On the calm b-differentiability of projector onto circular cone and its applications

Use your smartphone to scan this QR code and download this article ABSTRACT In this paper, we study a concept on the calm B-differentiability, a new kind of generalized differentiabilities for a given vector function introduced by Ye and Zhou in 2017, of the projector onto the circular cone. Then, we discuss its applications inmathematical programming problemswith circular cone complementarity constraints. Here, this problem can be considered to be a generalization of mathematical programming problems with second-order cone complementarity constraints, and thus it includes a large class ofmathematical models in optimization theory. Consequently, the obtained results for this problem are generalized, and then corresponding results for some special mathematical problems can be implied from them directly. For more detailed information, we will first prove the calmly B-differentiable property of the projector onto the circular cone. This result is not easy to be shownby simply resorting to those of the projection operator onto the second-order cone. By virtue of exploiting variational techniques, we next establish the exact formula for the regular (Fréchet) normal cone (this concept was proposed by Kruger and Mordukhovich in 1980) to the circular cone complementarity set. Note that this set can be considered to be a generalization of the second-order cone complementarity set. In finally, the exact formula for the regular (Fréchet) normal cone to the circular cone complementarity set would be useful for us to study first-order necessary optimality conditions for mathematical programming problems with circular cone complementarity constraints. Our obtained results in the paper are new, and they are generalized to some existing ones in the literature.


INTRODUCTION
The second-order cone programming (SOCP) problem plays an important role in the optimization theory and has attracted much attention from mathematicians, see, e.g., [1][2][3][4][5][6][7] . We refer the reader to 1,2,[4][5][6][7] and the references therein for some remarkable results on optimality conditions and stability analysis of (SOCP). Inspired by the second-order cone, many researchers have investigated optimization and complementarity problems where their constraints are involved in second-order cones. It is called the second-order cone complementarity problem (SOCCP), which includes a large class of optimization problems such as quadratically constrained problems (see 8 ), the second-order cone programming, and nonlinear complementarity problem (see 9 ). In particular, recent attention is paid to the second-order cone complementarity set. Let us now mention some existing results concerning this set. In 10 , Liang et al. provided formulations for Fréchet normal cone to the second-order cone complementarity set. Unfortunately, the obtained results were shown to be inexact in 11 . In that paper, Ye and Zhou gave exact formulas for the proximal/regular (Fréchet)/limiting normal cone to the second-order cone complementarity set by using the projection operator onto second-order cones and the generalized differentiability called the calm B-differentiability. Some first-order optimality conditions for mathematical programs with second-order cone complementarity constraints were obtained in 12 and sufficient conditions for error bound property of second-order cone complementarity problems were established in 13 . To obtain these results, the authors used the symmetric and self-dual property of the second-order cone. Recently, generalizations of second-order cones and second-order cone complementarity sets have been examined by many authors 5,[14][15][16][17][18][19][20][21][22] . For example, authors in 14,[19][20][21][22] considered circular cones, which are generalizations of second-order cones and are, in general, nonsymmetric and non-self-dual cones. The generalized differentiability of the projection operator onto the circular cone was provided in 14,22 . Moreover, the differentiability and calmness of vectorvalued functions associated with the circular cone were also studied in 19,23 . In particular, authors in 21 showed that the results of the projection operator onto a circular cone could not be shown by simply resorting to the results of the projection operator onto the second-order cone, and hence, it is necessary to study the results of circular cone directly. To the best of our knowledge, there is no result on the calmly B-differentiable property concerning the circular cone and its extension. In this paper, inspired by 11,13,22 , we first study in Section 3, the calm B-differentiability of the circular cone. We then provide in Section 4 the formula for the Fréchet normal cone to a circular cone complementarity set, which can be considered as a generalization of the secondorder cone complementarity set. This formula would be useful for us to study optimality conditions for mathematical programming problems with circular cone complementarity constraints.

PRELIMINARIES
Throughout the paper, if not otherwise specified, stands for the closed ball centered at x ∈ R n with radius r > 0. Given x, y ∈ R n , x T y stands for the scalar product of x and y. For x := (x 0 , x r ) ∈ R × R n−1 , we use the following notation x ⊥ := {y ∈ R n |x T y = 0} and if otherwise. any unit vector e ∈ R n−1 Let C⊂R n be a nonempty subset, clC denotes its closure. The polar cone C • and the dual cone C ⋆ of C are respectively. The Fréchet normal cone to C at x ∈ clC are defined respectively by, see 24 , Lemma 2.1 ( 24 , Theorem 1.14) Let D={x | h(x)∈ C} and let ∇h(x) be surjective. Then The indicator function of a set C ⊂ R n is denoted by It is known from [ 25 , Proposition 1.18] that ∂ δ C (x) = N C (x) for any x ∈ C. Let F:R n ⇒ R m be a set-valued mapping, the domain and the graph of F are The Fréchet coderivative of F at (x, y) ∈ gphF are respectively defined by, see [ 24 , Definition 1.32], for each y * ∈R m , When F(x) is single-valued, y can be omitted in the above notations. Moreover, if F is continuously differentiable, then for all y * ∈R m , we get The derivative in the directionh ∈ R n of F at x is defined by The circular cone is defined (cf. 14,19-23 ) by . When θ = π 4 , it reduces to the second-order cone defined by K θ := {x = (x 0 , x r ) ∈ R × R m |x 0 ≥ ||x r ||}. In this case, the set is called the second-order cone complementarity set. If θ ̸ = π 4 then K θ is a nonsymmetric and non-self-dual cone. The boundary and the interior of K θ are given respectively by The positive dual cone and the polar cone of K θ are defined respectively by, see [ 20 , Theorem 2.1], A relation between the boundary of K θ and that of K * θ is established as follows.
which implies the existence of κ ∈ R ++ such that x r = −κy r . Consequently, we obtain i.e., x 0 tan 2 θ = κy 0 . Hence, x = κ(y 0 cot 2 θ , −y r ) with κ = x 0 y 0 tan 2 θ , and the proof is completed. □ We recall that for any given x : where the spectral values λ 1 (x), λ 2 (x) and the spectral vectors u 1 x , u 2 x are defined respectively by The metric projection of x onto K θ , denoted by Π K θ (x), is defined as follows From 22 and the convexity of K θ , we get that Π K θ (x) is a single-valued set and . Let us define the circular cone complementarity set as which is a generalized type of (2.2). Given (x,y)∈Γ and an arbitrary u ∈ K θ , it holds that . Similarly, we get that y ∈ Π K * θ (y − x). The above observation allows us to obtain a relation between the complementarity set Γ, and the projection onto K θ as follows. Proposition 2.3 Let Γ be as in (2.4). Then, we get where I m+1 is the unit matrix of the degree m+1, has full rank. It follows from [ 27 , Exercise 6.7] that From the above discussion, one obtains the following result, which plays an important role in computing the Fréchet normal cone to complementarity set. Proposition 2.4 Let Γ be as in (2.4) and (x,y)∈Γ. Then, we get (2.5)

CALM B-DIFFERENTIABILITY OF THE PROJECT MAPPING ONTO A CIRCULAR CONE
In this section, we first show that the projection operator Π K θ is calmly B-differentiable at any x ∈ R m+1 . Then, we provide a characterization for a proximal normal vector of Γ.
Proof.Given an arbitrary x ∈ R m+1 , it is enough to show that, for h sufficiently close to 0, We consider the following cases.

It follows from (2.3) and Lemma 3.2 in 22 that
Since the function (·) + is Lipschitz with modulus 1, from (3.11), we obtain Note that the last inequality holds by ∥h r ∥≤∥ h∥. Consequently, (3.1) is implied.
Moreover, we can check that φ is continuously differentiable on B(x,r), so ∇ φ(u) has the full rank whenever u ∈ B(x, r). It follows from [ 27 , Example 6.8] that Taking (4.6) with k = y 0 x 0 into account, we have On the other hand, for each n ∈ N, let x n := x, y n := (1 + 1 n )y = (1 + 1 n )k(x 0 tan 2 θ , −x r ), then (x n , y n ) ∈ Γ for all n ∈ N. Similarly to (4.5), one gets Passing to limsup, it follows from the definition of regular normal cone that ⟨v, y⟩ ≤ 0. Otherwise, if we take x ′ n := x, y ′ n = (1− 1 n )y for each n ∈ N then by the similar method, we obtain ⟨v, y⟩ ≥ 0. Therefore, ⟨v, y⟩ = 0, i.e., v⊥y. Similarly, one gets u⊥x. Consequently, For the inverse of the above inclusion, let (u, v) be satisfied u ⊥ x, v ⊥ y and Moreover, since u r − kv r = −αx r , we get (4.8) It follows from (4.7) and (4.8), we have equation 4.9 in Figure 5 We next show that By replacing ∥ x r ∥= x 0 tan θ and αx 0 tan 2 θ = u 0 x 0 0 + kv 0 tan 2 θ , the above equality is equivalent to which is always fulfilled by the fact that Taking (4.9) into account, one gets It follows from (4.3), (4.7) and (4.10) that y). Hence, in this case, we get Case 4: x = 0,y ∈ bd K * θ \{0}. In this case, we get x − y = −y = (−y 0 , −y r ). Thus, one has λ 1 (x − y) = −y 0 − ∥ y r ∥ cot θ < 0 and λ 2 (x − y) = −y 0 + ∥ y r ∥ tan θ = 0. By [ 22 , Theorem 3.5(c)], we obtain This is equivalent to v ∈ R_ y and u ∈ y • , which implies that Case 6: x = 0, y = 0. Then, we have λ 1 (x−y) = λ 2 (x− y) = 0, and thus from [ 22 , Theorem 3.5(d)], one gets It follows from (2.5) that (u, v) ∈ N Γ (x, y) if and only if −v ∈ K θ and −u ∈ K * θ . Consequently, we obtain In what follows, we present necessary conditions for the following mathematical program with circular cone complementarity constraints: 2 ) and f : R n → R, G, H : R n → R m+1 are continuously differentiable and K θ ⊂ R m+1 is a circular cone. The problem (MPCCC) is a generalization of the mathematical program with second-order cone complementarity constraints (MPSOCC) studied in 10,11,13 . The feasible set of (MPCCC) is defined by where Γ is given as (2.4) and F(x) = (G(x), H(x)).

−
x be a feasible solution of (MPCCC). We say that − x is a local optimal solution of (MPCCC) if there exist there exists δ > 0 such that Proof. It is easy to observe that − x is a local optimal solution of (MPCCC) if − x is a local optimal solution the function f (x)+δ Ξ (x). We have from [ 25 , Proposition 1.10] that . Using Lemma 2.1, we obtain ). □. Finally, we give an example to illustrate Theorem 4.3. It is easy to check that

CONCLUSION
In this paper, we have first shown the calmly Bdifferentiable property of the projector onto a circular cone. Then, we presented the exact formula for computing the Fréchet normal cones to the circular cone complementarity set. Finally, we have provided firstorder necessary conditions for local optimal solutions to mathematical programs with circular cone complementarity constraints. For possible developments, we are planning to employ the obtained results in calculating the directionally limiting normal cone of the circular cone complementarity set. Moreover, inspired by 13 , sufficient conditions for the error bound property of circular cone complementarity problems would be established by using the current approach.