Lipschitz Continuity of the Solution Mapping of Symmetric Cone Complementarity Problems

Abstract

This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (linear or nonlinear) complementarity problems (SCLCP or SCCP, resp.) over Euclidean Jordan algebras. We show that if the transformation has uniform Cartesian P-property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultra P-property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultra P-property, the Cartesian P-property, and the Lipschitz continuity of the solutions are all equivalent to each other.

1. Introduction

Let (𝕍,∘,⟨⋅,⋅⟩) (we use 𝕍 for short in subsequent content) be a Euclidean Jordan algebra and 𝒦 be the symmetric cone in 𝕍. Given a continuous transformation 𝐹∶𝕍→𝕍 and 𝑞∈𝕍, the symmetric cone complementarity problem denoted by SCCP(𝐹,𝒦,𝑞) is to find a vector 𝑥∈𝕍 such that 𝑥∈𝒦,𝐹(𝑥)+𝑞∈𝒦,𝑥,𝐹(𝑥)+𝑞=0. (1.1) When 𝐹 reduces to a linear transformation 𝐿, the above problem is called the symmetric cone linear complementarity problem and is denoted by SCLCP(𝐿,𝒦,𝑞), that is, the symmetric cone linear complementarity problem is to find a vector 𝑥∈𝕍 such that 𝑥∈𝒦,𝐿(𝑥)+𝑞∈𝒦,𝑥,𝐿(𝑥)+𝑞=0. (1.2) These classes of symmetric cone complementarity problems provide a unified framework for the linear or nonlinear complementarity problems (LCP or NCP, resp.) over the nonnegative orthant cone in ℝ^𝑛, that is, 𝕍=ℝ^𝑛 and 𝒦=ℝ^𝑛₊ (see [1–4]), the second-order cone (linear or nonlinear) complementarity problems (SOCLCP or SOCCP, resp.), that is, 𝕍=ℝ^𝑛 and 𝒦=𝒦^𝑛 (see [5–8]), and the semidefinite (linear or nonlinear) complementarity problems (SDLCP or SDCP, resp.), that is, 𝕍=𝑆^𝑛 and 𝒦=𝑆^𝑛₊ (see [9–12]). It is also known that the complementarity problem is special case of variational inequality problem which has a wide range of applications, see [3, 9].

One of the important issues in complementarity problems is to characterize the Lipschitz continuity of its solutions (or called the Lipschitz continuity of solution mapping) with respect to 𝑞. For 𝑞∈𝕍, let 𝜙_𝐹(𝑞) be the set of all solutions to SCCP(𝐹,𝒦,𝑞). Then, we intend to know under what conditions the multivalued solution mapping 𝜙_𝐹∶𝑞↦𝜙_𝐹(𝑞) of SCCP(𝐹,𝒦,𝑞) is Lipschitz continuous. In other words, under what conditions, there will exist 𝜅>0 such that 𝜙_𝐹𝑞₁⊆𝜙_𝐹𝑞₂+𝜅𝑞₁−𝑞₂𝐵 (1.3) for all 𝑞₁, 𝑞₂∈𝕍 satisfying 𝜙_𝐹(𝑞₁)≠∅ and 𝜙_𝐹(𝑞₂)≠∅, where 𝐵 is the closed unit ball in 𝕍. That is, if 𝑥₁∈𝜙_𝐹(𝑞₁) there exists 𝑥₂∈𝜙_𝐹(𝑞₂) such that 𝑥₁−𝑥₂≤𝜅𝑞₁−𝑞₂. (1.4) Note that the Lipschitz constant 𝜅 depends only on the continuous transformation 𝐹. Below is a brief history regarding this issue. For LCP(𝑀,𝑞), it is well known that the Lipschitz continuity of the solution mapping with respect to 𝑞∈𝕍 can be described in any one of the following ways:(i)the matrix 𝑀 is 𝑃-matrix (see [13, 14]);(ii)LCP(𝑀,𝑞) has a unique solution for all 𝑞∈ℝ^𝑛 (i.e., GUS-property of 𝑀);(iii)for any 𝑞∈ℝ^𝑛, the solution set 𝜙_𝑀(𝑞)≠∅ and the set-valued mapping 𝑞→𝜙_𝑀(𝑞) are Lipschitzian.In particular, Mangasarian and Shiau [14] showed that if 𝑀 is a 𝑃-matrix, then solutions of linear inequalities, programs, and LCP are Lipschitz continuous. Murthy et al. [15] showed that 𝑀 is a 𝑃-matrix if and only if the LCP(𝑀,𝑞) has a solution for all 𝑞∈ℝ^𝑛 and the solution mapping is Lipschitzian. Gowda and Sznajder [16] generalized the above result to affine variational inequality problems, while Yen [17] studied Lipschitz continuity of the solution mapping of variational inequalities with a parametric polyhedral constraint. As for NCP, Levy [18] obtained that the solution mapping is locally single-valued and Lipschitz continuous under suitable conditions. How about when 𝒦 is nonpolyhedral? Balaji et al. [19] proved that 𝐿 being monotone and the Lipschitz continuity of the solution mapping of SDLCP imply the GUS-property, while Chen and Qi in [9] employed Cartesian 𝑃-property to guarantee the GUS-property and the locally Lipschitzian property of the solution mapping of SDLCP. These make a complete extension of (i)–(iii) to their counterparts in SDLCP. A natural question arises here: can the above results be extended to a general symmetric cone case which is a unified framework?

In fact, there has been some papers dealing with the SCLCP over Euclidean Jordan algebras. For example, Balaji [20] established the result that if 𝐿 has the Lipschitzian 𝑄-property, then 𝐿 has the positive principal minor property. Gowda et al. [21] showed that if 𝐿 has 𝑃-property, then SCLCP(𝐿,𝒦,𝑞) has a nonempty compact set for all 𝑞∈𝕍. In addition, Tao and Gowda [22] used degree-theoretic arguments to show that under a certain 𝑅₀-type condition, every 𝑃₀ symmetric cone nonlinear complementarity problem SCCP(𝐹,𝒦,𝑞) has a solution. However, it still remains open under what conditions the solution map 𝜙_𝐹∶𝑞↦𝜙_𝐹(𝑞) of SCCP(𝐹,𝒦,𝑞) is Lipschitz continuous. In this paper, we explore new results regarding Lipschitz continuity of the solution mapping of the SCLCP(𝐿,𝒦,𝑞) or SCCP(𝐹,𝒦,𝑞) over Euclidean Jordan algebras. In Theorem 3.1, we show that if the transformation 𝐹 has the uniform Cartesian 𝑃-property with modulus 𝜌>0, then the solution mapping 𝜙_𝐹 is Lipschitz continuous with respect to 𝑞∈𝕍. Meanwhile, we give examples to show that the solution mapping of nonstrong monotone SCLCP(𝐿,𝒦,𝑞) is not Lipschitz continuous with respect to 𝑞, and GUS-property does not imply Lipschitz continuity of the solution mapping.

On the other hand, various 𝑃-properties and GUS-property have been investigated in the literature [4, 9, 10, 13, 16, 19, 21–24]. Relations among them are well studied as well. In [19, Theorem 2.2], it is proved that if the linear transformation 𝐿 in SDLCP has the monotonicity property and 𝜙_𝐿 is Lipschitzian, then 𝐿 has the 𝑃₂-property and the GUS-property. The concept of 𝑃₂-property in 𝑆^𝑛 was extended to a general Euclidean Jordan algebra, called ultra 𝑃-property [23]. Hence, it is desirable to know whether [19, Theorem 2.2] can be true or not in SCLCP(𝐿,𝒦,𝑞) if 𝑃₂-property is replaced by ultra 𝑃-property. In this paper, we answer this question positively, see Theorem 3.8. Further, for the Lyapunov transformation 𝐿_𝑎, we present several equivalent conditions for the ultra 𝑃-property of 𝐿_𝑎.

Next are a few words about notations and some basic concepts employed. For a vector 𝑥∈𝕍, the norm is denoted by ‖𝑥‖∶=⟨𝑥,𝑥⟩, where ⟨⋅,⋅⟩ denotes the Euclidean inner product. For the Euclidean Jordan algebra 𝕍, let ℒ(𝕍) denote the set of all continuous linear transformation 𝐿∶𝕍→𝕍, and Aut(𝒦) denote the set of all (invertible) linear transformations Γ∶𝕍→𝕍 such that Γ(𝒦)=𝒦. For the convex set 𝒦, let int(𝒦) denote the interior of the 𝒦. 𝐿^𝑇 means the adjoint operator of 𝐿. The identical transformation on 𝕍 will be denoted by 𝐼. For the SCCP(𝐹,𝒦,𝑞), the solution set of SCCP(𝐹,𝒦,𝑞) is denoted by 𝜙_𝐹(𝑞). For the SCLCP(𝐿,𝒦,𝑞), the solution set of SCLCP(𝐿,𝒦,𝑞) is denoted by SOL(𝐿,𝒦,𝑞) or 𝜙_𝐿(𝑞).

2. Preliminaries

In this section, we briefly recall some basic concepts and background materials in Euclidean Jordan algebras, which will be used in the subsequent analysis. More details can be found in [21–23, 25].

An Euclidean Jordan algebra is a triple (𝕍,∘,⟨⋅,⋅⟩) (𝕍 for short), where 𝕍 is a finite-dimensional inner product over ℝ and (𝑥,𝑦)↦𝑥∘𝑦∶𝕍×𝕍→𝕍 is a bilinear mapping satisfying the following three conditions:(i)𝑥∘𝑦=𝑦∘𝑥 for all 𝑥, 𝑦∈𝕍; (ii)𝑥∘(𝑥²∘𝑦)=𝑥²∘(𝑥∘𝑦) for all 𝑥, 𝑦∈𝕍, where 𝑥²=𝑥∘𝑥; (iii)⟨𝑥∘𝑦,𝑧⟩=⟨𝑥,𝑦∘𝑧⟩ for all 𝑥, 𝑦, 𝑧∈𝕍. We call 𝑥∘𝑦 the Jordan product of 𝑥 and 𝑦. In addition, if there is an element 𝑒∈𝕍 such that 𝑥∘𝑒=𝑥 for all 𝑥∈𝕍, the element 𝑒 is called the identity element in 𝕍. In a given Euclidean Jordan algebra 𝕍, the set of squares 𝒦∶={𝑥²∶𝑥∈𝕍} is a symmetric cone [25, Theorem III.2.1]. In other words, 𝒦 is a self-dual closed convex cone, and, for any two elements 𝑥, 𝑦∈int(𝒦), there exists an invertible linear transformation Γ∶𝕍→𝕍 such that Γ(𝑥)=𝑦 and Γ(𝒦)=𝒦. For any 𝑥∈𝕍, we write 𝑥∈𝒦(𝑥∈int(𝒦))⟺𝑥≽0(𝑥≻0). (2.1) An element 𝑐∈𝕍 such that 𝑐²=𝑐 is called an idempotent in 𝕍; it is a primitive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempotents. We say that a finite set {𝑒₁,𝑒₂,…,𝑒_𝑟} of primitive idempotents in 𝕍 is a Jordan frame if 𝑒_𝑖∘𝑒_𝑗=0for𝑖≠𝑗,^𝑟_𝑖=1𝑒_𝑖=𝑒, (2.2) where 𝑟 is called the rank of 𝕍. Now, we recall the spectral and Peirce decompositions of an element 𝑥 in 𝕍.

Theorem 2.1 ((spectral decomposition) [25, Theorem III.1.2]). Let 𝕍 be an Euclidean Jordan algebra. Then, there is a number 𝑟 such that for every 𝑥∈𝕍, there exists a Jordan frame {𝑒₁,𝑒₂,…,𝑒_𝑟} and real numbers 𝜆₁,𝜆₂,…,𝜆_𝑟 with 𝑥=𝜆₁𝑒₁+⋯+𝜆_𝑟𝑒_𝑟. (2.3) Here, the numbers 𝜆_𝑖 for 𝑖=1,…,𝑟 are the eigenvalues of 𝑥 and the expression 𝜆₁𝑒₁+⋯+𝜆_𝑟𝑒_𝑟 is the spectral decomposition (or the spectral expansion) of 𝑥.

In an Euclidean Jordan algebra 𝕍, corresponding to the convex cone 𝒦, let Π_𝒦 denote the metric projection onto 𝒦, namely, for an 𝑥∈𝕍, 𝑥^∗=Π_𝒦(𝑥) if and only if 𝑥^∗∈𝒦 and ‖𝑥−𝑥^∗‖≤‖𝑥−𝑦‖ for all 𝑦∈𝒦. It is well known that 𝑥^∗ is unique. For any 𝑥∈𝕍, combining the spectral decomposition of 𝑥 with the metric projection of 𝑥 onto 𝒦, we have the expression of metric projection Π_𝒦(𝑥) as follows (see [21]): Π_𝒦(𝑥)=max0,𝜆₁𝑒₁+⋯+max0,𝜆_𝑟𝑒_𝑟. (2.4)

The Peirce Decomposition
Fix a Jordan frame {𝑒₁,𝑒₂,…,𝑒_𝑟} in an Euclidean Jordan algebra 𝕍. For 𝑖,𝑗∈{1,2,…,𝑟}, we define the following eigenspaces: 𝕍_𝑖𝑖∶=𝑥∈𝕍∣𝑥∘𝑒_𝑖=𝑥=ℝ𝑒_𝑖,𝕍_𝑖𝑗∶=𝑥∈𝕍∣𝑥∘𝑒_𝑖=12𝑥=𝑥∘𝑒_𝑗for𝑖≠𝑗. (2.5)

Theorem 2.2 (see [25, Theorem IV.2.1]). The space 𝕍 is the orthogonal direct sum of spaces 𝕍_𝑖𝑗  (𝑖≤𝑗). Furthermore, 𝕍_𝑖𝑗∘𝕍_𝑖𝑗⊂𝕍_𝑖𝑖+𝕍_𝑗𝑗,𝕍_𝑖𝑗∘𝕍_𝑗𝑘⊂𝕍_𝑖𝑘𝕍,if𝑖≠𝑘,_𝑖𝑗∘𝕍_𝑘𝑙={0},if𝑖,𝑗∩{𝑘,𝑙}=∅. (2.6) Hence, given any Jordan frame {𝑒₁,𝑒₂,…,𝑒_𝑟}, we can write any element 𝑥∈𝕍 as 𝑥=^𝑟_𝑖=1𝑥_𝑖𝑒_𝑖+_𝑖<𝑗𝑥_𝑖𝑗, (2.7) where 𝑥_𝑖∈ℝ and 𝑥_𝑖𝑗∈𝕍_𝑖𝑗. The expression ∑^𝑟_𝑖=1𝑥_𝑖𝑒_𝑖+∑_𝑖<𝑗𝑥_𝑖𝑗 is called the Peirce decomposition of 𝑥.

Next, we recall concept of Lyapunov transformation and its relevant conclusions which will be used in our analysis later. In an Euclidean Jordan algebra 𝕍, for any 𝑥∈𝕍, we define the corresponding Lyapunov transformation 𝐿_𝑥∶𝕍→𝕍 by 𝐿_𝑥(𝑧)=𝑥∘𝑧 for any 𝑧∈𝕍. As remarked in [21, page 209], traditionally, the notation 𝐿(𝑥) has been used the Lyapunov transformation [25]. As employed in [21], we also reserve the notation 𝐿_𝑥 for the Lyapunov transformation and write 𝐿(𝑥) to denote the image of an element 𝑥∈𝕍 under a linear transformation 𝐿∶𝕍→𝕍. We say that elements 𝑥 and 𝑦  operator commute if 𝐿_𝑥𝐿_𝑦=𝐿_𝑦𝐿_𝑥. It is well known that 𝑥 and 𝑦 operator commute if and only if 𝑥 and 𝑦 have their spectral decompositions with respect to a common Jordan frame [25, Lemma X.2.2].

Property 1 (see [21, Proposition 6]). For 𝑥, 𝑦∈𝕍, the following conditions are equivalent:(a)𝑥≽0, 𝑦≽0, and ⟨𝑥,𝑦⟩=0;(b)𝑥≽0, 𝑦≽0, and 𝑥∘𝑦=0. Moreover, in this case, elements 𝑥 and 𝑦 operator commute. That is, 𝑥 and 𝑦 have their spectral decompositions with respect to a common Jordan frame.

In fact, from Property 1 and definition of (1.1), it can be seen that SCCP(𝐹,𝒦,𝑞) is equivalent to find a 𝑥∈𝕍 such that 𝑥∈𝒦,𝐹(𝑥)+𝑞∈𝒦,𝑥∘𝐹(𝑥)+𝑞=0. (2.8) In addition, if 𝑥 is a solution of SCCP(𝐹,𝒦,𝑞), then 𝑥 and 𝐹(𝑥)+𝑞 operator commute. Now, we review various monotonicity and 𝑃-property for a continuous transformation 𝐹∶𝕍→𝕍.

Definition 2.3. Let 𝕍 be an Euclidean Jordan algebra. A continuous transformation 𝐹∶𝕍→𝕍 is said to be(a)monotone if ⟨𝐹(𝑥)−𝐹(𝑦),𝑥−𝑦⟩≥0, for all 𝑥, 𝑦∈𝕍;(b)strictly monotone if ⟨𝐹(𝑥)−𝐹(𝑦),𝑥−𝑦⟩>0, for all 𝑥≠𝑦∈𝕍;(c)strongly monotone if there is 𝛼>0 such that 𝐹(𝑥)−𝐹𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝕍. (2.9)
It is said to have(d) GUS-property if SCCP(𝐹,𝒦,𝑞) has a unique solution for any 𝑞∈𝕍;(e)𝑃-property if (𝑥)𝑦∘𝐹(𝑥)𝑦𝑥−𝑦and𝐹−𝐹operatorcommute𝑥−𝑦−𝐹⪯0⇒𝑥=𝑦; (2.10)(f)𝑄-property if 𝜙_𝐹(𝑞)≠∅ for any 𝑞∈𝕍.

Remark 2.4. (i) When 𝐹 is linear, strict monotonicity and strong monotonicity coincide. When 𝐹 is nonlinear, strong monotonicity implies strict monotonicity.
(ii) Whether 𝐹 is linear or nonlinear, we have the following implications [22–24]: strongmonotonicity⇒strictmonotonicity⇒𝑃-property⇒𝑄-property,strongmonotonicity⇒GUS-property⇒𝑃-property. (2.11)
(iii) When 𝕍=ℝ^𝑛 and 𝒦=ℝ^𝑛₊, GUS-property and 𝑃-property coincide. But, once 𝕍 and 𝒦 are the other cases, for example, 𝕍=ℝ^𝑛 and 𝒦=𝒦^𝑛, where 𝒦^𝑛 denotes the second-order cone, or 𝕍=𝕊^𝑛 and 𝒦=𝑆^𝑛₊, and so forth. GUS-property is not equivalent to 𝑃-property.

Given an Euclidean Jordan algebra 𝕍 with dim(𝕍)=𝑛>1, from [25, Proposition III 4.4-4.5 and Theorem V.3.7], we know that any Euclidean Jordan algebra 𝕍 and its corresponding symmetric cone 𝒦 are, in a unique way, a direct sum of simple Euclidean Jordan algebras and the constituent symmetric cone therein, respectively, that is, 𝕍=𝕍₁×⋯×𝕍_𝑚,𝒦=𝒦₁×⋯×𝒦_𝑚, (2.12) where every 𝕍_𝑖 is a simple Euclidean Jordan algebra (which cannot be direct sum of two Euclidean Jordan algebras) with the corresponding symmetric cone 𝒦_𝑖 for 𝑖=1,…,𝑚, and 𝑛=∑^𝑚_𝑖=1𝑛_𝑖 (𝑛_𝑖 is the dimension of 𝕍_𝑖). Therefore, for any 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇, 𝑦=(𝑦₁,…,𝑦_𝑚)^𝑇∈𝕍 with 𝑥_𝑖,𝑦_𝑖∈𝕍_𝑖, there exist 𝑥∘𝑦=𝑥₁∘𝑦₁,…,𝑥_𝑚∘𝑦_𝑚^𝑇∈𝕍,𝑥,𝑦=𝑥₁,𝑦₁+⋯+𝑥_𝑚,𝑦_𝑚. (2.13)

Through the above description and Cartesian 𝑃-properties proposed by Chen and Qi [9] in the setting of semidefinite matrices, Kong et al. [26] introduced the concept of uniform Cartesian 𝑃-property for the general transformation 𝐹 in the setting of Euclidean Jordan algebra. This concept is used to study the Lipschitz continuity of the solution mapping in SCCP.

Definition 2.5. Consider a linear or nonlinear transformation 𝐹∶𝕍→𝕍. We say that 𝐹 has the uniform Cartesian 𝑃-property if for any 𝑥, 𝑦∈𝕍 and 𝑥≠𝑦, there exist an index 𝜈∈{1,2,…,𝑚} and a scalar 𝜌>0 such that 𝑥−𝑦_𝜈,𝐹(𝑥)−𝐹𝑦_𝜈≥𝜌𝑥−𝑦². (2.14)

Remark 2.6. It is easy to observe that when 𝑚=1, the uniform Cartesian 𝑃-property becomes the strong monotonicity of transformation 𝐹. If 𝑚=𝑛 and 𝕍=ℝ^𝑛, it becomes the 𝑃-property in the context of NCP.

When the continuous transformation 𝐹∶𝕍→𝕍 is linear (i.e., 𝐹=𝐿), we will introduce another concept, the ultra 𝑃-property of 𝐿, which is a new concept recently developed for linear transformations on Euclidean Jordan algebra. In fact, the ultra 𝑃-property is an equivalently straightforward extension of 𝑃₂-property in the setting of the semidefinite matrices [23]. Since 𝑃₂-property involves the ordinary (associative) product of three square matrices and there may not have an associative (triple) product in an Euclidean Jordan algebra, for this reason, 𝑃₂-property cannot be extended in a natural way to an Euclidean Jordan algebra [23]. However, the 𝑃₂-property is introduced in Euclidean Jordan algebra using the concepts of principal subtransformation and cone automorphisms of 𝕍 [23].

Given a Jordan frame {𝑒₁,𝑒₂,…,𝑒_𝑟} in Euclidean Jordan algebra 𝕍, we define 𝕍^(𝑙)=𝕍𝑒₁+⋯+𝑒_𝑙,1∶=𝑥∈𝕍∣𝑥∘𝑒₁+⋯+𝑒_𝑙=𝑥for1≤𝑙≤𝑟. (2.15) It is known that 𝕍^(𝑙) is a subalgebra of 𝕍 with rank 𝑙, see [25, Proposition IV.1.1]. By means of Peirce decomposition, we have the following representation [21]: 𝕍^(𝑙)=ℝ𝑒₁+⋯+ℝ𝑒_𝑙+_{𝑖<𝑗≤𝑙}𝕍_𝑖𝑗. (2.16) Let 𝑃^(𝑙) denote the orthogonal projection from 𝕍 onto 𝕍^(𝑙). For a linear transformation 𝐿∶𝕍→𝕍, let 𝐿_𝑙=𝐿_{{𝑒1,…,𝑒𝑙}}∶=𝑃^(𝑙)𝐿∶𝕍^(𝑙)→𝕍^(𝑙). (2.17) We call 𝐿_𝑙 a principal subtransformation of 𝐿. The determinant of 𝐿_𝑙 is called a principal minor of 𝐿.

Definition 2.7 (see [23]). Consider a linear transformation 𝐿∶𝕍→𝕍. We say that 𝐿 has the ultra 𝑃-property if for any Γ∈Aut(𝒦), every principal subtransformation of 𝐿=Γ^𝑇𝐿Γ has the 𝑃-property.

3. Main Results

In this section, we first give several sufficient conditions for the Lipschitz continuity of the solution mapping 𝜙_𝐿 in the SCLCP(𝐿,𝒦,𝑞). For the classical LCP and SDLCP, the Lipschitz continuity results have been studied in [9, 13, 14, 19]. Along this direction, we generalize them to general SCCP(𝐹,𝒦,𝑞) case where a weaker condition, uniform Cartesian 𝑃-property, is used. Furthermore, we also establish relationship between the Lipschitz continuity of the solution mapping and the ultra 𝑃-property.

Theorem 3.1. Let 𝐹∶𝕍→𝕍 be a continuous linear or nonlinear transformation. If 𝐹 has the uniform Cartesian 𝑃-property, then 𝜙_𝐹 is Lipschitz continuous.

Proof. Suppose that 𝐹 has uniform Cartesian 𝑃-property. From [26, Theorem 6.2], we know that for any 𝑞∈𝕍, the problem (1.1) has a unique solution, that is, 𝜙_𝐹(𝑞) is a single point set. Thus, we let {𝑥}=𝜙_𝐹(𝑞₁) and {𝑦}=𝜙_𝐹(𝑞₂) for any 𝑞₁, 𝑞₂∈𝕍. If 𝑥=𝑦, the inequality ‖𝑥−𝑦‖≤𝜅‖𝑞₁−𝑞₂‖ is obvious, where 𝜅>0. If 𝑥≠𝑦, from definition of uniform Cartesian 𝑃-property, there exists an index 𝜈∈{1,…,𝑚} such that 𝜌𝑥−𝑦²≤𝑥−𝑦_𝜈,𝐹(𝑥)−𝐹𝑦_𝜈=𝑥_𝜈−𝑦_𝜈,𝐹(𝑥)_𝜈−𝐹𝑦_𝜈=𝑥_𝜈−𝑦_𝜈,𝐹(𝑥)+𝑞_1𝜈−𝐹𝑦+𝑞_2𝜈−𝑥_𝜈−𝑦_𝜈,𝑞_1𝜈−𝑞_2𝜈=𝑥_𝜈−𝑦_𝜈,𝑞_2𝜈−𝑞_1𝜈−𝑥_𝜈,𝐹𝑦+𝑞_2𝜈−𝑦_𝜈,𝐹(𝑥)+𝑞_1𝜈≤𝑥_𝜈−𝑦_𝜈,𝑞_2𝜈−𝑞_1𝜈≤𝑥_𝜈−𝑦_𝜈𝑞_1𝜈−𝑞_2𝜈≤𝑥−𝑦𝑞₁−𝑞₂, (3.1) where the third equality follows from ⟨𝑥_𝜈,(𝐹(𝑥)+𝑞₁)_𝜈⟩=0=⟨𝑦_𝜈,(𝐹(𝑦)+𝑞₂)_𝜈⟩ because 𝑥 and 𝑦 are the solution of the problem (1.1) for 𝑞₁, 𝑞₂∈𝕍, respectively. The second inequality is due to 𝑥_𝜈, 𝑦_𝜈, (𝐹(𝑥)+𝑞₁)_𝜈, and (𝐹(𝑦)+𝑞₂)_𝜈∈𝒦_𝜈. This implies that 𝜌‖𝑥−𝑦‖≤‖𝑞₁−𝑞₂‖. Letting 𝜅=1/𝜌 gives ‖𝑥−𝑦‖≤𝜅‖𝑞₁−𝑞₂‖. Hence, 𝜙_𝐹 is Lipschitzian.

Remark 3.2. In Theorem 3.1, if the transformation 𝐹 is linear, the condition of uniform Cartesian 𝑃-property reduces to the Cartesian 𝑃-property [26]. However, if we weaken the condition of uniform Cartesian 𝑃-property to the monotonicity for the linear transformation 𝐿, the conclusion of Theorem 3.1 is not true. The following example shows that the monotonicity property is not sufficient to conclude that the 𝜙_𝐿 is Lipschitz continuous with respect to 𝑞∈𝕍.

Example 3.3. Let 𝐿∶ℝ³→ℝ³ be defined as 𝑥𝑦𝑧0𝐿∶=𝑦000010002,where𝐿∶=2𝑧. (3.2) It is obvious that 𝐿 has the monotonicity property. It can be seen that SOL(𝐿,𝒦³,𝑒)={0}, where 𝒦³⊂ℝ³ is a second-order cone, and 𝑒 is identity element in Euclidean Jordan algebra ℝ³. Moreover, it is easy to verify that (𝛼,0,0)^𝑇∶0<𝛼∈ℝ⊆SOL𝐿,𝒦³,0. (3.3) It is an unbounded solution set. However, if the solution mapping 𝜙_𝐿 of SCLCP(𝐿,𝒦³,0) is Lipschitz continuous, then SOL(𝐿,𝒦³,0) must be a bounded set, which is clearly a contradiction.

Kong et al. [26] proved that the strong monotonicity implies the uniform Cartesian 𝑃-property whether the transformation 𝐹 is linear or nonlinear. Moreover, when 𝐹=𝐿 is linear transformation, by [21, Theorem 21], if 𝐿 is self-adjoint and has 𝑃-property, then 𝐿 is strongly monotone. Hence, we have the following corollary.

Corollary 3.4. Consider Euclidean Jordan algebra 𝕍.(a)Let 𝐹∶𝕍→𝕍 be a nonlinear transformation. If 𝐹 is strongly monotone, then 𝜙_𝐹 is Lipschitz continuous.(b)Let 𝐿∶𝕍→𝕍 be a linear transformation. If 𝐿 is either(i)strictly monotone, or (ii)self-adjoint and has 𝑃-property, or (iii)𝑃-property and 𝒦 is polyhedral,     then 𝜙_𝐿 is Lipschitz continuous.

Remark 3.5. Even the transformation 𝐹 is linear, the condition of uniform Cartesian 𝑃-property in Theorem 3.1 or strong monotonicity in Corollary 3.4 cannot be weakened to the GUS-property, otherwise the conclusion is not true. Example 4.2 will illustrate this point.

In the following theorem, we prove that if 𝜙_𝐿 is Lipschitz continuous, then 𝐿 has the ultra 𝑃-property provided the linear transformation 𝐿 is monotone. To establish another main result of this paper, the following lemmas play important roles.

Lemma 3.6. (a) Suppose that 𝜙_𝐿 is Lipschitz continuous, and SOL(𝐿,𝒦,𝑞)={0} for some 𝑞≻0. Then, SOL(𝐿,𝒦,𝑞)={0} for all 𝑞≽0.
(b) If SOL(𝐿,𝒦,𝑒)={0} and if 𝐿 has 𝑅₀-property (i.e., SOL(𝐿,𝒦,0)={0}), then 𝐿 has 𝑄-property.
(c) If 𝜙_𝐿 is Lipschitz continuous and 𝐿 has 𝑄-property, then for the every principal subtransformation 𝐿_𝑙 of 𝐿, 𝜙_𝐿𝑙 is the Lipschitz continuous with respect to any Jordan frame of 𝕍.

Proof. Please see [20, Lemma 5] for part (a), [20, Proposition 3] for part (b), and [20, Lemma 4] for part (c).

Lemma 3.7. If 𝜙_𝐿 is Lipschitz continuous and 𝐿 has 𝑄-property, then(a)the linear transformation 𝐿 is invertible;(b)SOL(𝐿,𝒦,𝑞)={0} for some 𝑞≻0.

Proof. Part (a) is from [20, Lemma 6], while part (b) is from [20, Lemma 1].

Theorem 3.8. Let 𝐿∶𝕍→𝕍 be a linear transformation. Suppose 𝐿 is monotone and the solution mapping 𝜙_𝐿 of SCLCP(𝐿,𝒦,𝑞) is Lipschitz continuous. Then,(a)𝐿 has the ultra 𝑃-property;(b)𝐿 has the GUS-property.

Proof. (a) Consider any Jordan frame {𝑒₁,…,𝑒_𝑟} of Euclidean Jordan algebra 𝕍 and the principal subtransformation 𝐿_𝑙∶=𝐿_{{𝑒1,…,𝑒𝑙}}∶𝕍^(𝑙)→𝕍^(𝑙), where 𝐿=Γ^𝑇𝐿Γ for any Γ∈Aut(𝒦). Note that 𝑧,𝐿(𝑧)=𝑧,Γ^𝑇(𝑧=⟨Γ(𝑧)(𝑧𝑧𝐿Γ),𝐿Γ)⟩forany𝑧∈𝕍,₁,𝐿_𝑙𝑧₁=𝑧₁,𝑃^(𝑙)𝐿𝑧₁=𝑧₁,𝐿𝑧₁,where𝑧₁∈𝕍^(𝑙)⊆𝕍. (3.4) Since 𝐿 is monotone, it follows that the linear transformation 𝐿 and 𝐿_𝑙 are both monotone. Thus, we have SOL(𝐿,𝒦,𝑒)={0} and SOL(𝐿_𝑙,𝒦^(𝑙),𝑒^(𝑙))={0}, where 𝒦^(𝑙) and 𝑒^(𝑙) denote the symmetric cone and the identity element in 𝕍^(𝑙), respectively. Furthermore, by direct calculation, it is not hard to prove that the solution mapping 𝜙_𝐿 of SCLCP(𝐿,𝒦,𝑞) is Lipschitz continuous if and only if the solution mapping 𝜙𝐿 of the corresponding SCLCP is Lipschitz continuous for the linear transformation 𝐿. Applying Lemma 3.6(a) and (b) yields that 𝐿 has 𝑄-property. Then using Lemma 3.6(c), we obtain that the solution mapping 𝜙𝐿_𝑙 of the corresponding SCLCP is Lipschitz continuous for the linear transformation 𝐿_𝑙. It follows from SOL(𝐿_𝑙,𝒦^(𝑙),𝑒^(𝑙))={0} and Lemma 3.6(a) again that 𝐿_𝑙 has 𝑄-property. This together with Lemma 3.7 says that the transformation 𝐿_𝑙 is invertible.
Next, we want to prove that the transformation 𝐿_𝑙 has 𝑃-property. Suppose that an element 0≠𝑥∈𝕍^(𝑙) operator commute with 𝐿_𝑙(𝑥) and 𝑥∘𝐿_𝑙(𝑥)⪯0. Since 𝐿_𝑙 is monotone by the above analysis, we have 0≤𝑥,𝐿_𝑙(𝑥)=𝑥∘𝐿_𝑙(𝑥),𝑒^(𝑙)≤0, (3.5) which means that ⟨𝐿_𝑙(𝑥)∘𝑥,𝑒^(𝑙)⟩=0. Together with Property 1, it is easy to verify that 𝐿_𝑙(𝑥)∘𝑥=0, and 𝐿_𝑙(𝑥) and 𝑥 have the same Jordan frame. Since 𝐿_𝑙(𝑥)∘𝑥=0, we write 𝑥=^𝑘_𝑖=1𝜆_𝑖𝑓_𝑖,𝐿_𝑙(𝑥)=^𝑙_𝑖=𝑘+1𝜇_𝑖𝑓_𝑖, (3.6) where {𝑓₁,𝑓₂,…,𝑓_𝑙} is a Jordan frame in 𝕍^(𝑙), 𝜆_𝑖≠0(𝑖=1,…,𝑘) and 1≤𝑘≤𝑙. Let 𝑄^(𝑘) denote the projection operator from 𝕍^(𝑙) onto the eigenspace 𝕎^(𝑘) of 𝑓₁+⋯+𝑓_𝑘. Then, 0=𝑄^(𝑘)𝐿_𝑙(𝑥)=𝑄^(𝑘)𝐿_{{𝑒1,…,𝑒𝑙}}(𝑥)=𝑄^(𝑘)𝑃^(𝑙)𝐿(𝑥). (3.7) Let 𝑇_𝑘∶=𝑄^(𝑘)𝐿_𝑙∶𝕎^(𝑘)→𝕎^(𝑘) be the principal subtransformation of 𝐿_𝑙 corresponding to {1,…,𝑘}. From the definition of 𝑇_𝑘, it follows that 𝑇_𝑘(𝑥)=𝑄^(𝑘)𝐿_𝑙(𝑥)=0. By the same arguments as above, we know that 𝑇_𝑘 has 𝑄-property, and the solution mapping 𝜙_𝑇𝑘 of the corresponding SCLCP is Lipschitz continuous for the transformation 𝑇_𝑘. Hence, from Lemma 3.7, we get that 𝑇_𝑘 is invertible. This together with 𝑇_𝑘(𝑥)=0 yields 𝑥=0, which gives a contradiction to 𝑥≠0. Therefore, we have proved that 𝐿 has the ultra 𝑃-property.
(b) This is immediate by [23, Theorem 6.2].

It was shown in [19, Theorem 2.2] that if 𝐿∶𝑆^𝑛→𝑆^𝑛 is monotone and 𝜙_𝐿 is Lipschitz continuous, then 𝐿 has the 𝑃₂-property. Note that 𝑃₂-property in 𝑆^𝑛 is equivalent to the ultra 𝑃-property in 𝑆^𝑛 (see [23]). Therefore, the result of Theorem 3.8 is a natural extension of [19, Theorem 2.2] to the setting of Euclidean Jordan algebra.

4. A Special Linear Transformation

In this section, we specialize to a special linear transformation which is studied in the SCLCP setting, see [19, 23]. For 𝑎∈𝕍, we consider the corresponding Lyapunov transformation 𝐿_𝑎. We will give several equivalent conditions regarding the ultra 𝑃-property of Lyapunov transformation 𝐿_𝑎.

Theorem 4.1. For the Lyapunov transformation 𝐿_𝑎(𝑎∈𝕍), the following statements are equivalent:(a)𝑎≻0;(b)𝐿_𝑎 is strongly monotone;(c)𝐿_𝑎 has (uniform) Cartesian 𝑃-property;(d)𝐿_𝑎 has GUS-property;(e)𝐿_𝑎 has 𝑃-property;(f)𝐿_𝑎 has the ultra 𝑃-property;(g)𝐿_𝑎 has 𝑄-property and the solution mapping 𝜙_𝐿𝑎 of the SCLCP(𝐿_𝑎,𝐾,𝑞) is Lipschitz continuous with respect to 𝑞∈𝕍.

Proof. (a)⇒(b) For any 0≠𝑥∈𝑉, we have ⟨𝑥,𝐿_𝑎(𝑥)⟩=⟨𝑥,𝑎∘𝑥⟩=⟨𝑎,𝑥²⟩. Since 𝑎≻0 and 𝑥²∈𝒦, ⟨𝑎,𝑥²⟩>0 (see [25, Proposition I.1.4]). Thus, 𝐿_𝑎 has the strong monotonicity property.
(b)⇒(c) It is straightforward by the definitions.
The implication (c)⇒(d) follows from [26, Theorem  6.2].
(d)⇒(e) This follows from [21, Theorem 14].
(e)⇒(a) Suppose that the Lyapunov transformation 𝐿_𝑎 has 𝑃-property. Let 𝑎=∑^𝑟_𝑖=1𝜆_𝑖(𝑎)𝑒_𝑖 and 𝐼={𝑖∶𝜆_𝑖(𝑎)≤0}, where {𝑒₁,…,𝑒_𝑟} is a Jordan frame of 𝕍. Note that 𝑎≻0 if and only if 𝐼=∅. Suppose that 𝐼≠∅. Let 𝑥=∑_𝑖∈𝐼𝑒_𝑖≠0. Then, 𝑥 and 𝐿_𝑎(𝑥) operator commute, and 𝑥∘𝐿_𝑎(𝑥)=∑_𝑖∈𝐼𝜆_𝑖(𝑎)𝑒_𝑖⪯0. Therefore, by the 𝑃-property of 𝐿_𝑎, we have 𝑥=0 which leads to 𝑎≻0.
(b)⇒(f) It follows from [23, Theorem  6.1].
(f)⇒(e) It is obvious.
(b)⇒(h) For any linear transformation, the strong monotonicity is equivalent to the strict monotonicity. Then, it follows from Corollary 3.4 that the solution mapping 𝜙_𝐿𝑎 of the SCLCP(𝐿_𝑎,𝒦,𝑞) is Lipschitz continuous with respect to 𝑞∈𝕍. Moreover, it is true that the strong monotonicity implies 𝑄-property for any linear transformation, see [21]. Hence, the conclusion of (ℎ) is obtained.
(h)⇒(b) Suppose that the solution mapping 𝜙_𝐿𝑎 of the SCLCP(𝐿_𝑎,𝒦,𝑞) is Lipschitz continuous with respect to 𝑞∈𝕍, and 𝐿_𝑎 has 𝑄-property. Let {𝑒₁,…,𝑒_𝑟} be a Jordan frame of 𝕍 and 𝑥=∑^𝑟_𝑖=1𝜆_𝑖(𝑥)𝑒_𝑖. Note that 𝑥,𝐿_𝑎(𝑥)=^𝑟_𝑖,𝑗=1𝜆_𝑖(𝑥)𝜆_𝑗(𝑥)𝑒_𝑖,𝐿_𝑎𝑒_𝑗=^𝑟_𝑖=1𝜆²_𝑖(𝑥)𝑒_𝑖,𝐿_𝑎𝑒_𝑖. (4.1) Since 𝐿_𝑎 has the 𝑄-property and the solution map 𝜙_𝐿𝑎 is Lipschitz continuous, ⟨𝐿_𝑎(𝑒_𝑖),𝑒_𝑖⟩>0 (see [27, Theorem  3.1]). It follows from (4.1) that ⟨𝐿_𝑎(𝑥),𝑥⟩>0 for all 0≠𝑥∈𝕍. Therefore, the linear transformation 𝐿_𝑎 has the strong monotonicity. The proof is complete.

In general, the above result may fail to hold. The following example shows that 𝜙_𝐿 is not Lipschitz continuous, but 𝐿 has the GUS-property. Meanwhile, this example also shows that for Theorem 3.1 and Corollary 3.4, if weaken the condition of strong monotonicity to GUS-property, the conclusions of Theorem 3.1 and Corollary 3.4 are not true.

Example 4.2. Let 𝕍=𝑆² and 𝒦=𝑆²₊. For 𝐴∶=0−333, (4.2) consider the corresponding Lyapunov transformation defined by 𝐿_𝐴(𝑋)∶=𝐴𝑋+𝑋𝐴^𝑇. (4.3) It is easy to prove that 𝐴 is positive stable and positive semidefinite, and 𝐿_𝐴 is a linear transformation. From [10, Theorem 9], we have that 𝐿_𝐴 has GUS-property. On the other hand, since 𝐴 is not a positive definite matrix, it follows from [19, Theorem 3.3] that 𝜙_𝐿𝐴 is not Lipschitz continuous.

5. Concluding Remarks

In this paper, we have studied the Lipschitz continuity of the solution mapping for symmetric cone linear or nonlinear complementarity problems over Euclidean Jordan algebras and provided several sufficient conditions for the Lipschitz continuity of the solution mapping. We have established the relationship between the Lipschitz continuity of the solution mapping and ultra 𝑃-property. Furthermore, for Lyapunov transformation, we have shown that the strong monotonicity property, the ultra 𝑃-property, GUS-property, the Lipschitz continuity of the solution mapping, and so forth are all equivalent to each other.

Acknowledgments

The authors are grateful to the referees for their constructive comments, which help to improve the paper a lot. The author’s work is supported by National Young Natural Science Foundation (No. 11101302) and The Seed Foundation of Tianjin University (No. 60302041). The author’s work is supported by National Science Council of Taiwan.