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First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition

The well known constant rank constraint qualification [Math. Program. Study 21:110--126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of our approach is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition that is stronger than the classical one obtained under Robinson's constraint qualification, in the sense that it holds for every Lagrange multiplier, even though our condition is independent of Robinson's condition.


Introduction
In the classical nonlinear programming (NLP) context, the so-called constant rank constraint qualification (CRCQ) [37] was first presented as a tool for stability analysis, which stood out for being independent of the usual Mangasarian-Fromovitz constraint qualification (MFCQ) and strictly weaker than the linear independence constraint qualification (LICQ). For instance, it has been applied with this purpose in NLP [29,37,47,48,50], mathematical programs with equilibrium constraints (MPEC) [34], generalized equations [35], and bilevel optimization [45,60]. Also, it is the origin of several other constant rank-type conditions, such as the constant positive linear dependence [10,12,52] and the constant rank of the subspace component [11], which have been successfully applied in the convergence analysis of iterative algorithms.
To name a few algorithms whose convergence theory relies on CRCQ and its variants, we point out: an augmented Lagrangian method [3,13], a regularized interior point method [53], sequential quadratic programming methods for NLP [42,52,59] and MPEC [39], and some relaxation schemes for MPEC [36,58]. In fact, a particularly interesting aspect of CRCQ that makes it suitable for supporting practical algorithms is the fact it can be roughly interpreted as a relaxation of LICQ that is able to separate the core information of the problem, ignoring redundant constraints. Moreover, all linear programming problems satisfy CRCQ, in contrast with LICQ and MFCQ.
Besides convergence of algorithms and stability analysis, CRCQ was used in several contexts, such as NLP [4,13,46], MPEC [33], vector optimization [44], and continuous-time NLP [49], for studying necessary second-order optimality conditions. One of the main goals of this paper is to bring such results to more general conic programming contexts, namely nonlinear second-order cone programming (NSOCP) and nonlinear semidefinite programming (NSDP). As far as we know, the best second-order results for these problems have appeared in a well-known paper by Bonnans, Cominetti, and Shapiro [21]. Basically, they derived no-gap second-order optimality conditions for problems over second-order regular cones [21,Definition 3], such as NSDP and NSOCP, under the well-known Robinson's CQ (see (7) on page 5, or [54]), which is the natural extension of MFCQ to conic programming. In particular, their second-order necessary condition states that every local solution that satisfies Robinson's CQ must also satisfy the following: for every critical direction, there exists a Lagrange multiplier (possibly depending on this direction), such that a certain quadratic form is nonnegative with respect to such direction and multiplier. However, the second-order condition that is obtained under CRCQ in NLP replaces "there exists a Lagrange multiplier" with "for every Lagrange multiplier", which is stronger than the one of [21]. Although this stronger condition can be obtained from [21] after assuming that the Lagrange multiplier is unique, which is ensured by stronger constraint qualifications such as the nondegeneracy condition (see (8) on page 5), this assumption is often regarded as too stringent. To the best of our knowledge, no second-order result concerning every Lagrange multiplier, without assuming its uniqueness, has been presented so far in the literature of nonlinear conic programming. Moreover, no extension of CRCQ has been proposed for nonlinear conic programming until very recently.
In 2019, Zhang and Zhang [61] proposed an extension of CRCQ and its relaxed version [47] for NSOCP, but it was later discovered that their results were incorrect [5]. This event has motivated us to investigate other possible extensions of CRCQ to conic problems, and their properties. The first step in this direction was made in [6], for NSOCP and NSDP problems with multiple constraints. The idea of [6] is to rewrite some of the conic constraints as locally equivalent NLP constraints, whenever possible, and then jointly applying nondegeneracy and the NLP version of CRCQ to the resulting problem. Later, in [8], based on the ideas from [7], we improved this strategy by exploiting the eigenvector structure of the semidefinite cone to deal with the conic constraints that could not be rewritten as NLP constraints. This approach was also extended to NSOCP problems in [9]. In simple terms, the condition of [8,9] demands the rank of some families of functions to remain constant along every sequence converging to the point of interest -roughly speaking, a constant rank "by paths" -therefore, this extension is highly specialized to deal with sequences generated by iterative algorithms, but since this rank may vary between paths, it is likely unsuitable for other purposes. Indeed, the focus of [8,9] was the global convergence of a large class of algorithms to first-order stationary points, and no second-order results were provided in it. Nevertheless, it is reasonable to expect that CRCQ may have multiple independent and correct extensions, each one of them generalizing at least one important aspect of it, but perhaps not all of them.
A common feature of all previous attempts of extending CRCQ to a conic context is an approach based on re-characterizing the conic program and the nondegeneracy condition, trying to make them as similar to NLP and LICQ as possible, so the extension of CRCQ would come out straightforwardly. This is somehow understandable because, even in NLP, the CRCQ condition has never received a geometrical interpretation before. In this paper, we present a new geometrical characterization of CRCQ for NLP in terms of the faces of the nonnegative orthant, which suggests a natural extension of it to NSOCP and NSDP. A point that we should stress is that contrary to our previously mentioned works, the definition of CRCQ that we present here is very simple. We prove that this extension is a constraint qualification strictly weaker than nondegeneracy and independent of Robinson's CQ, as it should be, and we also compare it with the condition of [8,9]. Then, as an application, we show that every local solution of the problem satisfies the strong second order optimality condition, provided our extension of CRCQ holds. Moreover, just as it happens in NLP, our result does not demand a priori any specific condition over the Lagrange multiplier set, besides nonemptiness.
The structure of this paper is as follows: Section 2 consists of a nonlinear conic programming review emphasizing some aspects of the theory that are not commonly discussed in the literature; in Section 3, we analyze CRCQ for NLP and we show how it can be interpreted in terms of the faces of the nonnegative orthant. In Sections 4 and 5, we propose extensions of CRCQ for NSOCP and NSDP, respectively, and we prove some of its properties. Finally, in Section 6, we conclude this paper with a short discussion and some ideas of prospective work.
We end this section by introducing some of our basic notation: throughout this paper, E will denote a finite-dimensional linear space equipped with the inner product ·, · ; and for a given set S ⊆ E, we will denote the polar of S by S • := {z ∈ E | z, y ≤ 0, ∀y ∈ S} and the orthogonal complement of S will be denoted by S ⊥ . The notations cl(S), int(S), bd(S), and bd + (S) stand for the topological closure, interior, boundary, and boundary excluding the origin of S in E, respectively. The smallest cone that contains S will be denoted by cone(S), and the smallest linear space that contains S will be denoted by span(S). Moreover, we denote by Dg(x) the derivative of a twice continuously differentiable function g : R n → E at a given point x ∈ R n , and by Dg(x) T the adjoint of Dg(x), which by definition satisfies Dg(x)d, z = d, Dg(x) T z for all d ∈ R n and z ∈ E. Similarly, D 2 g(x) denotes the second-order derivative of g at x, and the action of D 2 g(x) over d1, d2 ∈ R n will be denoted by 2 Common framework: nonlinear conic programming In this section, we will review some classical results of convex analysis, and first-and second-order optimality conditions and constraint qualifications for NSOCP and NSDP. These problems are the cornerstones of two independent research fields, but they can also be seen as particular cases of a nonlinear conic programming (NCP) problem, given by where f : R n → R and g : R n → E are twice continuously differentiable, and K ⊆ E is a closed convex pointed cone that is assumed to be nonempty. We will use (NCP) as a framework to discuss the common traits of NSOCP and NSDP simultaneously, before moving to specific traits. Throughout the whole paper, we will denote the feasible set of (NCP) by Ω := {x ∈ R n | g(x) ∈ K}. Let us begin with two key ideas that underlie all the results of this paper: reducibility and faces. Recall from [24,Definition 3.135] that for any given linear spaces E and F, a cone K ⊆ E is said to be reducible (more precisely, C 2 -reducible) at a point y ∈ K, to a closed convex pointed cone C ⊆ F, if there exists a neighborhood N of y and a twice continuously differentiable reduction function Ξ : N → F (possibly depending on y) such that Ξ(y) = 0, DΞ(y) is surjective, and In general, reductions are meant to be used as a simplification tool that allows one to interpret any point of K as a vertex of some other cone C, and then extend the results obtained at C to K in a smooth way. In this work, we are also interested in the geometrical properties of the reduced cone C as well; in particular, in its faces.
To make a brief revision, we recall that F is a face of C if every open line segment that contains a point of F also has its extrema in F ; that is, if for every y ∈ F and every z, w ∈ C such that y = αz + (1 − α)w for some α ∈ (0, 1), we have that z, w ∈ F . Further, when there exists some η ∈ C • such that that is, when F is the intersection between C and one of its supporting hyperplanes, we say that F is an exposed face of C. Some cones, like the nonnegative orthant, the semidefinite cone, and the second-order cone, are facially exposed, meaning all of their faces are exposed. We use the notation F C to say that F is a face of C. Now, to contextualize our results, we will revisit the classical theory of NCP in the next section, with a special emphasis in the work of Guignard [32], and Bonnans, Cominetti, and Shapiro [21]. In particular, we stress some aspects of the NCP theory that are often disregarded in the literature.

Review of first-order optimality conditions
For any set S ⊆ E and any z ∈ S, recall the (Bouligand) tangent cone to S at z, defined as TS(z) := y ∈ E | ∃{t k } k∈N → 0 + , ∃{y k } k∈N → y such that z + t k y k ∈ S for all k ∈ N .
Our review of first-order constraint qualifications for (NCP) revolves around two particular cones: the tangent cone TΩ(x) to Ω at a feasible pointx ∈ Ω, and the linearized tangent cone where TK(g(x)) is the tangent cone to K at g(x). The importance of these cones for our analyses lies on the necessary optimality conditions for (NCP) associated with them. Namely, given any local minimizerx ∈ Ω of (NCP), it is easy to see that ∇f (x), d ≥ 0 for all d ∈ TΩ(x); that is, This is one of the simplest necessary optimality conditions, sometimes called the first-order geometric necessary condition for the optimality ofx. However, it may be difficult to use (1) when Ω does not admit an explicit characterization since TΩ(x) • may not be easily computable in this case. The polar of LΩ(x), on the other hand, admits a practical description, as it is shown in the following lemma, extracted from the proof of [32, Theorem 2] by Guignard: and NK(g(x)) := TK(g(x)) • is the normal cone to K at g(x).
Proof. By the bipolar theorem (see e.g. [24,Proposition 2.40]), it suffices to prove that LΩ(x) = H(x) • . Take any direction d ∈ LΩ(x) and let z ∈ TK(g(x)) • . By definition, Dg(x)d ∈ TK(g(x)) and then Thus, since z is arbitrary, we obtain that d ∈ H(x) • ; and since d is also arbitrary, it follows that LΩ(x) ⊆ (H(x)) • . Conversely, assume that there exists a vector v ∈ H(x) • such that v / ∈ LΩ(x), that is, Dg(x)v / ∈ TK(g(x)). By the strong separation theorem (see e.g. [24,Theorem 2.14]), there exists a vector y such that y, Dg(x)v > 0 and y, z < 0, for all z ∈ TK(g(x)), that is, y ∈ NK(g(x)). Therefore, Recall that since K is a closed convex cone, we have Then, combining the first-order geometric necessary condition and Lemma 2.1 yields the following theorem, also by Guignard: [32]). Letx ∈ Ω be a local minimizer of (NCP). If TΩ(x) • = LΩ(x) • and H(x) is closed, then there exists someλ ∈ K • such that ∇f (x) + Dg(x) Tλ = 0 and g(x),λ = 0. ( Theorem 2.1 can be seen as the "dual form" of the first-order geometric condition (1), and any vector λ ∈ K • that satisfies the Karush-Kuhn-Tucker conditions (3) is called a Lagrange multiplier associated with x. Moreover, the collection of all Lagrange multipliers associated withx will be denoted by Λ(x), and when Λ(x) = ∅ we say thatx is a KKT point of (NCP).
The hypothesis of Theorem 2.1, is known in the literature as Guignard's CQ, and it is the weakest assumption that makes the KKT conditions necessary for the local optimality ofx, in the sense of: if Λ(x) = ∅ for every continuously differentiable function f that has a local minimizer constrained to Ω atx, then Guignard's CQ must also hold atx [ which is equivalent to (4) due to Lemma 2.1. In NLP, Guignard's CQ is usually stated in the form TΩ(x) • = LΩ(x) • , since the closedness of H(x) follows from the polyhedricity of R m + . However, as it can be seen in the following example, the equality TΩ(x) • = LΩ(x) • on its own may not ensure that Λ(x) = ∅ when H(x) is not closed.

The condition
TΩ(x) = LΩ(x) and H(x) is closed, which implies Guignard's CQ, is known as Abadie's CQ (see also Börgens et al. [25,Definition 5.5]), and Example 2.1 tells us that the closedness of H(x) cannot be omitted in this case, either. The reason why we emphasize this point is that, as far as we know, it appears that Abadie's CQ and Guignard's CQ are rarely seen in the literature of finite-dimensional conic programming problems other than NLP, and the closedness of H(x) is rarely regarded in the study of constraint qualifications. In contrast, H(x) plays an important role in our results.
In finite-dimensional conic contexts, the focus is usually on constraint qualifications that already imply H(x) is closed without requiring it explicitly, such as Robinson's CQ, that holds at a given pointx ∈ Ω when 0 ∈ int(Im(Dg(x)) − K + g(x)).
In particular, if K has nonempty interior, then Robinson's CQ holds atx if, and only if, there exists some d ∈ R n such that Robinson's CQ is stronger than Abadie's CQ, and it implies that Λ(x), besides being closed and convex, is also nonempty and bounded [24,Theorem 3.9] whenx is a local minimizer of (NCP). If K is reducible at the point g(x) to a cone C by the reduction function Ξ, then the constraint g(x) ∈ K is locally equivalent to the reduced constraint G(x) ∈ C, where G := Ξ • g. In this case, Robinson's CQ holds atx for the original constraint if, and only if, it holds for the reduced constraint at the same point. Another well-known constraint qualification in the context of conic programming is the nondegeneracy condition, which holds atx when Im(Dg(x)) + lin(TK(g(x))) = E, where lin(TK(g(x))) = TK(g(x)) ∩ −TK(g(x)) denotes the largest linear space contained in TK(g(x)); that is, its lineality space. This CQ has first appeared in Shapiro and Fan's article [57] for NSDP, by the name transversality, and then it was generalized to NCP by Shapiro, in [56]. Nondegeneracy is strictly stronger than Robinson's CQ and it is known that ifx is a local minimizer of (NCP) that satisfies nondegeneracy, then Λ(x) is a singleton (see, for instance, [24,Proposition 4.75]). Moreover, if K is reducible, nondegeneracy is equivalent to the surjectivity of DG(x), as it can be easily deduced from the equality lin(TK(g(x))) = Ker(DΞ(g(x))); see [24, Section 4.6.1]. Due to their implications over the Lagrange multiplier set, nondegeneracy and Robinson's CQ are currently the most important CQs in the study of second-order optimality conditions for (NCP), which will be reviewed in the next subsection.

Second-order optimality conditions
Before starting, recall that the (inner) second-order tangent set to a nonempty set S ⊆ E, at a point z ∈ S, in a direction y ∈ TS(z), is defined by which is closed for all such z, y, and S. In addition, if S is convex, then T 2 S (z, y) is also convex [24,Page 163]; and if S is second-order regular, as it is the case of the semidefinite cone and the second-order cone, then T 2 S (z, y) is nonempty [24,Page 202]. The role of second-order necessary optimality conditions is to provide additional information when firstorder conditions are not meaningful enough; that is, along the directions in the cone which is often called the cone of critical directions, or simply, the critical cone of (NCP) atx. Ben-Tal and Zowe [19] presented a geometric second-order necessary optimality condition for (NCP), stating that ifx is a local minimizer of the problem, then for every d ∈ C(x) and every s ∈ T 2 Ω (x, d). Then, Kawasaki [41, Theorem 5.1] made the first advances to derive a "dual form" of (10) under Robinson's CQ assuming that K is a closed convex cone with nonempty interior. This result was later generalized and refined by Cominetti [26,Theorem 4.2] to the case where K is assumed to be a closed convex set. An important improvement was made afterwards by Bonnans, Cominetti, and Shapiro [21], who clarified several key points of the previous works, and obtained no-gap 1 second-order conditions, in particular, for second-order regular cones [21,Section 4]. Let us recall Bonnans, Cominetti, and Shapiro's necessary condition in the context of second-order regular cones: [21]). Letx ∈ Ω be a local minimizer of (NCP) that satisfies Robinson's CQ. Then, for every direction d ∈ C(x), there exists someλ d ∈ Λ(x), such that where is the support function of The term σ(d,x,λ d ) characterizes a possible curvature of the set K at g(x) along Dg(x)d, and it is often called the "sigma-term" in the classical literature (for instance, in the book [24]). Sinceλ d ∈ Λ(x) and K is convex, σ(d,x,λ d ) is always nonnegative; and if K is polyhedral, as in NLP, then the sigma-term is zero everywhere. See also the discussion on polyhedricity and extended polyhedricity in [24, Section 3.2.3]. It is also worth mentioning that the second-order optimality condition of Theorem 2.2 can be derived without constraint qualifications, using Fritz John (generalized) multipliers [24,Theorem 3.50].
Although the condition of Theorem 2.2 is generally considered very natural and useful in the conic programming context and in NLP, a stronger condition where the Lagrange multiplierλ does not depend on d has several potential uses, in view of the NLP literature. This motivates the following definition: Definition 2.1. Letx ∈ Ω be a KKT point and letλ ∈ Λ(x) be given. We say that the pair (x,λ) satisfies the second-order condition (SOC) when for every d ∈ C(x).
In NLP, the existence of someλ ∈ Λ(x) such that SOC holds for the pair (x,λ) is known as the semistrong second-order necessary optimality condition [20]. Moreover, when SOC holds for everyλ ∈ Λ(x), then we obtain what is known as the strong second-order necessary optimality condition [4]. However, while the condition of Theorem 2.2 is necessary for optimality under Robinson's CQ, this is not true, in general, for the strong and semi-strong conditions. In fact, there is a counterexample published by Baccari [17,Section 3] (see also Anitescu [14] and Arutyunov [15]), that shows that Robinson's CQ does not guarantee the existence of aλ ∈ Λ(x) such that the pair (x,λ) satisfies SOC (see also the extended version of [18] for details). Under nondegeneracy, the set Λ(x) is a singleton and, in this case, the semi-strong and the strong second-order conditions both coincide with the condition of Theorem 2.2.
As far as we know, there is no result concerning the semi-strong and strong second-order conditions without assuming uniqueness of Lagrange multipliers in the literature of conic programming, except for NLP. In NLP, this has been addressed by means of constant rank-type constraint qualifications, which is also the path we will follow in this paper.

Revisiting constant rank CQs in NLP
In this section we will revisit some constant rank-type conditions for NLP from a geometrical point of view, in order to extend it to a more general conic context later on. Consider the standard NLP problem which is a particular case of (NCP) with E = R m+p , K = R m + × {0} p , and g(x) := (g1(x), . . . , gm+p(x)). As usual in NLP, given a feasible pointx of (NLP), we will denote the set of active inequality constraints atx as Now, let us recall Janin's constant rank constraint qualification as it was first presented in [37].
Definition 3.1 (CRCQ [37]). Letx be a feasible point of (NLP). We say that the constant rank constraint qualification for NLP (CRCQ) holds atx if there exists a neighborhood V ofx such that, for every subset To prove that CRCQ is a constraint qualification, Janin proved that it implies LΩ(x) ⊆ TΩ(x), which in turn implies Abadie's CQ in NLP. His proof is what motivates the requirement to consider every subset J of A(x) ∪ {m + 1, . . . , m + p} in Definition 3.1; indeed, after picking a direction in order to prove that d ∈ TΩ(x), it is sufficient to have the constant rank assumption for the constraints that correspond to the indices j ∈ A(x) such that ∇gj(x) T d = 0. Since those indices depend on d, and they are not determined a priori, one considers all possibilities. However, as it was noted several years later by Minchenko and Stakhovski [47], taking subsets of the equality constraints is completely superfluous, even for Janin's proof. The "correct" definition of CRCQ was then presented in [47] as a relaxed version of CRCQ.
Definition 3.2 (RCRCQ [47]). Letx be a feasible point of (NLP). We say that relaxed constant rank constraint qualification for NLP (RCRCQ) holds atx if there exists a neighborhood V ofx such that, for every subset J ⊆ A(x), the rank of the family {∇gj(x)} j∈J ∪{m+1,...,m+p} remains constant for all x ∈ V.
In order to bring these CQs to the conic setting, our approach in this manuscript consists first in generalizing two key ideas of NLP: the notion of "active constraints" and the notion of "subsets of indices of active constraints". The former can be interpreted in the general context as a consequence of reducibility. Indeed, for any givenx ∈ Ω, let s := |A(x)| and note that R m + × {0} p is reducible at g(x) to the cone by the mapping Ξ : N → R s+p such that Ξ(y) := (yj) j∈A(x)∪{m+1,...,m+p} for every y ∈ N , and in this case the reduced constraint function of (NLP) atx takes the form Therefore, in NLP, reducing the problem is essentially the same as simply disregarding inactive constraints around the pointx. The notion of "subsets of indices of the active constraints", on the other hand, can be interpreted in terms of faces. It is easy to see that every face of R s + can be written in terms of a unique subset of the canonical vectors of R s , which we will denote by c1, . . . , cs. That is, F R s + if, and only if, there exists some J ⊆ {1, . . . , s} such that where F and J are clearly in a one-to-one correspondence. Thus, considering all subsets of active constraints atx is the same as considering all faces of the reduced cone C = R s + × {0} p . This discussion suggests a natural characterization of RCRCQ in terms of the faces of the reduced cone, as follows: Letx be a feasible point of (NLP). Then, RCRCQ holds atx if, and only if, there exists a neighborhood V ofx such that, for each F R remains constant for every x ∈ V, where G is as defined in (14).
Proof. Let s := |A(x)| and, without loss of generality, let us assume that A(x) = {1, . . . , s}. Moreover, let c1, . . . , cs+p be the canonical basis of R s+p , and let F R s Consequently, The conclusion follows from the one-to-one correspondence between F and J.
The equivalent form of RCRCQ presented in Proposition 3.1 allows us to visualize what it actually describes, geometrically. Indeed, recall that R n = DG(x) −1 (span(F )) + (DG(x) −1 (span(F ))) ⊥ and it is elementary to see that This implies the following relation: Thus, RCRCQ can be equivalently stated as the constant dimension of DG( , on the other hand, can be regarded as a "linear approximation" of G −1 (C) aroundx, since DG(x) is the best linear approximation of G at x ∈ V and, similarly, the faces of C can also be seen as "linear approximations" of it at G(x). In fact, each face induces a potentially different linear approximation of G −1 (C), which in turn coincides with Ω aroundx. So roughly speaking: RCRCQ holds atx when the dimension of every linear approximation of the feasible set Ω atx is invariant to small perturbations. In particular, defining gJ (x) := (gj(x)) j∈J ∪{m+1,...,m+p} for every J ⊆ A(x), this characterization is equivalent to the constant dimension of Ker(DgJ (x)) for all x in a neihborhood ofx at every J ⊆ A(x), which can also be trivially seen from the original definition of RCRCQ.
Note that the characterization of RCRCQ from Proposition 3.1 and the discussion above do not appear to be limited to the context of NLP, contrary to its original definition. In the next two sections, we will prove that the same idea can be applied to NSOCP and NSDP, respectively, giving rise to new constraint qualifications.
Remark 3.1. It is possible to obtain a characterization of CRCQ in the same style of Proposition 3.1. To do this, it suffices to reformulate the equality constraints gj (x) = 0 as a pair of inequality constraints gj(x) ≥ 0 and −gj(x) ≥ 0, for j ∈ {m + 1, . . . , m + p}. That is, consider In view of Remark 3.1, we see that there are multiple ways of dealing with equality constraints in our approach, and they are not all equivalent. The suitability of each approach may depend on the application, but we highlight that our approach is able to deal with equality constraints regardless of how they are modelled. For simplicity, equality constraints are omitted in our exposition. See also Remarks 4.2 and 5.3. In the following two sections, we extend the ideas of this section to NSOCP and NSDP.

Nonlinear second-order cone programming
In this section, we consider the following problem: where and only if, −z ∈ Km j , for any j = 1, . . . , q. Also, note that (NSOCP) can be seen as a particular case of (NCP) with Given a feasible pointx ∈ Ω, let us define the following index sets: which consist of the indices of the constraints that hit the interior, the boundary excluding zero, and the vertex of their respective cones. For simplicity, we will omit equality constraints; we should mention, nevertheless, that our results can be easily adapted to deal with equality constraints -see Remark 4.2 for details. As another measure to avoid cumbersome notation, we will assume that IB(x) = {1, . . . , |IB(x)|}; this assumption will often be recalled throughout this section.
Following Bonnans and Ramírez [22], for any givenx ∈ Ω, we see that K is reducible to where Ξj : Nj → R m j is the identity function for every j ∈ I0(x), and Ξj : Nj → R is given by for every j ∈ IB(x), and every y ∈ R m j . This leaves us with the reduced constraint and φ : R n → R |I B (x)| has its j-th component given by Note that g(x) ∈ K if, and only if, G(x) ∈ C for every x sufficiently close tox. By [22,Lemma 25], we see that the linearized cone of the original constraints of (NSOCP) at a given x ∈ Ω can be computed as and that it coincides with the linearized cone of the reduced constraint atx. Moreover, it follows from [1, Lemma 15] that for each j = Iint(x) ∪ IB(x), we have λ j , gj(x) = 0, if, and only if, where Rm j is a matrix defined as and Im j −1 is the (mj − 1) × (mj − 1) identity matrix. Therefore, still following [22], the pointx satisfies the KKT conditions with respect to the constraint g(x) ∈ K if, and only if, there exist some vectorsλj ∈ K • m j , j ∈ I0(x) ∪ IB(x), such that: which also coincides with the KKT conditions with respect to the reduced constraint G(x) ∈ C. In fact, note that for each j ∈ IB(x), the reduced Lagrange multiplier with respect to the reduced constraint φj(x) ≥ 0 is simply [λj ]0. With this in mind, we are ready to present our extension of CRCQ (and RCRCQ) to NSOCP inspired by the characterization of Proposition 3.1.

A facial constant rank constraint qualification for NSOCP
Recall that, for each j = 1, . . . , q, the cone Km j is facially exposed, meaning every F Km j can be written as the intersection of one of its supporting hyperplanes, say {η} ⊥ with η ∈ Km j . In fact, although Km j has infinitely many faces when mj > 2, they are limited to only three types: • The vertex, {0}, which can be characterized by any η ∈ int(Km j ); • The cone Km j itself, which is characterized by η = 0; • A ray at the boundary of Km j , starting at the vertex and passing through a point z ∈ bd + (Km j ), which can be characterized by any η ∈ cone(Rm j z) \ {0}.
Moreover, every F C has the form where Fj Km j for every j ∈ I0(x), and R R |I B (x)| + . Then, for every x ∈ R n , sufficiently close tox, we have where φ(x) := (φj(x)) j∈I B (x) . This motivates the following definition: Definition 4.1. Letx be a feasible point of (NSOCP). We say that the facial constant rank property holds atx if there exists a neighborhood V ofx such that for each F C, the dimension of DG(x) T [F ⊥ ] remains constant for all x ∈ V, where G is given by (19) and C is given by (17).
Recall the discussion after Proposition 3.1 and note that Definition 4.1 can be equivalently stated in terms of the constant dimension of DG(x) −1 (span(F )) for all x ∈ V and every F C. That is, the facial constant rank property holds atx when the dimension of every linear approximation of the feasible set remains locally invariant aroundx. Although this characterization is somewhat more intuitive than Definition 4.2, the latter is easier to use.
The facial constant rank condition is sufficient for the equality TΩ(x) = LΩ(x) to hold. To prove this, we employ the main result of Janin's paper [37], but the version we use is a slightly different characterization found in [4,Proposition 3.1]. Despite the fact we work in a context more general than NLP, we use the same result that was used in NLP. Then, there exists some neighborhoods V1 and V2 ofx, and a diffeomorphism ψ : V1 → V2, such that: (iii) ζi(ψ −1 (x + y)) = ζi(ψ −1 (x)) for every y ∈ S ∩ (V2 −x) and every i ∈ I.
Moreover, the degree of differentiability of ψ is the same as of ζi, for all i ∈ I. Proof. It suffices to show that LΩ(x) ⊆ TΩ(x). Let d ∈ LΩ(x) and suppose thatx satisfies the facial constant rank property. Let where Fj Km j , j ∈ I0(x), are defined as and R R |I B (x)| is given by where cj is the j-th vector of the canonical basis of R |I B (x)| , and J := {j ∈ IB(x) | ∇φj(x) T d = 0}. Recall that we are assuming for simplicity that IB(x) = {1, . . . , |IB(x)|}, and note that DG(x)d ∈ F . Now, for every j ∈ I0(x) such that Dgj(x)d ∈ bd + (Km j ), let Aj ∈ R m j ×m j −1 be any matrix with full column rank such that Im(Aj) = {Dgj (x)d} ⊥ , and observe that Finally, observe that R ⊥ = span({cj }j∈J ) and then Therefore, for every x ∈ V, where V is the neighborhood ofx given by Definition 4.1, the linear space is generated by the family of vectors: which implies that the dimension of (27) equals the rank of (28), for every x ∈ V. Since this dimension remains constant in V, so does the rank of (28). This means we can apply Proposition 4.1 to the family of functions where gj,i(x) denotes the i-th entry of gj(x) for j ∈ J. Then, consider the following subspace: and note that d ∈ S, so it follows that there exists a local diffeomorphism ψ for which items (i), (ii) and (iii) of Proposition 4.1 are satisfied. Now, define the arc ξ(t) by for t ∈ R small enough so thatx + td ∈ V2, where V2 is given by Proposition 4.1. Then, we obtain that To complete the proof, it suffices to show that ξ(t) remains feasible for every sufficiently small t ≥ 0, so this is our goal from this point onwards. Proposition 4.1 tells us that there exists some ε > 0 such that ζi,j(ξ(t)) = ζi,j(x) = 0 for every t ∈ (−ε, ε). In terms of F , this means that G(ξ(t)) ∈ span(F ) for every such t, which follows directly from (29). Now, let us analyse each case separately: 1. For each index j ∈ I0(x), consider the Taylor expansion of gj(ξ(t)) around t = 0, given by gj(ξ(t)) = gj (ξ(0)) + tDgj(ξ(0))ξ ′ (0) + o(t) Then, we split in three sub-cases: • If Dgj (x)d ∈ int(Km j ), then it follows from (30) that gj(ξ(t)) ∈ Km j for every t ∈ [0, ε), shrinking ε if necessary; • If Dgj (x)d ∈ bd + (Km j ), then gj(ξ(t)) ∈ span(Dgj(x)d) due to (25), and it follows from (30) that g(ξ(t)) ∈ cone(Dgj(x)d) for every t ∈ [0, ε), taking a smaller ε if needed; • If Dgj(x)d = 0, then g(ξ(t)) = 0 for every t ∈ [0, ε), due to (25). 2. Since φ(ξ(t)) ∈ R for every t ∈ [0, ε), for each index j ∈ J, we have φj(ξ(t)) = 0. On the other hand, for each j / ∈ J, consider the Taylor expansion of φj(ξ(t)) around t = 0: and since ∇φj (x) T d > 0 for every j / ∈ J, it also follows that φj(ξ(t)) > 0 for every t ∈ [0, ε), taking a smaller ε if necessary.
A useful information that can be extracted from the proof above is an equivalent characterization of the facial constant rank property (Definition 4.1) without faces: Corollary 4.1. Letx ∈ Ω. Then, the facial constant rank property holds atx if, and only if, there exists a neighborhood V ofx such that: for all subsets J1, J2 ⊆ I0(x), J3 ⊆ IB(x), such that mj > 1 for all j ∈ J1, and for all ηj ∈ bd + (Km j ), j ∈ J1, the rank of the family remains the same for all x ∈ V, where Aj ∈ R m j ×m j −1 can be any matrix with full column rank such that Im(Aj) = {ηj } ⊥ , for each j ∈ J1, and A i j denotes the i-th column of Aj. Before proceeding, we will make a short discussion about Theorem 4.1 and its implications: Note that if all constraints are affine, then every feasible point satisfies the facial constant rank property. Then, it follows from Theorem 4.2 that TΩ(x) = LΩ(x) in this case, for everyx ∈ Ω. We highlight this fact because when it is paired with Example 2.1, two things can be concluded: first, the facial constant rank property alone is not a CQ for (NSOCP); second, the only reason why constraint linearity is not a CQ for NSOCP is that H(x) may not be closed. When H(x) is closed, facial constant rank is a CQ, and so is constraint linearity. In other words, the above discussion, in view of the minimality of Guignard's CQ, allows us to conclude that the closedness of H(x) is the weakest CQ for linear second-order cone programming problems.
The discussion of Remark 4.1, together with Theorem 4.2, motivates our extension of CRCQ (and RCRCQ) to NSOCP: Definition 4.2. Letx be a feasible point of (NSOCP) and let H(x) be the set defined in (2). We say that the constant rank constraint qualification for NSOCP (CRCQ) holds atx, if it satisfies the facial constant rank property and, in addition, the set H(x) is closed.
When m1 = m2 = . . . = mq = 1, problem (NSOCP) reduces to a NLP problem. Moreover, since the faces of K1 are {0} and R+, the facial constant rank property (Definition 4.1) reduces to CRCQ in this case, and so does Definition 4.2. Moreover, as mentioned before, it follows directly from Theorem 4.1, that: Since the nondegeneracy condition for (NSOCP) holds at a givenx ∈ Ω if, and only if, DG(x) T is injective, then by continuity of DG, nondegeneracy implies that DG(x) T remains injective for every x close enough tox. Therefore, it follows that the nondegeneracy condition implies CRCQ as in Definition 4.2. However, the converse is not true, as it can be seen in the following example: at the feasible pointx = 0. Since K2 is a polyhedral cone and g is linear, then CRCQ as in Definition 4.2 holds atx. However, Robinson's CQ is not satisfied atx = 0, since for every d ∈ R. Consequently, nondegeneracy is not satisfied, either.
Observe that Example 4.1 also shows that CRCQ does not imply Robinson's CQ. Conversely, Robinson's CQ does not imply CRCQ either, meaning they are not related, just as it happens with CRCQ and MFCQ in NLP. Let us show this with an example:  This will lead to an extension of RCRCQ. An extension of the original CRCQ condition can be obtained by writing the equality constraint as a pair of inequality constraints in the form h(x) ∈ R p + and −h(x) ∈ R p + , just as in Remark 3.1, then reducing, and applying Definition 4.2 to the new reduced cone.

Strong second-order optimality conditions for NSOCP
In this subsection we will investigate second-order optimality conditions for (NSOCP) under the facial constant rank property; and, consequently, under CRCQ as well. Recall that the second-order condition of Definition 2.1 can be further specialized to the context of NSOCP by characterizing the sigma-term explicitly. Following Bonnans and Ramírez [22], we have for anyx ∈ Ω and any of its associate Lagrange multipliers λ := (λ1, . . . ,λq) ∈ Λ(x), that With this in mind, we can prove that SOC holds at (x,λ) under the facial constant rank property by means of analysing the problem along the curve ξ(t) from the proof of Theorem 4.1.
The above expression can be simplified using the relation which can be directly computed from the definition of φj, j ∈ IB(x), since in this case [gj(x)]0 = gj(x) and Dgj (x)d, Rm j gj(x) = 0. Then, we get Moreover, by the KKT conditions, we have which yields together with equation (35), the following: Therefore, sinceλj = 0 for every j ∈ Iint(x), Since d ∈ C(x) is arbitrary, we conclude thatx satisfies SOC with respect toλ, which was also chosen arbitrarily and remained fixed from the very beginning. Thus, the proof is complete.
Observe that Theorem 4.3 implies that the facial constant rank property ensures the fulfilment of the strong second-order necessary condition at a given pointx, in the sense that for everyλ ∈ Λ(x), and every d ∈ C(x), inequality (13) holds true. If, in addition, H(x) is closed (CRCQ), then Λ(x) = ∅, and as consequence, we obtain that the strong second-order condition is satisfied in the presence of CRCQ. It is also worth mentioning that since the strong necessary condition of Theorem 4.3 implies the classical condition of Theorem 2.2, then it also induces a sufficient second-order optimality condition after replacing ≥ by > in inequality (32).

Remark 4.3.
In contrast with the facial constant rank property, the condition presented in [61, Definition 2.1] fails to be a CQ even when H(x) is closed. In fact, let us recall the counterexample presented in [5]: The unique solution of this problem isx = 0. For this particular example, [61, Definition 2.1] holds if, and only if, {1, 1 + 2x} remain with constant rank in some neighborhood ofx (one may consider also all of its subfamilies, see [5]). Of course, this is verified, and since K2 is polyhedral, the set H(x) is closed. However, x does not satisfy the KKT conditions. On the other hand, to see that CRCQ as in Definition 4.2 does not hold atx, take F := cone((1, 1)) K2 and note that has dimension 1 for every x = 0, but has dimension zero atx. In particular, this example shows that CRCQ as in Definition 4.2 is not a mere correction of the condition presented in [61], and that the condition of [61] cannot be corrected by simply adding the closedness of H(x) to its definition.

About the sequential constant rank CQ
In [9], we introduced an alternative extension of CRCQ for (NSOCP) that was based on a special recharacterization of the nondegeneracy condition [7] in terms of the eigenvectors of some perturbations of g(x). Let us recall an equivalent characterization of it, which will be used here as a definition for simplicity.
This constraint qualification was used in [9] to achieve global convergence of a class of algorithms to KKT points, and some interesting properties were shown together with a weaker variant of Seq-CRCQ. Namely, it is also independent of Robinson's CQ, strictly weaker than nondegeneracy, and it implies the metric subregularity CQ (also known as error bound CQ). Moreover, note that if I0(x) = ∅, then Seq-CRCQ coincides with the facial constant rank property, which in turn coincides with CRCQ. However, this is not necessarily true otherwise. In the following example, we show that CRCQ according to Definition 4.2 does not imply Seq-CRCQ. g(x) = (x, −x, 0) ∈ K3, and letx = 0, a feasible point. Since g is affine, then the facial constant rank property holds atx (see Remark 4.1). Now, let us show that H(x) is closed: since g(x) = 0, it holds that Therefore, H(x) is a closed set, and CRCQ according to Definition 4.2 holds atx.
On the other hand, Seq-CRCQ does not hold atx, since for any w = (w1, w2) ∈ R 2 , then, takew = (1, 0) and any sequence {w k } k∈N →w such that w k 1 = 1 for all k ∈ N, to see that Dg(x) T (1, w k 1 ) = 0 for every k ∈ N, but Dg(x) T (1,w) = 0. We were not able to prove nor find a counterexample for the converse statement. However, with only Example 4.3 at hand, we already know that CRCQ is in the worst case independent of Seq-CRCQ, and in the best case, strictly weaker than it, meaning the results of this paper either improve or are parallel to the results of [9].

Nonlinear semidefinite programming
In this section, we will study constant rank conditions for nonlinear semidefinite programming problems, which can be stated in standard form as follows: This problem can be seen as a particular case of (NCP), letting E = S m be the space of m × m symmetric matrices with real entries, and be the cone of all m×m symmetric positive semidefinite matrices, with G : R n → E being twice continuously differentiable. The symbol denotes the partial order induced by S m + , meaning that A B if, and only if, A − B ∈ S m + . In this section, for any given A ∈ S m we will denote by µi(A) the i-th eigenvalue of A arranged in non-increasing order, and ui(A) will denote an associated unitary eigenvector.
Recall from Section 3 that the constant rank constraint qualification can be obtained in two steps: first, reduce the problem to consider only the locally relevant part of the constraint; then, analyse the image of the faces of the reduced cone by the derivative of the reduced constraint function. Therefore, we begin by recalling a matrix analysis lemma that will be useful for the reduction part.
Lemma 5.1. LetĀ 0 and denote by r the rank ofĀ. Also, letĒ ∈ R m×m−r be a matrix whose columns form an orthonormal basis of Ker(Ā). Then, there exists an analytic matrix function E : S m → R m×m−r such that E (Ā) =Ē and, for all A close enough toĀ, the columns of E (A) form an orthonormal basis of the space spanned by the eigenvectors associated with the m − r smallest eigenvalues of A. Now, letx ∈ Ω, denote by r the rank of G(x), and letĒ ∈ R m×m−r be an arbitrary matrix with orthonormal columns that span Ker(G(x)). Moreover, let E be the analytic matrix function with the properties described in Lemma 5.1, such that E (G(x)) =Ē. Observe that S m + is reducible to in a neighborhood N of G(x) by the mapping Ξ : N → S m−r given by for every Y ∈ N close enough to G(x) so that µi(Y ) > 0 for every i = 1, . . . , r. Then, define the function E := E • G, consider the reduced constraint function and for every x sufficiently close tox, we have that G(x) ∈ S m + if, and only if, G(x) ∈ S m−r + . Moreover, it is worth recalling that, since the function E is analytic, the degree of differentiability of G is the same as of G.
Following Bonnans and Shapiro [24,Equation 5.161], we see that the linearized cone of the original constraints of (NSDP) atx ∈ Ω can be written as which also coincides with the linearized cone of the reduced constraint atx, because E(x) =Ē and for each x close enough tox, we have Ē. For more details on this reduction approach, see [21,23].
In the next section, we will introduce a constant rank-type condition for NSDP by means of the faces of the reduced cone. Proof. Let d ∈ LΩ(x), denote by s the rank ofĒ T DG(x)dĒ, and letQ ∈ R m−r×m−r be an orthogonal matrix such thatQ
Remark 5.1. Similarly to Remark 4.1, we observe that if G is affine, then everyx ∈ Ω satisfies the facial constant rank property, which implies it is not a CQ on its own, unless H(x) as defined in (2) is closed. We remark this fact because it implies that the weakest CQ that guarantees zero duality gap in linear SDP problems is the closedness of H(x).
With this in mind, we present our extension of CRCQ (and RCRCQ) for NSDP: Definition 5.2 (CRCQ). Letx ∈ Ω. We say thatx satisfies the constant rank constraint qualification condition for NSDP (CRCQ) if it satisfies the facial constant rank property and, in addition, H(x) is closed.
And, as an immediate consequence of Theorem 5.1, we obtain the following: The only feasible point isx = 0, for which one has G(x) = 0 ∈ S 2 + . Since G is linear it is enough to show that H(x) is closed (see Remark 5.1). In this case, Since DG(x) T A = DG(x), A = a22 − a11, but a11 and a22 are both nonnegative, it follows that H(x) = R, which is closed. On the other hand, Robinson's CQ does not hold atx. In fact, given a real number d ∈ R, we have that which is not in int(S 2 + ) regardless of d ∈ R. The above example shows that CRCQ does not imply Robinson's CQ. Conversely, we will show in the next example, that Robinson's CQ does not imply CRCQ either. at the pointx = (0, 0). Then, for any direction d = (d1, d2) ∈ R 2 , it follows that and Robinson's CQ holds atx. In order to analyze whether CRCQ holds atx or not, letĒ ∈ R 2×2 be a given orthogonal matrix and let us define G(x) :=Ē T G(x)Ē. Then, take Then, since ∇G12(x) = [2x1, 0] T and ∇G22(x) = [0, 1] T , for every x ∈ R 2 , the dimension of the subspace above is 1 atx, but it is equal to 2 for every x close enough tox such that x1 = 0. Therefore, CRCQ does not hold atx. To extend CRCQ one should write the equality constraint as a pair of inequality constraints in the form h(x) ∈ R p + and −h(x) ∈ R p + , giving rise to a multifold NSDP problem where R+ is seen as a copy of S 1 + . For a last comparison, we should mention a constraint qualification presented in one of our previous works [8], which was called Sequential CRCQ (Seq-CRCQ) therein. As a matter of fact, Seq-CRCQ differs from Definition 5.2 in many aspects. For instance, Example 5.1 shows that Seq-CRCQ is not implied by CRCQ. This example has already appeared in [8,Example 4.1], where we show that Seq-CRCQ is not satisfied atx = 0; on the other hand, we showed in Example 5.1, that CRCQ holds atx. Thus, CRCQ is either strictly weaker than Seq-CRCQ, or completely independent of it. Despite our efforts to clarify the converse statement, we were not able to prove nor find a counterexample for it, so this is left as an open problem.

Strong second-order optimality conditions for NSDP
The earliest work that provides a practical characterization of the sigma-term in NSDP is Shapiro's [55], using second-order directional derivatives of the least eigenvalue function, µmin : and G(x) † denotes the Moore-Penrose pseudoinverse of G(x). Shapiro also proved that if a local minimizer x ∈ Ω satisfies nondegeneracy and its associated Lagrange multiplierλ ∈ Λ(x) is such that rank(λ) + rank(G(x)) = m (strict complementarity), thenx satisfies SOC with respect toλ. Later, other authors provided creative ways of obtaining SOC via some local reformulation of (NSDP) with no curvature. For instance, Lourenço et al. [43] wrote G(x) 0 in the form G(x) − Z 2 = 0 with an additional variable Z ∈ S m , and then obtained SOC for NSDP out of SOC for NLP -under nondegeneracy and strict complementarity. Forsgren [27] rediscovered Shapiro's characterization of the sigma-term and obtained SOC (under nondegeneracy, but without strict complementarity) using a special reformulation of the problem. Jarre [38] provided an elementary construction of SOC via a certain Schur complement, under nondegeneracy, strict complementarity, and assuming that the tangent cone of the linearized constraint G(x) + DG(x)d ∈ K coincides with TΩ(x). Fukuda et al. [28] used the characterization S m + = {Z ∈ S m | Π S m + (−Z) 2 = 0} combined with an external penalty method and the Clarke subdifferential of Π S m + , to achieve a weaker secondorder condition, which is stated only in terms of the lineality space of C(x). However, their results were obtained assuming only Robinson's CQ together with the so-called weak constant rank (WCR) property, which is not a CQ on its own.

Final remarks
The constant rank constraint qualification (CRCQ) is one of the most important regularity conditions in nonlinear programming (NLP), with several relevant applications regarding global convergence of algorithms, second-order optimality conditions, and some topics of stability theory. However, one of the main reasons why most of these interesting results still remain exclusive to NLP is that CRCQ itself seems intrinsic to NLP. Until very recently, there was no extension or analogue of it in the conic programming context. In a recent pair of papers [8,9], we presented an extension of CRCQ for nonlinear semidefinite and second-order cone programming using sequences and the eigenvector structure of their respective cones, which would allow us to adopt a strategy similar to the existing nonlinear programming literature. See also [7]. While this is interesting from the point of view of algorithms, it may not be an appropriate tool for other uses. Therefore, in this paper we adopted a more innovative approach: we first characterized CRCQ for NLP in a geometrical way, by means of the faces of a reduced cone, and then we showed this geometrical characterization could carry the essence of CRCQ to more general contexts. As far as we know, this is also the first time an intuitive interpretation of CRCQ was ever presented, and it is surprisingly simple: CRCQ describes the situations where every possible linear approximation of the feasible set (around a point of interest) preserves its dimension under small perturbations. As a side note, we should mention that this definition is either independent or strictly weaker than the ones presented in [8,9]. As an application of our results, we obtained a strong second-order necessary optimality condition under CRCQ, in terms of any given Lagrange multiplier. This improves the classical result that is obtained under nondegeneracy, and serves as an alternative for the condition that can be obtained under Robinson's CQ, where for each direction in the critical cone, there is a Lagrange multiplier satisfying the second-order condition. We expect CRCQ to be an alternative to Robinson's CQ in other situations, especially those related with stability analysis of parametric nonlinear conic optimization programs, in view of the nonlinear programming literature -see, for instance, references [34,37,48]. Since CRCQ is independent of Robinson's CQ, we believe that this work allows the development of a new parallel strand in the study of stability. In a recent work, Gfrerer and Mordukhovich [29], fully characterized tilt stable local minimizers of NLP problems under the so-called bounded extreme point property, which is implied by CRCQ (improving a previous work that assumed CRCQ and MFCQ [50]), and one of the possibilities of future work mentioned by them is an extension for conic programs. In a related work, still for NLP, Gfrerer and Outrata [30] obtained similar results to [29] as an application of the generalized derivative of a particular set-valued mapping, which was computed assuming the metric subregularity constraint qualification (MSCQ) at the point of interest plus Robinson's CQ in its neighborhood. The CRCQ condition as presented in this paper may replace these assumptions in a possible extension of their results to NSOCP and NSDP. Moreover, we expect CRCQ to be useful for supporting the convergence theory of some iterative algorithms, and also to encourage the development of algorithms that rely on faces for solving nonlinear conic problems.
Regarding prospective work, the techniques employed in this paper strongly suggest that a further extension of CRCQ, for general reducible cones, is possible. In this paper we adopted a more pragmatic approach by working explicitly with NSOCP and NSDP for clarity, leaving the investigation of a more general result to future works. In fact, it would also be interesting to not rely on reducibility at all, which should be possible by taking into account the faces of the tangent cone to K at g(x), or perturbations of it, instead of the faces of the reduced cone C. Furthermore, this work may inspire extensions of weaker constant rank-type conditions from NLP (together with their applications) to the conic environment, with emphasis on the well-established constant positive linear dependence condition [10,12,52] and the constant rank of the subspace component condition [11].

Funding
This work has received financial support from CEPID-CeMEAI (