy introduced in [7], the nonsingularity of Clarke's Jacobian of 𝐸_FB at the KKT point, and the strong regularity of the KKT point are equivalent to each other. This, on the one hand, gives a new characterization for the strong regularity of the KKT point and, on the other hand, provides a mild condition to guarantee the quadratic convergence rate of the semismooth Newton method [5, 6] for the FB system. Note that parallel results are obtained recently for the FB system of the nonlinear semidefinite programming (see [11]); however, we do not duplicate them. As will be seen in Sections 3 and 4, the analysis techniques here are totally different from those in [11]. It seems hard to put them together in a unified framework under the Euclidean Jordan algebra. The main reason causing this is due to completely different analysis when dealing with the Clarke Jacobians associated with FB SOC complementarity function and FB semidefinite cone complementarity function.

Throughout this paper, 𝐼 denotes an identity matrix of appropriate dimension, ℝ^𝑛  (𝑛>1) denotes the space of 𝑛-dimensional real column vectors, and ℝ^𝑛1×⋯×ℝ^𝑛𝑟 is identified with ℝ^{𝑛1+⋯+𝑛𝑟}. Thus, (𝑥₁,…,𝑥_𝑟)∈ℝ^𝑛1×⋯×ℝ^𝑛𝑟 is viewed as a column vector in ℝ^{𝑛1+⋯+𝑛𝑟}. The notations 𝐢𝐧𝐭𝒦^𝑛, 𝐛𝐝𝒦^𝑛, and 𝐛𝐝⁺𝒦^𝑛 denote the interior, the boundary, and the boundary excluding the origin of 𝒦^𝑛, respectively. For any 𝑥∈ℝ^𝑛, we write 𝑥≽_𝒦^𝑛0 (resp., 𝑥≻_𝒦^𝑛0) if 𝑥∈𝒦^𝑛 (resp., 𝑥∈𝐢𝐧𝐭𝒦^𝑛). For any given real symmetric matrix 𝐴, we write 𝐴≽0 (resp., 𝐴≻0) if 𝐴 is positive semidefinite (resp., positive definite). In addition, 𝒥_𝜔𝑓(𝜔) and 𝒥²_𝜔𝜔𝑓(𝜔) denote the derivative and the second-order derivative, respectively, of a twice differentiable function 𝑓 with respect to the variable 𝜔.

2. Preliminary Results

First we recall from [12] the definition of Jordan product and spectral factorization.

Definition 1. The Jordan product of 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑛−1 is given by 𝑥∘𝑦∶=𝑥,𝑦,𝑥₁𝑦₂+𝑦₁𝑥₂.(13)

Unlike scalar or matrix multiplication, the Jordan product is not associative in general. The identity element under this product is 𝑒∶=(1,0,…,0)^𝑇∈ℝ^𝑛, that is, 𝑒∘𝑥=𝑥 for all 𝑥∈ℝ^𝑛. For each 𝑥=(𝑥₁,𝑥₂)∈ℝ×ℝ^𝑛−1, we define the associated arrow matrix by 𝐿_𝑥𝑥∶=¹𝑥^𝑇₂𝑥₂𝑥₁𝐼.(14) Then it is easy to verify that 𝐿_𝑥𝑦=𝑥∘𝑦 for any 𝑥,𝑦∈ℝ^𝑛. Recall that each 𝑥=(𝑥₁,𝑥₂)∈ℝ×ℝ^𝑛−1 admits a spectral factorization, associated with 𝒦^𝑛, of the form 𝑥=𝜆₁(𝑥)𝑢_𝑥⁽¹⁾+𝜆₂(𝑥)𝑢_𝑥⁽²⁾,(15) where 𝜆₁(𝑥),𝜆₂(𝑥)∈ℝ and 𝑢_𝑥⁽¹⁾,𝑢_𝑥⁽²⁾∈ℝ^𝑛 are the spectral values and the associated spectral vectors of 𝑥, respectively, with respect to the Jordan product, defined by 𝜆_𝑖(𝑥)∶=𝑥₁+(−1)^𝑖𝑥₂,𝑢_𝑥^(𝑖)1∶=21(−1)^𝑖𝑥₂,for𝑖=1,2,(16) with 𝑥₂=𝑥₂/‖𝑥₂‖ if 𝑥₂≠0 and otherwise being any vector in ℝ^𝑛−1 satisfying ‖𝑥₂‖=1.

Definition 2. The determinant of a vector 𝑥∈ℝ^𝑛 is defined as det(𝑥)∶=𝜆₁(𝑥)𝜆₂(𝑥), and a vector 𝑥 is said to be invertible if its determinant det(𝑥) is nonzero.

By the formula of spectral factorization, it is easy to compute that the projection of 𝑥∈ℝ^𝑛 onto the closed convex cone 𝒦^𝑛, denoted by Π_𝒦^𝑛(𝑥), has the expression Π_𝒦^𝑛(𝑥)=max0,𝜆₁(𝑥)𝑢_𝑥⁽¹⁾+max0,𝜆₂(𝑥)𝑢_𝑥⁽²⁾.(17) Define |𝑥|∶=2Π_𝒦^𝑛(𝑥)−𝑥. Then, using the expression of Π_𝒦^𝑛(𝑥), it follows that |𝑥|=𝜆₁(𝑥)𝑢_𝑥⁽¹⁾+𝜆₂(𝑥)𝑢_𝑥⁽²⁾.(18)

The spectral factorization of the vectors 𝑥, 𝑥², √𝑥 and the matrix 𝐿_𝑥 have various interesting properties (see [13]). We list several properties that we will use later.

Property 3. For any 𝑥=(𝑥₁,𝑥₂)∈ℝ×ℝ^𝑛−1 with spectral factorization (15), we have the following.(a)𝑥²=𝜆²₁(𝑥)𝑢_𝑥⁽¹⁾+𝜆²₂(𝑥)𝑢_𝑥⁽²⁾∈𝒦^𝑛.(b)If 𝑥∈𝒦^𝑛, then 0≤𝜆₁(𝑥)≤𝜆₂(𝑥) and √𝑥=𝜆₁(𝑥)𝑢_𝑥⁽¹⁾+𝜆₂(𝑥)𝑢_𝑥⁽²⁾.(c)If 𝑥∈𝐢𝐧𝐭𝒦^𝑛, then 0<𝜆₁(𝑥)≤𝜆₂(𝑥) and 𝐿_𝑥 is invertible with 𝐿_𝑥⁻¹=1(𝑥)𝑥det¹−𝑥^𝑇₂−𝑥₂det(𝑥)𝑥₁𝑥𝐼+²𝑥^𝑇₂𝑥₁.(19)(d)𝐿_𝑥≽0 (resp., 𝐿_𝑥≻0) if and only if 𝑥∈𝒦^𝑛 (resp., 𝑥∈𝐢𝐧𝐭𝒦^𝑛).

The following lemma states a result for the arrow matrices associated with 𝑥,𝑦∈ℝ^𝑛 and 𝑧≽_𝒦^𝑛𝑥²+𝑦², which will be used in the next section to characterize an important property for the elements of Clarke's Jacobian of 𝜙_FB at a general point.

Lemma 4. For any given 𝑥,𝑦∈ℝ^𝑛 and 𝑧≻_𝒦^𝑛0, if 𝑧²≽_𝒦^𝑛𝑥²+𝑦², then 𝐿_𝑧⁻¹𝐿_𝑥𝐿_𝑧⁻¹𝐿_𝑦2≤1,(20) where ‖𝐴‖₂ means the spectral norm of a real matrix 𝐴. Consequently, it holds that 𝐿_𝑧⁻¹𝐿_𝑥Δ𝑢+𝐿_𝑧⁻¹𝐿_𝑦Δ𝑣≤‖Δ𝑢‖²+‖Δ𝑣‖²,∀Δ𝑢,Δ𝑣∈ℝ^𝑛.(21)

Proof. Let 𝐴=[𝐿_𝑧⁻¹𝐿_𝑥𝐿_𝑧⁻¹𝐿_𝑦]. From [13, Proposition 3.4], it follows that 𝐴𝐴^𝑇=𝐿_𝑧⁻¹𝐿²_𝑥+𝐿²_𝑦𝐿_𝑧⁻¹⪯𝐿_𝑧⁻¹𝐿²_𝑧𝐿_𝑧⁻¹=𝐼.(22) This shows that ‖𝐴‖₂≤1, and the first part follows. Note that, for any 𝜉∈ℝ^2𝑛, 𝐴𝜉²=𝜉^𝑇𝐴^𝑇𝐴𝜉≤𝜆_max𝐴^𝑇𝐴𝜉²≤𝜉².(23) By letting 𝜉=(Δ𝑢,Δ𝑣)∈ℝ^𝑛×ℝ^𝑛, we immediately obtain the second part.

The following two lemmas state the properties of 𝑥,𝑦 with 𝑥²+𝑦²∈𝐛𝐝𝒦^𝑛 which are often used in the subsequent sections. The proof of Lemma 5 is given in [3, Lemma 2].

Lemma 5. For any 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑛−1 with 𝑥²+𝑦²∈𝐛𝐝𝒦^𝑛, one has 𝑥²₁=𝑥₂²,𝑦²₁=𝑦₂²,𝑥₁𝑦₁=𝑥^𝑇₂𝑦₂,𝑥₁𝑦₂=𝑦₁𝑥₂.(24)

Lemma 6. For any 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑛−1, let 𝑤=(𝑤₁,𝑤₂)∶=𝑥²+𝑦². (a)If 𝑤∈𝐛𝐝𝒦^𝑛, then for any 𝑔=(𝑔₁,𝑔₂),ℎ=(ℎ₁,ℎ₂)∈ℝ×ℝ^𝑛−1, it holds that 𝑥₁𝑥₂+𝑦₁𝑦₂^𝑇𝑥₁𝑔₂+𝑔₁𝑥₂+𝑦₁ℎ₂+ℎ₁𝑦₂=𝑥²₁+𝑦²₁𝑥^𝑇𝑔+𝑦^𝑇ℎ.(25)(b)If 𝑤∈𝐛𝐝⁺𝒦^𝑛, then the following four equalities hold 𝑥₁𝑤₂𝑤₂=𝑥₂,𝑥^𝑇₂𝑤₂𝑤₂=𝑥₁,𝑦₁𝑤₂𝑤₂=𝑦₂,𝑦^𝑇₂𝑤₂𝑤₂=𝑦₁;(26) and consequently the expression of 𝜙_FB(𝑥,𝑦) can be simplified as 𝜙_FB=𝑥𝑥,𝑦¹+𝑦₁−𝑥²₁+𝑦²₁𝑥₂+𝑦₂−𝑥₁𝑥₂+𝑦₁𝑦₂𝑥²₁+𝑦²₁.(27)

Proof. (a) The result is direct by the equalities of Lemma 5 since 𝑥²+𝑦²∈𝐛𝐝𝒦^𝑛.
(b) Since 𝑤∈𝐛𝐝⁺𝒦^𝑛, we must have 𝑤₂=2(𝑥₁𝑥₂+𝑦₁𝑦₂)≠0. Using Lemma 5, 𝑤₂=2(𝑥₁𝑥₂+𝑦₁𝑦₂) and ‖𝑤₂‖=𝑤₁=2(𝑥²₁+𝑦²₁), we easily obtain the first part. Note that 𝜙_FB(𝑥,𝑦)=(𝑥+𝑦)−√𝑤. Using Property 3(b) and Lemma 5 yields (27).

When 𝑥,𝑦∈𝐛𝐝𝒦^𝑛 satisfies the complementary condition, we have the following result.

Lemma 7. For any given 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑛−1, if 𝑥,𝑦∈𝐛𝐝𝒦^𝑛 and ⟨𝑥,𝑦⟩=0, then there exists a constant 𝛼>0 such that 𝑥₁=𝛼𝑦₁ and 𝑥₂=−𝛼𝑦₂.

Proof. Since 𝑥,𝑦∈𝐛𝐝𝒦^𝑛, we have that 𝑥₁=‖𝑥₂‖ and 𝑦₁=‖𝑦₂‖, and consequently, 0=⟨𝑥,𝑦⟩=𝑥₁𝑦₁+𝑥^𝑇₂𝑦₂=𝑥₂𝑦₂+𝑥^𝑇₂𝑦₂.(28) This means that there exists 𝛼>0 such that 𝑥₂=−𝛼𝑦₂, and then 𝑥₁=𝛼𝑦₁.

Next we recall from [14] the strong regularity for a solution of generalized equation 0∈𝜙(𝑧)+𝒩_𝐷(𝑧),(29) where 𝜙 is a continuously differentiable mapping from a finite dimensional real vector space ℤ to itself, 𝐷 is a closed convex set in ℤ, and 𝒩_𝐷(𝑧) is the normal cone of 𝐷 at 𝑧. As will be shown in Section 4, the KKT condition (7) can be written in the form of (29).

Definition 8. We say that 𝑧 is a strongly regular solution of the generalized equation (29) if there exist neighborhood ℬ of the origin 0∈ℤ and 𝒱 of 𝑧 such that for every 𝛿∈ℬ, the linearized generalized equation 𝛿∈𝜙(𝑧)+𝒥_𝑧𝜙(𝑧)(𝑧−𝑧)+𝒩_𝐷(𝑧) has a unique solution in 𝒱, denoted by 𝑧_𝒱(𝛿), and the mapping 𝑧_𝒱∶ℬ→𝒱 is Lipschitz continuous.

To close this section, we recall from [15] Clarke's (generalized) Jacobian of a locally Lipschitz mapping. Let 𝑆⊂ℝ^𝑛 be an open set and Ξ∶𝑆→ℝ^𝑛 a locally Lipschitz continuous function on 𝑆. By Rademacher's theorem, Ξ is almost everywhere 𝐹(réchet)-differentiable in 𝑆. We denote by 𝑆_Ξ the set of points in 𝑆 where Ξ is 𝐹-differentiable. Then Clarke's Jacobian of Ξ at 𝑦 is defined by 𝜕Ξ(𝑦)∶=conv{𝜕_𝐵Ξ(𝑦)}, where “conv” means the convex hull, and 𝐵-subdifferential 𝜕_𝐵Ξ(𝑦), a name coined in [16], has the form 𝜕_𝐵Ξ𝑦∶=𝑉∶𝑉=lim_𝑘→∞𝒥_𝑦Ξ𝑦^𝑘,𝑦^𝑘→𝑦,𝑦^𝑘∈𝑆_Ξ.(30) For the concept of (strong) semismoothness, please refer to the literature [5, 6].

Unless otherwise stated, in the rest of this paper, for any 𝑥∈ℝ^𝑛  (𝑛>1), we write 𝑥=(𝑥₁,𝑥₂), where 𝑥₁ is the first component of 𝑥 and 𝑥₂ is a column vector consisting of the remaining 𝑛−1 entries of 𝑥. For any 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑛−1, let 𝑤=𝑤𝑥,𝑦∶=𝑥²+𝑦²,𝑤₂𝑤∶=²𝑤₂,if𝑤₂≠0,𝑧=𝑧𝑥,𝑦=𝑤𝑥,𝑦.(31)

3. Directional Derivative and 𝐵-Subdifferential

The function 𝜙_FB is directionally differentiable everywhere by [2, Corollary 3.3]. But, to the best of our knowledge, the expression of its directional derivative is not given in the literature. In this section, we derive its expression and then prove that the 𝐵-subdifferential of 𝜙_FB at a general point coincides with that of its directional derivative function at the origin. Throughout this section, we assume that 𝒦=𝒦^𝑛.

Proposition 9. For any given 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑛−1, the directional derivative 𝜙^′_FB((𝑥,𝑦);(𝑔,ℎ)) of 𝜙_FB at (𝑥,𝑦) with the direction (𝑔,ℎ) has the following form. (a)If (𝑥,𝑦)=(0,0), then 𝜙^′_FB((𝑥,𝑦);(𝑔,ℎ))=𝜙_FB(𝑔,ℎ).(b)If 𝑥²+𝑦²∈𝐢𝐧𝐭𝒦^𝑛, then 𝜙^′_FB((𝑥,𝑦);(𝑔,ℎ))=(𝐼−𝐿_𝑧⁻¹𝐿_𝑥)𝑔+(𝐼−𝐿_𝑧⁻¹𝐿_𝑦)ℎ.(c)If 𝑥²+𝑦²∈𝐛𝐝⁺𝒦^𝑛, then 𝜙^′_FB;=−𝜑𝑥,𝑦𝑔,ℎ𝑔+ℎ𝑔,ℎ21−𝑤₂+𝑥^𝑇₂𝑔₂+𝑦^𝑇₂ℎ₂2𝑥²₁+𝑦²₁0𝑤₂−12𝑥²₁+𝑦²₁𝑥^𝑇𝑔+𝑦^𝑇ℎ2𝑥₁𝑔₂+𝑔₁𝑥₂+2𝑦₁ℎ₂+ℎ₁𝑦₂,(32) where 𝑔=(𝑔₁,𝑔₂),ℎ=(ℎ₁,ℎ₂)∈ℝ×ℝ^𝑛−1, and 𝜑∶ℝ^𝑛×ℝ^𝑛→ℝ is defined by 𝜑𝑔,ℎ∶=𝑥₁𝑔₁−𝑥^𝑇₂𝑔₂+𝑦₁ℎ₁−𝑦^𝑇₂ℎ₂²+𝑥₁ℎ₂−ℎ₁𝑥₂+𝑔₁𝑦₂−𝑦₁𝑔₂²𝑥²₁+𝑦²₁.(33)

Proof. Part (a) is immediate by noting that 𝜙_FB is a positively homogeneous function. Part (b) is due to [13, Proposition 5.2]. We next prove part (c) by two subcases as shown in the following. In the rest of proof, we let 𝜆₁,𝜆₂ with 𝜆₁≤𝜆₂ denote the spectral values of 𝑤. Since 𝑤=𝑥²+𝑦²∈𝐛𝐝⁺𝒦^𝑛, we have 𝑤₂≠0, and from Lemma 6(b) it follows that 𝑤₁=𝑤₂=2𝑥₁𝑥₂+𝑦₁𝑦₂=2𝑥²₁𝑤₂+𝑦²₁𝑤₂=2𝑥²₁+𝑦²₁,𝜆₁=𝑤₁−𝑤₂=0,𝜆₂=𝑤₁+𝑤₂=4𝑥²₁+𝑦²₁.(34)
(c.1): (𝑥+𝑡𝑔)²+(𝑦+𝑡ℎ)²∈𝐛𝐝⁺𝒦^𝑛 for sufficiently small 𝑡>0. In this case, from Lemma 6(b), we know that 𝜙_FB(𝑥+𝑡𝑔,𝑦+𝑡ℎ) has the following expression: 𝑥₁+𝑦₁+𝑡𝑔₁+ℎ₁−𝑥₁+𝑡𝑔₁²+𝑦₁+𝑡ℎ₁²𝑥₂+𝑦₂+𝑡𝑔₂+ℎ₂−𝑥₁+𝑡𝑔₁𝑥₂+𝑡𝑔₂+𝑦₁+𝑡ℎ₁𝑦₂+𝑡ℎ₂𝑥₁+𝑡𝑔₁²+𝑦₁+𝑡ℎ₁².(35) Let [𝜙_FB(𝑥,𝑦)]₁ be the first element of 𝜙_FB(𝑥,𝑦) and [𝜙_FB(𝑥,𝑦)]₂ the vector consisting of the rest 𝑛−1 components of 𝜙_FB(𝑥,𝑦). By the above expression of 𝜙_FB(𝑥+𝑡𝑔,𝑦+𝑡ℎ), lim_𝑡↓0𝜙_FB𝑥+𝑡𝑔,𝑦+𝑡ℎ₁−𝜙_FB𝑥,𝑦₁𝑡=𝑔₁+ℎ₁−lim_𝑡↓0𝑥₁+𝑡𝑔₁²+𝑦₁+𝑡ℎ₁²−𝑥²₁+𝑦²₁𝑡=𝑔₁+ℎ₁−𝑥₁𝑔₁+𝑦₁ℎ₁𝑥²₁+𝑦²₁,lim_𝑡↓0𝜙_FB𝑥+𝑡𝑔,𝑦+𝑡ℎ₂−𝜙_FB𝑥,𝑦₂𝑡=𝑔₂+ℎ₂−lim_𝑡↓0𝑥₁+𝑡𝑔₁𝑥₂+𝑡𝑔₂+𝑦₁+𝑡ℎ₁𝑦₂+𝑡ℎ₂𝑡𝑥₁+𝑡𝑔₁²+𝑦₁+𝑡ℎ₁²−𝑥₁𝑥₂+𝑦₁𝑦₂𝑡𝑥²₁+𝑦²₁=𝑔₂+ℎ₂−𝑔₁𝑥₂+𝑥₁𝑔₂+𝑦₁ℎ₂+ℎ₁𝑦₂𝑥²₁+𝑦²₁−lim_𝑡↓0𝑥₁𝑥₂+𝑦₁𝑦₂𝑡𝑥₁+𝑡𝑔₁²+𝑦₁+𝑡ℎ₁²−𝑥₁𝑥₂+𝑦₁𝑦₂𝑡𝑥²₁+𝑦²₁=𝑔₂+ℎ₂−𝑔₁𝑥₂+𝑥₁𝑔₂+𝑦₁ℎ₂+ℎ₁𝑦₂𝑥²₁+𝑦²₁+𝑥₁𝑥₂+𝑦₁𝑦₂𝑥₁𝑔₁+𝑦₁ℎ₁𝑥²₁+𝑦²₁𝑥²₁+𝑦²₁=𝑔₂+ℎ₂−𝑥₁𝑔₂+𝑦₁ℎ₂𝑥²₁+𝑦²₁,(36) where the last equality is using 𝑥₁𝑦₂=𝑦₁𝑥₂ by Lemma 5. The above two limits imply 𝜙^′_FB;=−𝑥𝑥,𝑦𝑔,ℎ𝑔+ℎ¹𝑥²₁+𝑦²₁𝑦𝑔−¹𝑥²₁+𝑦²₁ℎ.(37)
(c.2): (𝑥+𝑡𝑔)²+(𝑦+𝑡ℎ)²∈𝐢𝐧𝐭𝒦^𝑛 for sufficiently small 𝑡>0. Let 𝑢=(𝑢₁,𝑢₂)∶=(𝑥+𝑡𝑔)²+(𝑦+𝑡ℎ)² with the spectral values 𝜇₁,𝜇₂. An elementary calculation gives 𝑢₁=𝑥+𝑡𝑔²+𝑦+𝑡ℎ²=𝑤₁+2𝑡𝑥^𝑇𝑔+𝑦^𝑇ℎ+𝑡²𝑔²+‖ℎ‖²,𝑢(38)₂=2𝑥₁+𝑡𝑔₁𝑥₂+𝑡𝑔₂+2𝑦₁+𝑡ℎ₁𝑦₂+𝑡ℎ₂=𝑤₂+2𝑡𝑥₁𝑔₂+𝑔₁𝑥₂+𝑦₁ℎ₂+ℎ₁𝑦₂+2𝑡²𝑔₁𝑔₂+ℎ₁ℎ₂.(39) Also, since 𝑤₂≠0, applying the Taylor formula of ‖⋅‖ at 𝑤₂ and Lemma 6(a) yields 𝑢₂=𝑤₂+𝑤^𝑇₂𝑢₂−𝑤₂𝑤₂(𝑡)=𝑤+𝑜₂+2𝑡𝑥^𝑇𝑔+𝑦^𝑇ℎ+𝑜(𝑡).(40) Now using the definition of 𝜙_FB and noting that 𝜆₁=0 and 𝑤₂≠0, we have that 𝜙_FB𝑥+𝑡𝑔,𝑦+𝑡ℎ−𝜙_FB=−√𝑥,𝑦𝑥+𝑡𝑔+𝑦+𝑡ℎ𝑢−𝑥+𝑦+√𝑤−√=𝑡𝑔+ℎ𝜇₁+√𝜇₂−𝜆₂2√𝜇₂−√𝜇₁2𝑢₂𝑢₂−𝜆₂2𝑤₂𝑤₂,(41) which in turn implies that 𝜙^′_FB;=−𝑥,𝑦𝑔,ℎ𝑔+ℎlim_𝑡↓0√𝜇₁+√𝜇₂−𝜆₂2𝑡lim_𝑡↓0√𝜇₂−√𝜇₁𝑢2𝑡𝑢²₂−𝜆₂𝑤2𝑡𝑤²₂.(42) We first calculate lim_𝑡↓0((√𝜇₂−𝜆₂)/𝑡). Using (38) and (40), it is easy to see that 𝜇₂−𝜆₂=𝑢₁−𝑤₁+𝑢₂−𝑤₂=4𝑡𝑥^𝑇𝑔+𝑦^𝑇ℎ+𝑜(𝑡),(43) and consequently, lim_𝑡↓0√𝜇₂−𝜆₂𝑡=lim_𝑡↓0𝜇₂−𝜆₂𝑡⋅1√𝜇₂+𝜆₂=𝑥^𝑇𝑔+𝑦^𝑇ℎ2𝜆₂=𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑥²₁+𝑦²₁.(44) We next calculate lim_𝑡↓0(√𝜇₁/𝑡). Since 𝑤₁−‖𝑤₂‖=0, using (38)-(39) and Lemma 6(a), 𝜇₁=𝑢₁−𝑤₁−𝑢₂−𝑤₂=𝑢₁−𝑤₁−𝑢₂²−𝑤₂²𝑢₂+𝑤₂=2𝑡𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑤−4𝑡^𝑇2𝑥₁𝑔₂+𝑔₁𝑥₂+𝑦₁ℎ₂+ℎ₁𝑦₂𝑢₂+𝑤₂+𝑡²𝑔²+‖ℎ‖²−4𝑡²𝑔₁𝑥₂+𝑥₁𝑔₂+𝑦₁ℎ₂+ℎ₁𝑦₂²𝑢₂+𝑤₂−4𝑡²𝑤^𝑇₂𝑔₁𝑔₂+ℎ₁ℎ₂𝑢₂+𝑤₂+𝑜𝑡²=2𝑡𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑥−8𝑡²¹+𝑦²₁𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑢₂+𝑤₂+𝑡²𝑔²+‖ℎ‖²+𝑜𝑡²−4𝑡²𝑔₁𝑥₂+𝑥₁𝑔₂+𝑦₁ℎ₂+ℎ₁𝑦₂²𝑢₂+𝑤₂−8𝑡²𝑥₁𝑥₂+𝑦₁𝑦₂^𝑇𝑔₁𝑔₂+ℎ₁ℎ₂𝑢₂+𝑤₂.(45) Using ‖𝑤₂‖=2(𝑥²₁+𝑦²₁) and (40), we simplify the sum of the first two terms in (45) as 2𝑡𝑥^𝑇𝑔+𝑦^𝑇ℎ−4𝑡𝑤₂𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑢₂+𝑤₂=2𝑡𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑢₂−𝑤₂𝑢₂+𝑤₂=4𝑡²𝑥^𝑇𝑔+𝑦^𝑇ℎ²𝑢₂+𝑤₂+𝑜𝑡².(46) Then, from (45) and ‖𝑤₂‖=2(𝑥²₁+𝑦²₁), we obtain that lim_𝑡↓0𝜇₁𝑡²=𝑥²₁+𝑦²₁𝑔²+‖ℎ‖²−𝑔₁𝑥₂+𝑥₁𝑔₂+𝑦₁ℎ₂+ℎ₁𝑦₂²𝑥²₁+𝑦²₁+𝑥^𝑇𝑔+𝑦^𝑇ℎ²−2𝑥₁𝑥₂+𝑦₁𝑦₂^𝑇𝑔₁𝑔₂+ℎ₁ℎ₂𝑥²₁+𝑦²₁.(47) We next make simplification for the numerator of the right hand side of (47). Note that 𝑥²₁+𝑦²₁𝑔²+‖ℎ‖²−𝑔₁𝑥₂+𝑥₁𝑔₂+𝑦₁ℎ₂+ℎ₁𝑦₂²=𝑥²₁+𝑦²₁𝑔²+‖ℎ‖²−𝑔₁𝑥₂+𝑥₁𝑔₂²−𝑦₁ℎ₂+ℎ₁𝑦₂²−2𝑔₁𝑥₂+𝑥₁𝑔₂^𝑇𝑦₁ℎ₂+ℎ₁𝑦₂=𝑥²₁‖ℎ‖²+𝑦²₁𝑔²−2𝑥₁𝑔₁𝑥^𝑇₂𝑔₂−2𝑦₁ℎ₁𝑦^𝑇₂ℎ₂−2𝑔₁𝑥₂+𝑥₁𝑔₂^𝑇𝑦₁ℎ₂+ℎ₁𝑦₂,𝑥^𝑇𝑔+𝑦^𝑇ℎ²−2𝑥₁𝑥₂+𝑦₁𝑦₂^𝑇𝑔₁𝑔₂+ℎ₁ℎ₂=𝑥₁𝑔₁+𝑥^𝑇₂𝑔₂²+𝑦₁ℎ₁+𝑦^𝑇₂ℎ₂²+2𝑥^𝑇𝑔𝑦^𝑇ℎ−2𝑥₁𝑥₂+𝑦₁𝑦₂^𝑇𝑔₁𝑔₂+ℎ₁ℎ₂=𝑥₁𝑔₁²+𝑥^𝑇₂𝑔₂²+𝑦₁ℎ₁²+𝑦^𝑇₂ℎ₂²+2𝑥^𝑇𝑔𝑦^𝑇ℎ−2𝑥₁ℎ₁𝑥^𝑇₂ℎ₂−2𝑔₁𝑦₁𝑔^𝑇₂𝑦₂.(48) Therefore, adding the last two equalities and using Lemma 5 yield that 𝑥²₁+𝑦²₁𝑔²+‖ℎ‖²−𝑔₁𝑥₂+𝑥₁𝑔₂+𝑦₁ℎ₂+ℎ₁𝑦₂²+𝑥^𝑇𝑔+𝑦^𝑇ℎ²−2𝑥₁𝑥₂+𝑦₁𝑦₂^𝑇𝑔₁𝑔₂+ℎ₁ℎ₂=𝑥²₁‖ℎ‖²−2𝑥₁ℎ₁𝑥^𝑇₂ℎ₂+𝑦²₁𝑔²−2𝑔₁𝑦₁𝑔^𝑇₂𝑦₂+𝑥₁𝑔₁²+𝑥^𝑇₂𝑔₂²−2𝑥₁𝑔₁𝑥^𝑇₂𝑔₂+𝑦₁ℎ₁²+𝑦^𝑇₂ℎ₂²−2𝑦₁ℎ₁𝑦^𝑇₂ℎ₂+2𝑥^𝑇𝑔𝑦^𝑇ℎ−2𝑔₁𝑥₂+𝑥₁𝑔₂^𝑇𝑦₁ℎ₂+ℎ₁𝑦₂=𝑥₁ℎ₂−ℎ₁𝑥₂²+𝑔₁𝑦₂−𝑦₁𝑔₂²+𝑥₁𝑔₁−𝑥^𝑇₂𝑔₂²+𝑦₁ℎ₁−𝑦^𝑇₂ℎ₂²+2𝑔₁𝑥₁+𝑔^𝑇₂𝑥₂𝑦₁ℎ₁+𝑦^𝑇₂ℎ₂−2𝑔₁𝑥₂+𝑥₁𝑔₂^𝑇𝑦₁ℎ₂+ℎ₁𝑦₂=𝑥₁ℎ₂−ℎ₁𝑥₂²+𝑔₁𝑦₂−𝑦₁𝑔₂²+𝑥₁𝑔₁−𝑥^𝑇₂𝑔₂²+𝑦₁ℎ₁−𝑦^𝑇₂ℎ₂²+2𝑥₁ℎ₂−ℎ₁𝑥₂^𝑇𝑔₁𝑦₂−𝑔₂𝑦₁+2𝑥₁𝑔₁−𝑥^𝑇₂𝑔₂𝑦₁ℎ₁−𝑦^𝑇₂ℎ₂=𝑥₁ℎ₂−ℎ₁𝑥₂+𝑔₁𝑦₂−𝑦₁𝑔₂²+𝑥₁𝑔₁−𝑥^𝑇₂𝑔₂+𝑦₁ℎ₁−𝑦^𝑇₂ℎ₂².(49) Combining this equality with (47) and using the definition of 𝜑 in (33), we readily get lim_𝑡↓0√𝜇₁𝑡=𝜑𝑔,ℎ.(50) We next calculate lim_𝑡↓0[((√𝜇₂−√𝜇₁)/2𝑡)(𝑢₂/‖𝑢₂‖)−((𝜆₂/2𝑡)(𝑤₂/‖𝑤₂‖))]. To this end, we also need to take a look at ‖𝑤₂‖𝑢₂−‖𝑢₂‖𝑤₂. From (38)-(39) and (40), it follows that 𝑤₂𝑢₂−𝑢₂𝑤₂=2𝑡𝑤₂𝑥₁𝑔₂+𝑔₁𝑥₂+𝑦₁ℎ₂+ℎ₁𝑦₂−𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑤₂+𝑜(𝑡).(51) Together with (44) and (50), we have that lim_𝑡↓0√𝜇₂−√𝜇₁𝑢2𝑡𝑢²₂−𝜆₂𝑤2𝑡𝑤²₂=−lim_𝑡↓0√𝜇₁𝑢2𝑡𝑢²₂+lim_𝑡↓0√𝜇₂𝑢2𝑡𝑢²₂−𝜆₂𝑤2𝑡𝑤²₂=−lim_𝑡↓0√𝜇₁𝑢2𝑡𝑢²₂+lim_𝑡↓0√𝜇₂−𝜆₂𝑢2𝑡𝑢²₂+lim_𝑡↓0𝜆₂𝑤₂𝑢₂−𝑢₂𝑤₂2𝑡𝑢₂𝑤₂𝜑=−𝑔,ℎ2𝑤₂+𝑥₁𝑔₂+𝑔₁𝑥₂+𝑦₁ℎ₂+ℎ₁𝑦₂𝑥²₁+𝑦²₁−𝑥^𝑇𝑔+𝑦^𝑇ℎ2𝑥²₁+𝑦²₁𝑤₂𝜑=−𝑔,ℎ2𝑤₂+2𝑥₁𝑔₂+𝑔₁𝑥₂+2𝑦₁ℎ₂+ℎ₁𝑦₂2𝑥²₁+𝑦²₁−𝑥^𝑇₂𝑔₂+𝑦^𝑇₂ℎ₂2𝑥²₁+𝑦²₁𝑤₂,(52) where the last equality is using 𝑥₁𝑤₂=𝑥₂ and 𝑦₁𝑤₂=𝑦₂. Combining with (42), (44), and (50), a suitable rearrangement shows that 𝜙^′_FB((𝑥,𝑦);(𝑔,ℎ)) has the expression (32).
Finally, we show that, when (𝑥+𝑡𝑔)²+(𝑦+𝑡ℎ)²∈𝐛𝐝⁺𝒦^𝑛 for sufficiently small 𝑡>0, the formula in (32) reduces to the one in (37). Indeed, an elementary calculation yields 𝜆₁𝑥+𝑡𝑔²+𝑦+𝑡ℎ²=𝑥+𝑡𝑔²+𝑦+𝑡ℎ²²−4𝑥₁+𝑡𝑔₁𝑥₂+𝑡𝑔₂+𝑦₁+𝑡ℎ₁𝑦₂+𝑡ℎ₂²=4𝑡²𝜑𝑔,ℎ𝑥²₁+𝑦²₁+4𝑡³𝑥^𝑇𝑔+𝑦^𝑇ℎ𝑔²+‖ℎ‖²−8𝑡²𝑥₁𝑔₂+𝑔₁𝑥₂+𝑦₁ℎ₂+ℎ₁𝑦₂^𝑇𝑔₁𝑔₂+ℎ₁ℎ₂+𝑡⁴𝑔²+‖ℎ‖²²−2𝑔₁𝑔₂+ℎ₁ℎ₂²=4𝑡²𝜑𝑔,ℎ𝑥²₁+𝑦²₁+𝑜𝑡².(53) This implies that if (𝑥+𝑡𝑔)²+(𝑦+𝑡ℎ)²∈𝐛𝐝⁺𝒦^𝑛 for sufficiently small 𝑡>0, that is, 𝜆₁((𝑥+𝑡𝑔)²+(𝑦+𝑡ℎ)²)=0 for sufficiently small 𝑡>0, then 𝜑(𝑔,ℎ)=0, and hence 𝑥₁𝑔₁+𝑦₁ℎ₁−(𝑥^𝑇₂𝑔₂+𝑦^𝑇₂ℎ₂)=0. Thus, 𝜙^′_FB((𝑥,𝑦);(𝑔,ℎ)) in (32) can be simplified as +𝑥𝑔+ℎ¹𝑔₁+𝑦₁ℎ₁2𝑥²₁+𝑦²₁0𝑤₂−12𝑥²₁+𝑦²₁×2𝑥₁𝑔₁+2𝑦₁ℎ₁2𝑥₁𝑔₂+𝑔₁𝑥₂+2𝑦₁ℎ₂+ℎ₁𝑦₂=−𝑥𝑔+ℎ¹𝑥²₁+𝑦²₁𝑦𝑔−¹𝑥²₁+𝑦²₁ℎ,(54) where the equality is using 𝑔₁𝑥₂=𝑔₁𝑥₁𝑤₂,ℎ₁𝑦₂=ℎ₁𝑦₁𝑤₂. The proof is complete.

As a consequence of Proposition 9, we have the following sufficient and necessary characterizations for the (continuously) differentiable points of 𝜙_FB and 𝜗(𝑥)∶=|𝑥|.

Corollary 10. (𝑎) The function 𝜙_FB is (continuously) differentiable at (𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑛 if and only if 𝑧(𝑥,𝑦)∈𝐢𝐧𝐭𝒦^𝑛, where 𝑧(𝑥,𝑦) is defined in (31). Also, when 𝑧(𝑥,𝑦)∈𝐢𝐧𝐭𝒦^𝑛, one has 𝒥𝜙_FB𝑥,𝑦=𝐼−𝐿⁻¹_{𝑧(𝑥,𝑦)}𝐿_𝑥𝐼−𝐿⁻¹_{𝑧(𝑥,𝑦)}𝐿_𝑦.(55)
(𝑏) The function 𝜗(𝑥)∶=|𝑥| is (continuously) differentiable at 𝑥∈ℝ^𝑛 if and only if 𝑥 is invertible. Also, when 𝑥 is invertible, 𝒥𝜗(𝑥)=𝐿⁻¹_|𝑥|𝐿_𝑥.

Proof. (a) The “if” part is direct by [13, Proposition 5.2]. We next prove the “only if” part. If 𝜙_FB is differentiable at (𝑥,𝑦), then 𝜙^′_FB((𝑥,𝑦);(𝑔,ℎ)) is a linear function of (𝑔,ℎ), which by Proposition 9 holds only if 𝑧(𝑥,𝑦)∈𝐢𝐧𝐭𝒦^𝑛. The formula of 𝒥𝜙_FB(𝑥,𝑦) is given in [13].
(b) Since 𝜗(𝑥)=√𝑥², by part (a) 𝜗 is (continuously) differentiable at 𝑥∈ℝ^𝑛 if and only if |𝑥|∈𝐢𝐧𝐭𝒦^𝑛, which is equivalent to requiring that 𝑥 is invertible since |𝑥|∈𝒦^𝑛 always holds. When 𝑥 is invertible, the formula of 𝒥𝜗(𝑥) follows from part (a).

For any given 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑛−1 with 𝑥²+𝑦²∈𝐛𝐝⁺𝒦^𝑛, define 𝜃₁𝑔,ℎ∶=𝑥₁𝑔₁−𝑥^𝑇₂𝑔₂+𝑦₁ℎ₁−𝑦^𝑇₂ℎ₂,𝜃₂𝑔,ℎ∶=𝑥₁ℎ₂−ℎ₁𝑥₂+𝑔₁𝑦₂−𝑦₁𝑔₂(56) for any 𝑔=(𝑔₁,𝑔₂),ℎ=(ℎ₁,ℎ₂)∈ℝ×ℝ^𝑛−1, and let 𝜃(𝑔,ℎ)∶=(𝜃₁(𝑔,ℎ),𝜃₂(𝑔,ℎ)). Then, comparing with (33), we can rewrite the function 𝜑 as 𝜑=𝜃𝑔,ℎ𝑔,ℎ𝑥²₁+𝑦²₁,∀𝑔,ℎ∈ℝ^𝑛.(57) Note that the Euclidean norm ‖⋅‖ is globally Lipschitz continuous and strongly semismooth everywhere in ℝ^𝑛, and 𝜃(⋅,⋅) is a linear function. Then, (57) implies that 𝜑 is globally Lipschitz continuous and strongly semismooth everywhere in ℝ^𝑛×ℝ^𝑛 by [17, Theorem 19]. Also, the function 𝜑 is differentiable at (𝑔,ℎ) if and only if 𝜑(𝑔,ℎ)>0. The following lemma characterizes the 𝐵-subdifferential of the function 𝜑 at the origin.

Lemma 11. For any given (𝑥,𝑦) with 𝑥²+𝑦²∈𝐛𝐝⁺𝒦^𝑛, let 𝜑 be defined by (33). Then, the B-subdifferential of the function 𝜑 at (0,0) takes the following form: 𝜕_𝐵𝜑()=𝜉0,0^𝑇2𝑦₂+𝜉₁𝑥₁𝑥²₁+𝑦²₁,−𝑦₁𝜉^𝑇₂−𝜉₁𝑥^𝑇₂𝑥²₁+𝑦²₁,−𝜉^𝑇₂𝑥₂+𝜉₁𝑦₁𝑥²₁+𝑦²₁,𝑥₁𝜉^𝑇₂−𝜉₁𝑦^𝑇₂𝑥²₁+𝑦²₁∶𝜉₁,𝜉₂∈ℝ×ℝ^𝑛−1𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠𝜉²₁+𝜉₂²=1.(58)

Proof. Let (𝑢^𝑇,𝑣^𝑇)∈𝜕_𝐵𝜑(0,0). By the definition of the elements in 𝜕_𝐵𝜑(0,0), there exists a sequence {(𝑔^𝑘,ℎ^𝑘)} in ℝ^𝑛×ℝ^𝑛 converging to (0,0) with 𝜑(𝑔^𝑘,ℎ^𝑘)>0 such that 𝑢^𝑇,𝑣^𝑇=lim_𝑘→∞𝒥𝜑𝑔^𝑘,ℎ^𝑘.(59) By (57), a simple computation shows that such (𝑢^𝑇,𝑣^𝑇) belongs to the set on the right hand side of (58). Thus, 𝜕_𝐵𝜑(0,0) is included in the set on the right hand side of (58). In fact, 𝜉₁ and 𝜉₂ in (58) are the limit points of {𝜃₁(𝑔^𝑘,ℎ^𝑘)/‖𝜃(𝑔^𝑘,ℎ^𝑘)‖} and {𝜃₂(𝑔^𝑘,ℎ^𝑘)/‖𝜃(𝑔^𝑘,ℎ^𝑘)‖}, respectively.
Conversely, let (𝑢^𝑇,𝑣^𝑇) be an arbitrary element of the set on the right hand side of (58). Then, there exists a (𝜉₁,𝜉₂)∈ℝ×ℝ^𝑛−1 with 𝜉²₁+‖𝜉₂‖²=1 such that 𝜉𝑢=^𝑇2𝑦₂+𝜉₁𝑥₁𝑥²₁+𝑦²₁,−𝑦₁𝜉₂−𝜉₁𝑥₂𝑥²₁+𝑦²₁,𝑣=−𝜉^𝑇₂𝑥₂+𝜉₁𝑦₁𝑥²₁+𝑦²₁,𝑥₁𝜉₂−𝜉₁𝑦₂𝑥²₁+𝑦²₁.(60) Take the sequence {(𝑔^𝑘,ℎ^𝑘)}⊂ℝ^𝑛×ℝ^𝑛 with 𝑔^𝑘=(𝑥₁𝜉₁/𝑘,−𝑦₁𝜉₂/𝑘) and ℎ^𝑘=(𝑦₁𝜉₁/𝑘,𝑥₁𝜉₂/𝑘). Clearly, (𝑔^𝑘,ℎ^𝑘)→(0,0) as 𝑘→∞. Also, by Lemma 5, it is easy to verify that 𝜃₁𝑔^𝑘,ℎ^𝑘=1𝑘𝑥²₁+𝑦²₁𝜉₁,𝜃₂𝑔^𝑘,ℎ^𝑘=1𝑘𝑥²₁+𝑦²₁𝜉₂.(61) This shows that 𝜑(𝑔^𝑘,ℎ^𝑘)>0 and 𝒥𝜑(𝑔^𝑘,ℎ^𝑘)=(𝑢^𝑇,𝑣^𝑇). Hence, (𝑢^𝑇,𝑣^𝑇)∈𝜕_𝐵𝜑(0,0). Thus, the set on the right hand side of (58) is included in 𝜕_𝐵𝜑(0,0).

Now we may prove the equivalence between the 𝐵-subdifferential of 𝜙_FB at a general point (𝑥,𝑦) and that of its directional derivative function 𝜙^′_FB((𝑥,𝑦);(⋅,⋅)) at (0,0). This result corresponds to that of [18, Lemma 14] established for the NR SOC function 𝜙_NR.

Lemma 12. For any given (𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑛, let 𝜓_FB(⋅,⋅)≡𝜙^′_FB((𝑥,𝑦);(⋅,⋅)). Then, 𝜕_𝐵𝜙_FB𝑥,𝑦=𝜕_𝐵𝜓_FB(0,0).(62)

Proof. The result is direct by Proposition 9(a)-(b) and Lemma A.1 in the Appendix.

Using Lemma 12, we may present an upper estimation for Clarke's Jacobian of 𝜙_FB at the point (𝑥,𝑦) with 𝑥²+𝑦²∈𝐛𝐝⁺𝒦^𝑛, which will be used in the next section.

Proposition 13. For any given 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂) with 𝑥²+𝑦²∈𝐛𝐝⁺𝒦^𝑛, one has 𝜕𝜙_FB⊆1𝑥,𝑦𝐼−𝑋−21−𝑤₂𝑢^𝑇1𝐼−𝑌−21−𝑤₂𝑣^𝑇𝑓𝑜𝑟𝑢=𝑢₁,𝑢₂,𝑣=𝑣₁,𝑣₂𝑢𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔𝑡ℎ𝑒𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑖𝑒𝑠₁≤𝑢₂≤1,𝑣₁≤𝑣₂𝑢≤1,₁−𝑣₁≤𝑢₂−𝑣₂,𝑢₁+𝑣₁≤𝑢₂+𝑣₂,𝑢₁−𝑣₁²+𝑢₂+𝑣₂²𝑢≤2,₁+𝑣₁²+𝑢₂−𝑣₂²𝑤≤2,1,^𝑇₂𝑢=0,1,𝑤^𝑇₂𝑤𝑣=0,1,−^𝑇₂𝑢=2𝑢₁,1,−𝑤^𝑇₂𝑣=2𝑣₁,(63) where 𝑋 and 𝑌 are 𝑛×𝑛 real symmetric matrices defined as follows: 1𝑋=2𝑤₁𝑥₁𝑥^𝑇₂𝑥₂2𝑥₁𝐼−𝑤₂𝑥^𝑇₂,1𝑌=2𝑤₁𝑦₁𝑦^𝑇₂𝑦₂2𝑦₁𝐼−𝑤₂𝑦^𝑇₂.(64)

Proof. We first make simplifications for the last two terms in (32) by 𝑋,𝑌. Note that 𝑥^𝑇₂𝑔₂+𝑦^𝑇₂ℎ₂2𝑥²₁+𝑦²₁0𝑤₂−12𝑥²₁+𝑦²₁×𝑥^𝑇𝑔+𝑦^𝑇ℎ2𝑥₁𝑔₂+𝑔₁𝑥₂+2𝑦₁ℎ₂+ℎ₁𝑦₂=12𝑥²₁+𝑦²₁0𝑤₂0𝑥^𝑇₂𝑔+12𝑥²₁+𝑦²₁0𝑤₂0𝑦^𝑇₂ℎ−12𝑥²₁+𝑦²₁𝑥₁𝑥^𝑇₂𝑥₂2𝑥₁𝐼𝑔−12𝑥²₁+𝑦²₁𝑦₁𝑦^𝑇₂𝑦₂2𝑦₁𝐼ℎ=−𝑋𝑔−𝑌ℎ,(65) where the last equality is using 𝑤₁=2(𝑥²₁+𝑦²₁). Therefore, from (32), we have 𝜙^′_FB;=−𝜑𝑥,𝑦𝑔,ℎ𝑔+ℎ𝑔,ℎ21−𝑤₂−𝑋𝑔−𝑌ℎ,∀𝑔,ℎ∈ℝ^𝑛.(66) Now, applying Lemma 12, we immediately obtain that 𝜕𝜙_FB=1𝑥,𝑦𝐼−𝑋−21−𝑤₂𝑢^𝑇1𝐼−𝑌−21−𝑤₂𝑣^𝑇for𝑢=𝑢₁,𝑢₂,𝑣=𝑣₁,𝑣₂with𝑢^𝑇,𝑣^𝑇∈𝜕𝜑(0,0),(67) where, by Lemma 11 and the definition of Clarke's Jacobian, ()=𝜉𝜕𝜑0,0^𝑇2𝑦₂+𝜉₁𝑥₁𝑥²₁+𝑦²₁,−𝑦₁𝜉^𝑇₂−𝜉₁𝑥^𝑇₂𝑥²₁+𝑦²₁,−𝜉^𝑇₂𝑥₂+𝜉₁𝑦₁𝑥²₁+𝑦²₁,𝑥₁𝜉^𝑇₂−𝜉₁𝑦^𝑇₂𝑥²₁+𝑦²₁∶𝜉₁,𝜉₂∈ℝ×ℝ^𝑛−1satisfies𝜉²₁+𝜉₂²≤1.(68)
Let (𝑢^𝑇,𝑣^𝑇)∈𝜕𝜑(0,0) with 𝑢=(𝑢₁,𝑢₂),𝑣=(𝑣₁,𝑣₂)∈ℝ×ℝ^𝑛−1. Then, it suffices to prove that such 𝑢 and 𝑣 satisfy all inequalities and equalities in (63). By (68), there exists a vector (𝜉₁,𝜉₂)∈ℝ×ℝ^𝑛−1 with 𝜉²₁+‖𝜉₂‖²≤1 such that 𝑢₁=𝜉^𝑇₂𝑦₂+𝜉₁𝑥₁𝑥²₁+𝑦²₁,𝑢₂=−𝑦₁𝜉₂−𝜉₁𝑥₂𝑥²₁+𝑦²₁,𝑣₁=−𝜉^𝑇₂𝑥₂+𝜉₁𝑦₁𝑥²₁+𝑦²₁,𝑣₂=𝑥₁𝜉₂−𝜉₁𝑦₂𝑥²₁+𝑦²₁.(69) Using Lemma 5, it is immediate to obtain that 𝜉^𝑇₂𝑦₂+𝜉₁𝑥₁²=𝜉^𝑇₂𝑦₂²+𝜉₁𝑥₁²+2𝜉₁𝑥₁𝜉^𝑇₂𝑦₂≤𝜉₂²𝑦²₁+𝜉²₁𝑥²₁+2𝜉₁𝑦₁𝜉^𝑇₂𝑥₂=−𝑦₁𝜉₂−𝜉₁𝑥₂²≤𝜉₂²𝑦²₁+𝜉²₁𝑥²₁+𝜉₂²𝑥²₁+𝜉²₁𝑦₂²=𝜉₂²+𝜉²₁𝑥²₁+𝑦²₁≤𝑥²₁+𝑦²₁.(70) This means that |𝑢₁|≤‖𝑢₂‖≤1. Similarly, we also have |𝑣₁|≤‖𝑣₂‖≤1. We next prove that (𝑢₁−𝑣₁)²−‖𝑢₂−𝑣₂‖²≤0. By Lemma 5, it is easy to verify that 𝑥₂−𝑦₂𝑥₁+𝑦₁=𝑥₂+𝑦₂𝑥₁−𝑦₁,𝑥₁