Jacobian Nonsingularity in Nonlinear Symmetric Conic Programming Problems and Its Application

Wang, Yun; Kong, Dezhou

doi:https://doi.org/10.1155/2020/8824126

Mathematical Problems in Engineering

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Fractional Mathematical Modelling and Optimal Control Problems of Differential Equations

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Volume 2020 | Article ID 8824126 | https://doi.org/10.1155/2020/8824126

Jacobian Nonsingularity in Nonlinear Symmetric Conic Programming Problems and Its Application

Yun Wang¹and Dezhou Kong¹

Academic Editor: Xinguang Zhang

Received22 Sept 2020

Revised04 Nov 2020

Accepted18 Nov 2020

Published31 Dec 2020

Abstract

This 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. Then, 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.

1. Introduction

In this paper, we consider the nonlinear symmetric conic programming (NSCP) as follows:where and are two finite dimensional real vector spaces; , , and are twice continuously differentiable functions; and is a symmetric cone defined by Euclidean Jordan algebras. In the following part, unless otherwise specified, we denote , and 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. The 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 [2–4].

When 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 is the class of -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 KKT system (see [10–13] and the references therein).

For symmetric cone programming problem, Kong, Tunçel, and Xiu [14–16] 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. They consider the linear symmetric cone programming problem as follows: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 finite-dimensional 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). The 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.

The 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. The 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. Then, 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.

2. Preliminaries

In this section, some preliminaries used in the paper are given firstly. Then, we introduce the decomposition results in Euclidean Jordan algebras, which are vital to this paper.

For a locally Lipschitz continuous function , Clarke’s generalized Jacobian of at is defined bywhere is the set of F-differentiable points in and “conv” denotes the convex hull.

The next conclusion of the chain rule is given in [19], which is stronger than its first version in [9].

Lemma 1. Suppose that is a continuously differentiable function and is a locally Lipschitz continuous function. Denote and let be onto. Then, the composite function is F-differentiable at if and only if is F-differentiable at , where is an open neighborhood of and

The following conclusion of implicit functions can be obtained from [20] (Section 7.1) and [21] (Lemma 2) directly.

Lemma 2. Let be a locally Lipschitz continuous function and . Suppose that any element in is nonsingular. Then, there exists a locally Lipschitz continuous function satisfying andwhere is an open neighborhood of . Furthermore, if is (strongly) semismooth, then is (strongly) semismooth.

In the last part of this section, we provide some properties about the metric projector over a convex set in Banach space (see [22]).

Lemma 3. Suppose that is a convex set in a Banach space . Then, for any and , is self-adjoint. Furthermore, for any , and .

2.1. Euclidean Jordan Algebras

In this part, we show some useful notations and conclusions on Euclidean Jordan Algebras introduced in [23]. Suppose that is the real field and is a finite-dimensional vector space over .

For any , denote

The pair defined over the real field is called a Euclidean Jordan algebra, if, for all :(i)(ii), where (iii)

Here are some common concepts used in this paper.

An element is called to be the unit element of if for all . We always assume that there exists a unit element of in the following paper.

If is called to be idempotent. If two idempotents and satisfy , they are called orthogonal. And orthogonal idempotents are said to be a complete system if they satisfy

If a nonzero idempotent cannot be written as the sum of two other nonzero idempotents, we say is primitive. And Jordan frame is a complete system of orthogonal primitive idempotents.

The following theorem in [23] is very important to show the spectral decomposition.

Theorem 1. Let be a Euclidean Jordan algebra and . Then, for any ,where is a Jordan frame and , satisfying .

We say the numbers to be eigenvalues of . Then, has the spectral decomposition (8) and

Therefore, another associative inner product can be defined by using : . Let be the norm on induced by this inner product, then

For a scalar-valued function , we define Löwner’s operator associated with from [24] aswhere .

The metric projection operator oncan be described by Löwner’s operator using as follows:which is very important in the following research.

It is known from ([23], Theorem III.2.1) that the above cone 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 , we denote as follows:

Suppose that is the orthogonal projection onto . Then, has the following expression:where denotes orthonormal vectors and satisfies .

It follows from [25] that

Actually, all the eigenvectorsform an orthonormal basis of .

Assume that there exist two integers and such that

Let us introduce three index sets:

For , denote

For the merit projector onto the symmetric cone, Sun and Sun [25] has proved that is strongly semismooth everywhere. Wang [17] gave a characterization pertaining the structure of the B-subdifferential of . Recently, Kong, Tunçel, and Xiu presented an exact expression for B-subdifferential and Clarke’s generalized Jacobian of [14, 15]. Here, we rewrite the conclusion obtained by Kong et al. [15] as follows.

Proposition 1. Let and the indexsets be given by (19). Then, for any , there exists such thatwhere 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 , the linearity of , critical cone , and the affine space of , denoted by . As Wang [17] and Kong et al. [15] have obtained the characterizations independently, we only show the topological results in the following proposition.

Proposition 2. Let have eigenvalues as in (18). It holds that

2.2. A Linear-Quadratic Function

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

Definition 1. For any given , define a linear-quadratic function bywhere is the Moore–Penrose pseudoinverse of .

Here, the linear-quadratic function is different from Definition 3.2 in [15]. Even in the especial case, and , 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.

Proposition 3. If , then

Proof. As we haveand (28) comes from the definition of .

Lemma 4. Let and . Then, for any ,where satisfying .

Proof. Denote . Then,Thus, we assume that has the following spectral decomposition:satisfying (18), and and have the spectral decompositions as follows:From Proposition 1, we have, for any , there exists such thatTherefore, we have from thatwhich impliesWe can easily check thatThen, by the properties of the projection of the metric projector in Lemma 3,Hence, by equations (35)–(37), we obtain from thatThe proof is completed.

3. Optimality Conditions and Nonsingularity

We consider nonlinear conic problem (1). Let belong to the feasible set of problem (1). Ifwe say that Robinson’s constraint qualification holds at . Then, there exists a Lagrange multiplier satisfying the following KKT conditions:whereis the Lagrangian function of (1). Let be the set of all the Lagrangian multipliers.

It is easy to verify the KKT conditions (42) which can be equivalently expressed as

For a general constraint , where is a continuously differentiable function and is a closed convex set, if for a feasible point ,then the constraint nondegeneracy condition holds at . So, for conic optimization problem (1), the constraint nondegeneracy condition at has the following form:

It follows from [2] that, if a locally optimal solution satisfies (46), then is a singleton.

For a KKT point of problem (1), suppose that has the spectral decomposition:satisfying

Then,

According to (23) and (25), we have

Although the critical cone of problem (1) has an explicit formula is not easy to describe. Instead, we define the following outer approximation set to with respect to :

It is easy to get that, for any ,

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 is a SDP cone and a second-order cone.

Definition 2. Let be a feasible point of (1) such that constraint nondegeneracy condition (46) holds at . We say that the strong second-order sufficient condition holds at ifwhere and is defined by (52).

The following theorem establishes the relationship between the strong second-order sufficient condition and the nonsingularity of Clarke’s Jacobian of the mapping defined by (44).

Theorem 2. Let is a local minimizer of (1). Assume that is a KKT point to (1). If the constraint nondegeneracy condition and the second-order sufficient condition (53) hold at , then any element in is nonsingular.

Proof. Firstly, we assume that the strong second-order sufficient condition (53) holds at with the constraint nondegeneracy condition (46). We shall prove that any is nonsingular. Let satisfying the conditionSuppose that has the spectral decomposition (47) satisfying (48), then we can write and in the form of (49). From Lemma 1, there exists such thatFrom the third equality in (56), namely,it hasThen, it implies from (52) and (56) thatCombining with the last equation of (36) and Lemma 4, we obtainComparing with the strong second-order sufficient condition in the strong form (53), we get . Thus, (36) can be expressed in the following form:From , we haveThen, it follows from the nondegeneracy (46) that there exists a vector and satisfyingTherefore, we havewhere the last equation can be obtained by (50) and (62). Thus, Together with , we get that is nonsingular. The proof is completed.

4. Application

In this section, as an application of the nonsingularity in Theorem 2, we will study the sequential quadratic programming- (SQP-) type method to solving problem (1). Let be the current iteration point. The new iteration points will be generated by solving the following quadratic problem:

()where and is a matrix function satisfying . This model introduced in [18] to solve the classical nonlinear programming problems is used to solve the nonlinear SDP problems [26, 27] and nonlinear second-order cone programming problems [28, 29], where they may choose the different forms of .

Theorem 3. Suppose that is a KKT point to (1) and the second-order sufficient condition (53) with constraint nondegeneracy condition holds at . Let the matrix function satisfying be semismooth at . Then, for any , a neighborhood of , (39) has a KKT solution satisfying

Proof. Let be a KKT point of (). Then,namely,whereis a function with the variable . Let and . Let and . We can easily have the following equation:By replacing in (65) with , we obtain a new problem expressed as (), and is an KKT point of ().
Therefore, the strong second-order sufficient condition of (Sub) at under the constraint nondegeneracy condition (46) has the same expression as condition (53). It is concluded from Theorem 2 that any element is nonsingular. Then, there exists a strong semismooth function satisfyingwhere is a neighborhood of . Denote . Then, is a KKT point of () if . By the strong semismoothness of ,The proof is completed.
Now we present a local SQP method based on solving SQP-type model () in each iteration to solve (1).

Next, the primal-dual quadratic convergence of Algorithm 1 is demonstrated by using the semismoothness of M and Theorem 2.

	Step 0: for a given initial point , calculate the value of and . Set .
	Step 1: if , stop.
	Step 2: calculate and find a KKT point of () by solving the KKT system (67).
	Step 3: set .
	Step 4: calculate . Set . Go to step 1.

Theorem 4. Suppose that all the hypotheses of Theorem 3 still hold for Algorithm 1. Then, there exists a neighborhood of such that, for any , the sequence generated by Algorithm 1 converges to quadratically.

Proof. According to Theorem 3, it is easy to verify that the algorithm is well-defined. DenoteWe obtainwhere is a solution to (65), and is the associated multiplier.
As and is semismooth at ,It follows from the Taylor expansion to (67) at , , (74), and (75) thatBecause of the strongly semismoothness of the projection operator , we get satisfyingSinceit is known thatwhich, together with and , impliesFollowing from (76), (77), with (81), we obtainCombining (46) and (54) with Theorem 2, we show thatwhich means the quadratic convergence of the sequence . Then, we complete the proof.

5. 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 at the KKT point. In the special cases of NLP, SCOP, and SDP, the converses are also true. Then, 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

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the National Natural Science Foundation of China under project nos.61573228 and 11501080 and by Natural Science Foundation of Shandong Province of China (no.ZR2017MA034).

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Copyright © 2020 Yun Wang and Dezhou Kong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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