The dual Z-property for the Lorentz cone

TL;DR

This paper investigates the dual cone of linear maps with the Z-property specifically for the Lorentz cone, extending the understanding of cone-complementarity in this context.

Contribution

It provides a solution for characterizing the dual cone of Z-property maps with respect to the Lorentz cone, a specific case not previously addressed.

Findings

Derived the dual cone of Z-property maps for the Lorentz cone.

Extended the theory of cone-complementarity to Lorentz cones.

Clarified the structure of Z-property maps in this setting.

Abstract

The Z-property of a linear map with respect to a cone is an extension of the notion of Z-matrices. In a recent paper of Orlitzky (see Corollary 6.2 in M. Orlitzky. Positive and $\mathbf{Z}$-operators on closed convex cones, Electron. J Linear Algebra, 444--458, 2018) the characterisation of cone-complementarity is given in terms of the dual of the cone of linear maps satisfying the Z-property. Therefore, it is meaningful to consider the problem of finding the dual cone of the cone of linear maps which have the Z-property with respect to a cone. This short note will solve this problem in the particular case when the Z-property is considered with respect to the Lorentz cone.

Equations

$$⟨x,y⟩=x^{\top }y=x_1{y}_1+x_2{y}_2+\cdots +x_N{y}_N,$$

$$K^{\ast }=\{z\in {\mathbb{R}}^N:⟨z,v⟩\ge 0,\forall v\in K\}.$$

$$L=\{(x_1,x_2,\dots ,x_n{)}^{\top }\in {\mathbb{R}}^n:x_1\ge \sqrt{x_2^2+\cdots +x_n^2}\}$$

$$Z(L)=\{\gamma I-J(P+Q):\gamma \in \mathbb{R},P\in {\mathbb{S}}_{+}^n\text{ and }Q\in {\mathbb{A}}^n\}.$$

$$Z(L)=\{A\in {\mathbb{R}}^{n\times n}:\exists \gamma \in \mathbb{R}\text{ such that }\gamma J-(JA+A^{\top }J)\in {\mathbb{S}}_{+}^n\}.$$

$$\Gamma =\{\gamma I-J(P+Q):\gamma \in \mathbb{R},P\in {\mathbb{S}}_{+}^n\text{ and }Q\in {\mathbb{A}}^n\}.$$

$$Z(L{)}^{\ast }=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}B=0\text{ and }JB\text{ is negative semidefinite}\}.$$

$$Z(L{)}^{\ast }=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}(B^{\top }A)=⟨B,A⟩\ge 0,\forall A\in Z(L)\}$$

$$=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}\left(B^{\top }[\gamma I-J(P+Q)]\right)\ge 0,\forall \gamma \in \mathbb{R},\forall P\in {\mathbb{S}}_{+}^n\text{ and }\forall Q\in {\mathbb{A}}^n\}$$

$$=\{B\in {\mathbb{R}}^{n\times n}:\gamma \mathrm{tr}(B)-\mathrm{tr}\left(B^{\top }J(P+Q)\right)\ge 0,\forall \gamma \in \mathbb{R},\forall P\in {\mathbb{S}}_{+}^n\text{ and }\forall Q\in {\mathbb{A}}^n\}$$

$$=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}(B)=0,\mathrm{tr}\left(B^{\top }J(P+Q)\right)\le 0,\forall P\in {\mathbb{S}}_{+}^n\text{ and }\forall Q\in {\mathbb{A}}^n\}$$

$$=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}(B)=0,\mathrm{tr}\left((P+Q)B^{\top }J\right)\le 0,\forall P\in {\mathbb{S}}_{+}^n\text{ and }\forall Q\in {\mathbb{A}}^n\}$$

$$=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}(B)=0,⟨(P+Q{)}^{\top },B^{\top }J⟩\le 0,\forall P\in {\mathbb{S}}_{+}^n\text{ and }\forall Q\in {\mathbb{A}}^n\}$$

$$=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}(B)=0,⟨P+Q,B^{\top }J⟩\le 0,\forall P\in {\mathbb{S}}_{+}^n\text{ and }\forall Q\in {\mathbb{A}}^n\}.$$

$$-B^{\top }J\in ({\mathbb{S}}_{+}^n+{\mathbb{A}}^n{)}^{\ast }=({\mathbb{S}}_{+}^n{)}^{\ast }\cap ({\mathbb{A}}^n{)}^{\ast }=({\mathbb{S}}_{+}^n{)}^{\ast }\cap ({\mathbb{A}}^n{)}^{\perp }=\left[({\mathbb{S}}_{+}^n{)}_{\ast }+({\mathbb{S}}_{+}^n-{\mathbb{S}}_{+}^n{)}^{\perp }\right]\cap ({\mathbb{A}}^n{)}^{\perp }$$

$$=\left[{\mathbb{S}}_{+}^n+({\mathbb{S}}_{+}^n-{\mathbb{S}}_{+}^n{)}^{\perp }\right]\cap ({\mathbb{A}}^n{)}^{\perp }=\left[{\mathbb{S}}_{+}^n+({\mathbb{S}}^n{)}^{\perp }\right]\cap ({\mathbb{A}}^n{)}^{\perp }=\left[{\mathbb{S}}_{+}^n+{\mathbb{A}}^n\right]\cap {\mathbb{S}}^n={\mathbb{S}}_{+}^n,$$

$$Z(L{)}^{\ast }=\{B\in {\mathbb{R}}^{n\times n}:\mathrm{tr}B=0\text{ and }JB\text{ is negative semidefinite}\}.$$

Full text

1. Introduction

In the followings we will identify a linear map $A:\,\mathbb{R}^{n}\to\mathbb{R}^{n}$ with its matrix in $\mathbb{R}^{n\times n}$ with respect to the canonical basis. In $\mathbb{R}^{N}$ consider the standard inner product

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for any two column vectors $x=(x_{1},\dots,x_{N})^{\top}$, $y=(y_{1},\dots,y_{N})^{\top}$. In the case of $N=n\times n$ this can be expressed by the standard trace formula $\langle A,B\rangle=\operatorname{tr}{A^{\top}B}$, for any $A,B\in\mathbb{R}^{n\times n}$. All following duals will be considered with respect to the standard inner product.

Recall that a closed set $K\subseteq\mathbb{R}^{N}$ is called a closed convex cone if and only $\lambda u\in K$ and $u+v\in K$ for any $u,v\in K$ and any $\lambda>0$. We remind that the dual of $K$ is the closed convex cone $K^{*}$ defined by

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Denote $C(K)=\{(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}:\,x\in K,\textrm{ }y\in K^{*},\textrm{ and }\langle x,y\rangle=0\}$. One says that a linear map $A:\,\mathbb{R}^{n}\to\mathbb{R}^{n}$ has the Z-property with respect to $K$ (or shortly $K$-Z-property) if $(x,y)\in C(K)$ implies $\langle Ax,y\rangle\leq 0$. This notion has been introduced by Gowda and Tao in [5]. It is easy to check that a matrix $A$ has the Z-property with respect to the nonnegative orthant $\mathbb{R}^{n}_{+}$ if an only if it is a Z-matrix, that is, a matrix whose off-diagonal elements are nonpositive ([3],[7, Definition 2.5.1]).

Z-matrices are important in differential equations, dynamical systems, optimization, economics etc. [1]. The importance of Z-matrices in linear complementarity is clear from the equivalences (1)-(6) in the Introduction of [5] which come from Chapter 6 of [1].

The following papers justify the importance of the Z-property in finite dimensions: [5, 4, 6, 11, 8, 10, 9]. Theorems 6, 7, 9, 10 and 13 in [5] show the crucial role of Z-property in linear cone-complementarity, extending the fundamental role of Z-matrices in linear complementarity.

We also mention that important results have been obtained for the Z-property in infinite dimensions too [2].

Denote by $Z(K)$ the cone of linear mappings with $K$-Z-property. The dual of $Z(K)$ appears first time in the paper [9] of Orlitzky. From Corollary 6.2 of this paper it follows that $(x,y)\in C(K)$ if and only if $-yx^{\top}\in Z(K)^{*}$, which can also be easily checked by using the properties of the trace function for matrices. Although simple, this equivalence makes the study of $Z(K)^{*}$ an interesting topic in cone-complementarity. In the case of the nonnegative orthant $\mathbb{R}^{n}_{+}$ it is a straightforward exercise to check that the dual of $Z(\mathbb{R}^{n}_{+})$ is formed of nonpositive matrices with zero diagonal. In general finding $Z(K)^{*}$ is a highly nontrivial task. This short note will determine $Z(L)^{*}$ for the Lorentz cone L.

2. The main result

Let $J=\operatorname{diag}(1,-1,-1,\dots,-1)\in\mathbb{R}^{n\times n}$, $L\subset\mathbb{R}^{n}$ be the Lorentz cone defined by

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and $Z(L)$ the cone of linear maps $A:\,\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $L$-Z-property (that is, Z-property with respect to $L$).

Denote by $\mathbb{S}^{n}$, $\mathbb{S}^{n}_{+}$ and $\mathbb{A}^{n}$ the linear subspace of symmetric matrices of $\mathbb{R}^{n\times n}$, the cone of $n\times n$ symmetric positive semidefinite matrices and the linear subspace of skew-symmetric matrices of $\mathbb{R}^{n\times n}$, respectively.

Then, we have the following lemma:

Lemma 1

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Proof. According to Gowda and Tao [5, Example 4] we have

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Denote

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We need to show that $Z(L)=\Gamma$.

Let $A=\gamma I-J(P+Q)\in\Gamma$, where $\gamma\in\mathbb{R}$, $P\in\mathbb{S}^{n}_{+}$ and $Q\in\mathbb{A}^{n}$. Then, $J(P+Q)=\gamma I-A$. Multiplying this equation by $J$ from the left, and using $J^{2}=I$, we obtain $P+Q=\gamma J-JA$. Hence, $Q=\gamma J-JA-P$ and thereforee $0=Q+Q^{\top}=2\gamma J-(JA+A^{\top}J)-P$. Thus, $2\gamma J-(JA+A^{\top}J)\in\mathbb{S}^{n}_{+}$, which according to (1) implies that $A\in Z(L)$. This yields $\Gamma\subseteq Z(L)$.

Conversely, let that $A\in Z(L)$. Then, according to (1), we get $2\gamma J-(JA+A^{\top}J)=2P$, for some $\gamma\in\mathbb{R}$ and some positive semidefinite matrix $P\in\mathbb{R}^{n\times n}$. Hence, $JA+(JA)^{\top}=2(\gamma J-P)$, which implies $JA=\gamma J-P-Q$, for some skew-symmetric matrix $Q\in\mathbb{R}^{n\times n}$. Multiplying the last equality by $J$ from the left and using $J^{2}=I$, we obtain that $A=\gamma I-J(P+Q)$, which implies $A\in\Gamma$. This yields $Z(L)\subseteq\Gamma$.

In conclusion, $Z(L)=\Gamma$. $\Box$

In the next theorem we find the dual of $Z(L)$.

Theorem 1

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Proof. By using Lemma 1 we have

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The last equality in the formula of $Z(L)^{*}$ above shows that $B\in Z(L)^{*}$ if and only if $\operatorname{tr}(B)=0$ and $-B^{\top}J\in(\mathbb{S}^{n}_{+}+\mathbb{A}^{n})^{*}$, where the dual is considered as the dual of a cone in $\mathbb{R}^{n\times n}$.

In case of $\mathbb{S}^{n}_{+}$ we should make a distinction between the dual of this cone in $\mathbb{R}^{n\times n}$ and the dual in $\mathbb{S}^{n}$. In case of the former we will use the notation $(\mathbb{S}^{n}_{+})^{*}$ and in the case of the latter $(\mathbb{S}^{n}_{+})_{*}$.

In general for a closed convex cone $K\subseteq\mathbb{R}^{N}$ (in our case $N=n\times n$) we denote by $K^{*}$ the dual of $K$ in $\mathbb{R}^{N}$, by $K_{*}$ the dual of $K$ in the linear subspace $K-K$ spanned by $K$, and by $\perp$ the orthogonal complement operation for subspaces.

It is an easy exercise to check that the dual of a subspace (as a cone) is its orthogonal complement and $K^{*}=K_{*}+(K-K)^{\perp}$. It is well known that $\mathbb{S}^{n}_{+}$ is self-dual in $\mathbb{S}^{n}$, that is $(\mathbb{S}^{n}_{+})_{*}=\mathbb{S}^{n}_{+}$ and that $(\mathbb{S}^{n})^{\perp}=\mathbb{A}^{n}$. Hence, the membership $-B^{\top}J\in(\mathbb{S}^{n}_{+}+\mathbb{A})^{*}$, which together with $\operatorname{tr}(B)=0$ characterizes $B\in Z(L)^{*}$, is equivalent to

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or to $-JB\in\mathbb{S}_{n}^{+}$, as $-B^{\top}J=(-B^{\top}J)^{\top}=-JB$ and it is an easy exercise to check that for any two closed convex cones $K_{1}$, $K_{2}$ we have $(K_{1}+K_{2})^{*}=K_{1}^{*}\cap K_{2}^{*}$. In conclusion,

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$\Box$