Nonsingularity Conditions for Fb System of Reformulating Nonlinear Second-order Cone Programming

This paper is a counterpart of [2]. Specifically, for a locally optimal solution to the nonlinear second-order cone programming (SOCP), 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.


Introduction
The nonlinear second-order cone programming (SOCP) problem can be stated as min where  : R  → R, ℎ : R  → R  , and  : R  → R  are given twice continuously differentiable functions, and K is the Cartesian product of some second-order cones, that is, with  1 + ⋅ ⋅ ⋅ +   =  and K   being the second-order cone (SOC) in R   defined by By introducing a slack variable to the second constraint, the SOCP (1) is equivalent to min ,∈R   () s.t.ℎ() = 0,  () −  = 0,  ∈ K.
In this paper, we will concentrate on this equivalent formulation of problem (1).
With an SOC complementarity function  soc associated with K, we may reformulate the KKT optimality conditions in (7) as the following nonsmooth system: The most popular SOC complementarity functions include the vector-valued natural residual (NR) function and Fischer-Burmeister (FB) function, respectively, defined as  NR (, ) :=  − Π K ( − ) , ∀,  ∈ R  ,  FB (, ) := ( + ) − √  2 +  2 , ∀, ∈ R  , (10) where Π K (⋅) is the projection operator onto the closed convex cone K,  2 =  ∘  means the Jordan product of  and itself, and √ denotes the unique square root of  ∈ K.It turns out that the FB SOC complementarity function  FB enjoys almost all favorable properties of the NR SOC complementarity function  NR (see [2]).Also, the squared norm of  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: 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 They showed that the strong regularity of the KKT point, the nonsingularity of Clarke's Jacobian of  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 -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  FB .Note that up till now one even does not know whether the -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  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,  denotes an identity matrix of appropriate dimension, R  ( > 1) denotes the space of dimensional real column vectors, and R  1 × ⋅ ⋅ ⋅ × R   is identified with R  1 +⋅⋅⋅+  .Thus, ( 1 , . . .,   ) ∈ R  1 ×⋅ ⋅ ⋅×R   is viewed as a column vector in R  1 +⋅⋅⋅+  .The notations int K  , bd K  , and bd + K  denote the interior, the boundary, and the boundary excluding the origin of K  , respectively.For any  ∈ R  , we write  ⪰ K  0 (resp.,  ≻ K  0) if  ∈ K  (resp.,  ∈ int K  ).For any given real symmetric matrix , we write  ⪰ 0 (resp.,  ≻ 0) if  is positive semidefinite (resp., positive definite).In addition, J  () and J 2  () denote the derivative and the second-order derivative, respectively, of a twice differentiable function  with respect to the variable .

Preliminary Results
First we recall from [12] the definition of Jordan product and spectral factorization.
∈ R  .For each  = ( 1 ,  2 ) ∈ R × R −1 , we define the associated arrow matrix by Then it is easy to verify that    =  ∘  for any ,  ∈ R  .
The spectral factorization of the vectors ,  2 , √ and the matrix   have various interesting properties (see [13]).We list several properties that we will use later.
The following lemma states a result for the arrow matrices associated with ,  ∈ R  and  ⪰ K  √ 2 +  2 , which will be used in the next section to characterize an important property for the elements of Clarke's Jacobian of  FB at a general point.

Lemma 4.
For any given ,  ∈ R  and  ≻ where ‖‖ 2 means the spectral norm of a real matrix .
Consequently, it holds that From [13,Proposition 3.4], it follows that This shows that ‖‖ 2 ≤ 1, and the first part follows.Note that, for any By letting  = (Δ, Δ) ∈ R  × R  , we immediately obtain the second part.
The following two lemmas state the properties of ,  with  2 +  2 ∈ bd K  which are often used in the subsequent sections.The proof of Lemma 5 is given in [3, Lemma 2].When ,  ∈ bd K  satisfies the complementary condition, we have the following result.
Next we recall from [14] the strong regularity for a solution of generalized equation where  is a continuously differentiable mapping from a finite dimensional real vector space Z to itself,  is a closed convex set in Z, and N  () is the normal cone of  at .As will be shown in Section 4, the KKT condition (7) can be written in the form of (29).
Definition 8. We say that  is a strongly regular solution of the generalized equation (29) if there exist neighborhood B of the origin 0 ∈ Z and V of  such that for every  ∈ B, the linearized generalized equation  ∈ () + J  ()( − ) + N  () has a unique solution in V, denoted by  V (), and the mapping  V : B → V is Lipschitz continuous.
To close this section, we recall from [15] Clarke's (generalized) Jacobian of a locally Lipschitz mapping.Let  ⊂ R  be an open set and Ξ :  → R  a locally Lipschitz continuous function on .By Rademacher's theorem, Ξ is almost everywhere (réchet)-differentiable in .We denote by  Ξ the set of points in  where Ξ is -differentiable.Then Clarke's Jacobian of Ξ at  is defined by Ξ() := conv{  Ξ()}, where "conv" means the convex hull, and subdifferential   Ξ(), a name coined in [16], has the form For the concept of (strong) semismoothness, please refer to the literature [5,6].
Unless otherwise stated, in the rest of this paper, for any  ∈ R  ( > 1), we write  = ( 1 ,  2 ), where  1 is the first component of  and  2 is a column vector consisting of the remaining  − 1 entries of .For any (31)

Directional Derivative and 𝐵-Subdifferential
The function  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 -subdifferential of  FB at a general point coincides with that of its directional derivative function at the origin.Throughout this section, we assume that K = K  .(c) If  2 +  2 ∈ bd + K  , then

Proposition 9. For any given
where Let [ FB (, )] 1 be the first element of  FB (, ) and [ FB (, )] 2 the vector consisting of the rest −1 components of  FB (, ).By the above expression of  FB ( + ,  + ℎ), where the last equality is using  1  2 =  1  2 by Lemma 5.The above two limits imply 2 with the spectral values  1 ,  2 .An elementary calculation gives Also, since  2 ̸ = 0, applying the Taylor formula of ‖ ⋅ ‖ at  2 and Lemma 6(a) yields Now using the definition of  FB and noting that  1 = 0 and  2 ̸ = 0, we have that which in turn implies that We first calculate lim ↓0 (( √  2 − √ 2 )/).Using ( 38) and ( 40), it is easy to see that and consequently, lim We next calculate lim -( 39) and Lemma 6(a), Using ‖ 2 ‖ = 2( 2 1 + Then, from (45) and We next make simplification for the numerator of the right hand side of (47).Note that Therefore, adding the last two equalities and using Lemma 5 yield that (49 Combining this equality with (47) and using the definition of  in (33), we readily get To this end, we also need to take a look at ‖ 2 ‖ 2 − ‖ 2 ‖ 2 .From (38)-( 39) and (40), it follows that Together with (44) and (50), we have that lim where the last equality is using  1 w2 =  2 and  1 w2 =  2 .
As a consequence of Proposition 9, we have the following sufficient and necessary characterizations for the (continuously) differentiable points of  FB and () := ||.
(b) Since () = √  2 , by part (a)  is (continuously) differentiable at  ∈ R  if and only if || ∈ int K  , which is equivalent to requiring that  is invertible since || ∈ K  always holds.When  is invertible, the formula of J() follows from part (a).
Hence, (  ,   ) ∈   (0, 0).Thus, the set on the right hand side of (58) is included in   (0, 0).Now we may prove the equivalence between the subdifferential of  FB at a general point (, ) and that of its directional derivative function   FB ((, ); (⋅, ⋅)) at (0, 0).This result corresponds to that of [18,Lemma 14]  Using Lemma 12, we may present an upper estimation for Clarke's Jacobian of  FB at the point (, ) with  2 +  2 ∈ bd + K  , which will be used in the next section.

Proposition 13. For any given
where  and  are  ×  real symmetric matrices defined as follows: ) . (64) Proof.We first make simplifications for the last two terms in (32) by , .Note that where the last equality is using  1 = 2( 2 1 +  2 1 ).Therefore, from (32), we have Now, applying Lemma 12, we immediately obtain that with (  ,   ) ∈  (0, 0) } , where, by Lemma 11 and the definition of Clarke's Jacobian, ) , Let (  ,   ) ∈ (0, 0) with  = ( 1 ,  2 ),  = ( 1 ,  2 ) ∈ R × R −1 .Then, it suffices to prove that such  and  satisfy Abstract and Applied Analysis all inequalities and equalities in (63).By (68), there exists a vector Using Lemma 5, it is immediate to obtain that This means that | 1 | ≤ ‖ 2 ‖ ≤ 1.Similarly, we also have Using the two equalities, it is not hard to calculate that Similarly, we also have In addition, we have that A similar argument also yields The last four equalities in (63) are direct by Lemma 5 and the expression of  1 ,  2 ,  1 , and  2 .

Nonsingularity Conditions
This section studies the nonsingularity of Clarke's Jacobian of  FB at a KKT point.Let (, , , , ) ∈ R  × K × R  × R  × K be a KKT point of the SOCP (4), that is, Taking into account that − ∈ N K () if and only if  and  satisfy we introduce the following index sets associated with  and : From [19], we learn that the above six index sets form a partition of {1, 2, . . ., }.First of all, let us take a careful look at the properties of the elements in  FB (  ,   ) for  ∈  0 ∪  0 , as stated in the following.The proof of Lemma 15 is given in the Appendix.
We next prove the implication in (90).By the expressions of   ,   , and   , where the second equivalence is due to the dimension of the solution space for the system [− 2  1 ] = 0 equals 1.Note that   is a nonzero solution of this linear system.Therefore, Making an inner product with (Δ)  for the equality and using ⟨(Δ)  ,   ⟩ = 0, we get Using the similar arguments as above and noting that ⟨(Δ)  ,   ⟩ = 0, we may obtain The last two equalities show that the implication in (90) holds.
Case 4 ( ∈  0 ).By Lemma 15(b), there exists an   ×   orthogonal matrix   = [  Q    ] such that   =        and   =   Λ     , where   and    are given by (86), and   and Λ  take one of the form in (87)-(88).Thus, we have When   =  and Λ  = 0, we have    (Δ)  = 0, and then (Δ)  = 0.When   and Λ  take the form in (87), we have where the last equality is using the definition of    .When   and Λ  take the form in (88), we also have [ Using the same arguments as above, we have that (Δ)  has the form of (102).Thus, we prove that (Δ)  ∈ R( 1 , − 2 ).
Case 5 ( ∈  0 ).Using Lemma 15(a) and following the same arguments as in Case 4, the result can be checked routinely.So, we omit the proof.
The following lemma states an important property for the elements of Clarke's Jacobian of  FB at a general point, which will be used to prove Proposition 18.
Consequently, for any Δ, Δ ∈ R  , we have that From the above two equations, we immediately obtain the desired result.
Before stating the main result of this section, we also need to recall several concepts, including constraint nondegeneracy, Robinson's constraint qualification (CQ) (see [20]), and the strong second-order sufficient condition introduced in [ Then, we may rewrite the nonlinear SOCP (4) succinctly as follows: min Definition 19.A feasible vector  = (, ) of ( 4) is called constraint nondegenerate if where lin(T K ()) is the largest linear space of T K (), that is, lin(T K ()) = T K () ∩ −T K ().
Let (, , , , ) ∈ R  × K × R  × R  × K be a KKT point of the SOCP (4).From [7, Lemma 25], it follows that the tangent cone of K at  takes the form of which implies that the largest linear space in T K () has the following form: We next recall the critical cone of problem (4) at a feasible  0 which is defined as The critical cone C( 0 ) represents those directions for which the linearization of ( 4) does not provide any information about optimality of  0 and is very important in studying second-order optimality conditions.Particularly, if the set of Lagrange multipliers Λ( 0 ) at the point  0 is nonempty, then C( 0 ) can be rewritten as where  0 ∈ Λ( 0 ) and ( 0 ) ⊥ means the orthogonal complementarity space of  0 .Now let (, , ) ∈ Λ().Then, using g() =  and the expression of T K (), we have that and aff(C()) denotes the affine hull of C() and is now equivalent to the span of C(): Now we are in a position to prove the nonsingularity of Clarke's Jacobian of  FB under the strong second-order sufficient condition and constraint nondegeneracy.Proposition 22.Let (, , , , ) be a KKT point of the nonlinear SOCP (4).Suppose that the strong second-order sufficient condition (123) holds at  = (, ) and  is constraint nondegenerate, then any element in  FB (, , , , ) is nonsingular.

Conclusions
In this paper, for a locally optimal solution of the nonlinear SOCP, we established the equivalence between the nonsingularity of Clarke's Jacobian of the FB system and the strong regularity of the corresponding KKT point.This provides a new characterization for the strong regularity of the nonlinear SOCPs and extends the result of [22,Corollary 3.7] for the FB system of variational inequalities with the polyhedral cone R  + constraints to the setting of SOCs.Also, this result implies that the semismooth Newton method [5,6] applied to the FB system is locally quadratically convergent to a KKT point under the strong second-order sufficient condition and constraint nondegeneracy.We point it out that we have also established parallel (not exactly the same) results for SDP case in [11] recently.However, it seems hard to put them together in a unified framework under Euclidean Jordan algebra.The main reason causing this is due to that the analysis and techniques are totally different when dealing with the Clarke Jacobians associated with FB SOC complementarity function and FB SDP complementarity function.
(b) In view of the symmetry of  and  in   and   , the results readily follow by using similar arguments as in part (a).Thus, we complete the proof.
(4)ch is clear in the form of the generalized equation given by (29).Now using Proposition 22 and [9, Theorem 3.1], we may establish the main result of this paper, which states that Clarke's Jacobian of  FB at a KKT point is nonsingular if and only if the KKT point is a strongly regular solution to the generalized equation (141).Let (, ) be a locally optimal solution to the nonlinear SOCP(4).Suppose that Robinson's CQ holds at this point.Let (, , ) ∈ R  ×R  ×K be such that (, , , , ) is a KKT point of (4).Then the following statements are equivalent.(a) The strong second-order sufficient condition in Definition 21 holds at (, ) and (, ) is constraint nondegenerate.(b) Any element in  FB (, , , , ) is nonsingular.(c) Any element in  NR (, , , , ) is nonsingular.(d) (, , , , ) is a strongly regular solution of the generalized equation (141).Proof.First, Lemma 14 and the definition of  FB and  NR imply the following inclusion:  NR (, , , , ) ⊆  FB (, , , , ) .(142) Using this inclusion and Proposition 22, we have that (a) ⇒ (b) ⇒ (c).Since the SOCP (4) is obtained from (1) by introducing a slack variable, we know from [9, Theorem 3.1] that (a) ⇔ (c) ⇔ (d).Thus, we complete the proof of this theorem.