Jacobian Nonsingularity in Nonlinear Symmetric Conic Programming Problems and Its Application

.is paper considers the nonlinear symmetric conic programming (NSCP) problems. Firstly, a type of strong sufficient optimality condition for NSCP problems in terms of a linear-quadratic term is introduced. .en, a sufficient condition of the nonsingularity of Clarke’s generalized Jacobian of the Karush–Kuhn–Tucker (KKT) system is demonstrated. At last, as an application, this property is used to obtain the local convergence properties of a sequential quadratic programming(SQP-) type method.


Introduction
In this paper, we consider the nonlinear symmetric conic programming (NSCP) as follows: where X and Y are two finite dimensional real vector spaces; f: X ⟶ R, h: X ⟶ R m , and g: X ⟶ Y are twice continuously differentiable functions; and K ∈ Y is a symmetric cone defined by Euclidean Jordan algebras. In the following part, unless otherwise specified, we denote X, Y, and Z to represent finite dimensional real vector spaces with a scalar product 〈·, ·〉 and norm ‖ · ‖.
It is well-known that the Karush-Kuhn-Tucker (KKT) conditions of optimization problem (1) are equivalent to the KKT system, which is a nonsmooth system with the metric projector over the symmetric cone. e nonsingularity of Clarke's generalized Jacobian of the KKT system introduced by Pang and Qi [1] is not only one of the most important concepts in perturbation analysis of optimization problems but also plays a vital part in the design of the algorithms and the analysis of the convergence [2][3][4].
When K in problem (1) is a polyhedral set, Robinson [5] has showed that the strong second-order sufficient condition and the LICQ imply the nonsingularity of Clarke's generalized Jacobian of the KKT system. Interestingly, the converse is also true [2,6,7]. Bonnans and Ramírez [8] and Sun [9] demonstrate the equivalent conditions to the nonsingularity of the second-order cone programming problem (SOCP) and the semidefinite programming problem (SDP), respectively.
When K is the class of C 2 -cone reducible sets ( [3], Definition 3.135), there are lots of most important results about the Aubin property and the robust isolated calmness of the KKT solution mapping, which guarantee the nonsingularity of the KKTsystem (see [10][11][12][13] and the references therein).
For symmetric cone programming problem, Kong, Tunçel, and Xiu [14][15][16] use a triangular representation of the Jacobian of Löwner operator to characterize the structure of Clarke's generalized Jacobian of metric projection operator onto symmetric cone. ey consider the linear symmetric cone programming problem as follows: min 〈c, x〉, and present five equivalent conditions to the nonsingularity of Clarke's generalized Jacobian of KKT system in [15].
In this paper, we focus on the nonsingularity of Clarke's generalized Jacobian of the KKT system in the setting of the nonlinear symmetric cone programming problem (1). In order to present the optimality conditions of NSCP, we need the variational analysis of symmetric cones and some important sets such as tangent cone. We found that, almost at the same time, we [17] and Kong et al. [15] independently obtained the same expressions of the tangent cone and so on by different approaches (see Proposition 2). Importantly, we introduce a linear-quadratic function to establish the second-order optimality conditions. Using the Euclidean Jordan Algebras and the Peirce decomposition of a finitedimensional vector space, we obtain an upper bound of the linear-quadratic function. Under the constraint nondegeneracy condition, we demonstrate our main result that if a kind of strong second-order sufficient condition holds, any element in Clarke's generalized Jacobian of the KKT system is nonsingular.
In [18], the local convergence for an SQP-type method is ensured by the nonsingularity of Clarke's generalized Jacobian. In this paper, as an application, we give an SQP-type method to solve NSCP (1). e analysis of the local convergence is presented by using the above properties of the nonsingularity, and our proof is a natural extension of the nonlinear programming problem. e paper is organized as follows. In Section 2, it gives some preliminaries which are used in the paper, including the fundamental notations in Euclidean Jordan algebras. e properties of a linear-quadratic function are developed. In Section 3, we describe the KKT condition and a kind of second-order sufficient condition of NSCP (1) using the linear-quadratic function. en, we discuss the nonsingularity of Clarke's generalized Jacobian of the equation reformulation of the KKT system. Lastly, the local convergence of a SQP-type method is analyzed by using the nonsingularity in Section 4.

Preliminaries
In this section, some preliminaries used in the paper are given firstly. en, we introduce the decomposition results in Euclidean Jordan algebras, which are vital to this paper.
For a locally Lipschitz continuous function Ξ: O⊆Y ⟶ Z, Clarke's generalized Jacobian of Ξ at y is defined by where O Ξ is the set of F-differentiable points in O and "conv" denotes the convex hull. e next conclusion of the chain rule is given in [19], which is stronger than its first version in [9]. Lemma 1. Suppose that Ψ: X ⟶ Y is a continuously differentiable function and Ξ: O⊆Y ⟶ Z is a locally Lipschitz continuous function. Denote y * ≔ Ψ(x * ) ∈ O and let Ψ ′ (x * ): X ⟶ Y be onto. en, the composite function e following conclusion of implicit functions can be obtained from [20] (Section 7.1) and [21] (Lemma 2) directly.

Lemma 2.
Let Φ: X × Y ⟶ X be a locally Lipschitz continuous function and Φ(x, ty) � 0. Suppose that any element in Π X zΦ(x, ty) is nonsingular. en, there exists a locally Lipschitz continuous function where In the last part of this section, we provide some properties about the metric projector over a convex set C in Banach space (see [22]).

Lemma 3.
Suppose that C is a convex set in a Banach space Z. en, for any y ∈ Z and V ∈zΠ C (y), V is self-adjoint. Furthermore, for any d ∈ Z, 〈d, V d〉 ≥ 0 and 〈V d, d − V d〉 ≥ 0.

Euclidean Jordan Algebras.
In this part, we show some useful notations and conclusions on Euclidean Jordan Algebras introduced in [23]. Suppose that F is the real field R and V is a finite-dimensional vector space over F .
For any x ∈ V, denote e pair A ≔ (V, ·) defined over the real field R is called a Euclidean Jordan algebra, if, for all x, y ∈ V: Here are some common concepts used in this paper. An element c ∈ V is called to be the unit element of A if x · e � e · x � x for all x ∈ V. We always assume that there exists a unit element e ∈ V of A � (V, ·) in the following paper.
If is called to be idempotent. If two idempotents p and q satisfy p · q � 0, they are called orthogonal. And k orthogonal idempotents c 1 , c 2 , . . . , c k are said to be a complete system if they satisfy 2 Mathematical Problems in Engineering If a nonzero idempotent q cannot be written as the sum of two other nonzero idempotents, we say q is primitive. And Jordan frame is a complete system of orthogonal primitive idempotents.
e following theorem in [23] is very important to show the spectral decomposition.
We say the numbers λ 1 (x), λ 2 (x), . . . , λ r (x) to be eigenvalues of x. en, x has the spectral decomposition (8) and erefore, another associative inner product can be defined by using tr(x): 〈x, y〉 ≔ tr(x · y), x, y ∈ V. Let ‖ · ‖ be the norm on V induced by this inner product, then For a scalar-valued function ϕ: R ⟶ R, we define Löwner's operator associated with A � (V, ·) from [24] as can be described by Löwner's operator using ϕ(t) � max(0, t) as follows: which is very important in the following research.
It is known from ( [23], eorem III.2.1) that the above cone K � y 2 : y ∈ V is a self-dual homogeneous closed convex cone, we call it a symmetric cone, which is the constraint set of problem (1) For a Jordan frame of A c 1 , c 2 , . . . , c r , we denote V ij as follows: Suppose that C ij (x) is the orthogonal projection onto V ij . en, C ij (x) has the following expression: where ∈ V denotes orthonormal vectors and d Actually, all the eigenvectors form an orthonormal basis of V.
Assume that there exist two integers s and s 1 such that (18) Let us introduce three index sets:

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For the merit projector Π K onto the symmetric cone, Sun and Sun [25] has proved that Π K is strongly semismooth everywhere. Wang [17] gave a characterization pertaining the structure of the B-subdifferential of Π K . Recently, Kong, Tunçel, and Xiu presented an exact expression for B-subdifferential and Clarke's generalized Jacobian of Π K [14,15]. Here, we rewrite the conclusion obtained by Kong et al. [15] as follows. Proposition 1. Let x � r j�1 λ j (x)c j and the indexsets α, β, c be given by (19).
where K |β| is the symmetric cone in subspace In order to describe the optimality conditions of NSCP (1), we need the expressions of some important sets, such as the tangent cone of K at x + (T K (x + )), the linearity of T K (x + )(lin(T K (x + ))), critical cone C(x + ), and the affine space of C(x + ), denoted by aff(C(x + )). As Wang [17] and Kong et al. [15] have obtained the characterizations independently, we only show the topological results in the following proposition. (18). It holds that

A Linear-Quadratic Function. Inspired by the works by
Bonnans and Shapiro [2] and Sun [9], we define a linearquadratic function as follows, which is vital for establishing a kind of strong second-order sufficient condition.

Definition 1. For any given
where v † is the Moore-Penrose pseudoinverse of v.
Here, the linear-quadratic function is different from Definition 3.2 in [15]. Even in the especial case, v ∈ K and v · z � 0, the value of the linear-quadratic function (26) is twice the one in [15].
For the linear-quadratic function (16), we have the following conclusions.
us, we assume that x has the following spectral decomposition: satisfying (18), and b and r have the spectral decompositions as follows: We can easily check that en, by the properties of the projection of the metric projector in Lemma 3, e proof is completed.

Optimality Conditions and Nonsingularity
We consider nonlinear conic problem (1). Let x belong to the feasible set of problem (1). If we say that Robinson's constraint qualification holds at x. en, there exists a Lagrange multiplier (y, z) ∈ R m × Y satisfying the following KKT conditions: where is the Lagrangian function of (1). Let Λ(x) ⊂ R m × Y be the set of all the Lagrangian multipliers. It is easy to verify the KKT conditions (42) which can be equivalently expressed as (44)

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For a general constraint G(x) ∈ K ′ , x ∈ X, where G: X ⟶ Y ′ is a continuously differentiable function and K ′ ⊆Y ′ is a closed convex set, if for a feasible point x, then the constraint nondegeneracy condition holds at x. So, for conic optimization problem (1), the constraint nondegeneracy condition at x * has the following form: It follows from [2] that, if a locally optimal solution x * satisfies (46), then Λ(x * ) is a singleton.
For a KKT point (x * , y * , z * ) ∈ X × R m × Y of problem (1), suppose that u * ≔ g(x * ) − z * has the spectral decomposition: satisfying According to (23) and (25), we have Although the critical cone C(x * ) of problem (1) has an explicit formula aff C(x * ) is not easy to describe. Instead, we define the following outer approximation set to aff C(x * ) with respect to (y * , z * ): It is easy to get that, for any (y * , z * ) ∈ Λ(x * ), We now introduce a kind of strong second-order sufficient condition for problem (1), which is coincided with the strong second-order sufficient condition in [8,9] when K is a SDP cone and a second-order cone.
Definition 2. Let x * be a feasible point of (1) such that constraint nondegeneracy condition (46) holds at x * . We say that the strong second-order sufficient condition holds at where (y * , z * ) � Λ(x * ) ⊂ R m × Y and app (y * , z * ) is defined by (52).
e following theorem establishes the relationship between the strong second-order sufficient condition and the nonsingularity of Clarke's Jacobian of the mapping F defined by (44).

Theorem 2.
Let x * is a local minimizer of (1). Assume that (x * , y * , z * ) ∈ X × R m × Y is a KKT point to (1). If the constraint nondegeneracy condition and the second-order sufficient condition (53) hold at x * , then any element in zF(x * , y * , z * ) is nonsingular.
Step 2: calculate M k � M(x k , y k , z k ) and find a KKT point (Δx k , y k QP , z k QP ) of (Sub k ) by solving the KKT system (67).
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Conclusion
In this paper, we discuss the nonlinear symmetric conic programming problems. We show that a kind of strong second-order sufficient condition, together with constraint nondegeneracy condition, implies the nonsingularity of Clarke's generalized Jacobian of the mapping F at the KKT point. In the special cases of NLP, SCOP, and SDP, the converses are also true. en, we demonstrate the local quadratic convergence of the SQP-type method for solving the nonlinear symmetric conic programming problems.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.