Two wide neighborhood interior-point methods for symmetric cone optimization

Abstract

In this paper, we present two primal–dual interior-point algorithms for symmetric cone optimization problems. The algorithms produce a sequence of iterates in the wide neighborhood $\mathcal {N}(\tau ,\,\beta )$ of the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We derive that these two path-following algorithms have

$$\begin{aligned} \text{ O }\left( \sqrt{r\text{ cond }(G)}\log \varepsilon ^{-1}\right) , \text{ O }\left( \sqrt{r}\left( \text{ cond }(G)\right) ^{1/4}\log \varepsilon ^{-1}\right) \end{aligned}$$

iteration complexity bounds, respectively. The obtained complexity bounds are the best result in regard to the iteration complexity bound in the context of the path-following methods for symmetric cone optimization. Numerical results show that the algorithms are efficient for this kind of problems.

Appendix: Euclidean Jordan algebras and symmetric cones

In this section, we briefly recall some basic concepts and useful results from Euclidean Jordan algebras. For a comprehensive treatment of Euclidean Jordan algebras, the reader is referred to the monograph by Faraut and Korányi [4].

A Euclidean Jordan algebra is a triple $\left( \mathcal {J},\,\langle \cdot ,\,\cdot \rangle ,\,\circ \right)$, where $\left( \mathcal {J},\,\langle \cdot ,\,\cdot \rangle \right)$ is a finite dimensional inner product space over ${\mathbb {R}}$ endowed with a bilinear mapping $\circ :(x,\,y)\mapsto x\circ y$ from $\mathcal {J}\times \mathcal {J}$ to $\mathcal {J}$ satisfying following conditions:

• (i) $x\circ y=y\circ x$ for all $x,\,y \in \mathcal {J}$ (Commutativity);

• (ii) $x\circ (x^2\circ y)=x^2\circ (x\circ y)$ for all $x,\,y \in \mathcal {J},\,$ where $x^2:=x\circ x$ (Jordan’s Axiom);

• (iii) $\langle x\circ y,\,z\rangle =\langle y,\,x\circ z\rangle$ for all $x,\,y,\,z \in \mathcal {J}$ (Euclidean),

where the inner product $\langle \cdot ,\,\cdot \rangle$ is defined by $\langle x,\,y\rangle := \mathbf {Tr}(x\circ y)$ for any $x,\,y \in \mathcal {J}$.

Every Euclidean Jordan algebra has a multiplicative identity element e, such that $x\circ e = e\circ x= x$ for all $x \in \mathcal {J}$. For $x\in \mathcal {J},\,$ k is called the degree of x and denoted by $\deg (x),\,$ if k be the smallest integer such that the set $\left\{ e,\,x,\,\ldots ,\,x^k \right\}$ is linearly dependent. The rank of $\mathcal {J}$, denoted by $\mathrm{rank}\,(\mathcal {J})$, is defined by $\max \left\{ \deg (x):~x\in \mathcal {J}\right\}$. The set $\mathcal {K}:=\left\{ x^2:~x\in \mathcal {J}\right\}$ is called the cone of squares of Euclidean Jordan algebra $\left( \mathcal {J},\,\langle \cdot ,\,\cdot \rangle ,\,\circ \right)$. A cone is symmetric (i.e., self-dual and homogeneous) if and only if it is the cone of squares of some Euclidean Jordan algebra.

An element $c\in \mathcal {J}$ is idempotent if $c^2=c$. Two idempotents $c_i$ and $c_j$ are orthogonal if $c_i\circ c_j=0$. A complete system of orthogonal idempotents is a set $\left\{ c_1,\,c_2,\,\ldots ,\,c_k\right\}$ of orthogonal idempotents if all $c_i,\,i=1,\,2,\,\ldots ,\,k$ are idempotent, each two $c_j,\,c_l$ are orthogonal and $\sum _{j=1}^{k}c_j=e$. An idempotent element is said to be primitive if it can not be written as the sum of two other idempotents. A complete system of orthogonal primitive idempotents is called a Jordan frame.

An important property of Euclidean Jordan algebra is that each element x of an Euclidean Jordan algebra admits a spectral decomposition, as described in the following theorem.

Theorem 5.1

(Theorem III.1.2 in [4]) Let $\mathcal {J}$ be a Euclidean Jordan algebra of rank r. For any $x\in \mathcal {J}$, there exists a Jordan frame $\left\{ c_1,\,c_2,\,\ldots ,\,c_r\right\}$ and real numbers $\lambda _1,\,\ldots ,\,\lambda _r$ such that

$$\begin{aligned} x=\sum _{i=1}^{r}\lambda _ic_i. \end{aligned}$$

(36)

The numbers $\lambda _1,\,\lambda _2,\,\ldots ,\,\lambda _r$ are called the eigenvalues of x and (36) is the spectral decomposition of x. We can conclude that $x\in \mathcal {K}$ if and only if $\lambda _i\ge 0$ and $x\in \text{ int }~\mathcal {K}$ (the interior of $\mathcal {K}$) if and only if $\lambda _i> 0$, for all $i=1,\,2,\,\ldots ,\,r$.

Given the spectral decomposition $x=\sum _{j=1}^{r}\lambda _jc_j$, we obtain that:

• Square root $x^{1/2}:=\sum _{i=1}^{r}\lambda _i^{1/2}c_i$ if all $\lambda _i\ge 0$;

• Inverse $x^{-1}:=\sum _{i=1}^{r}\lambda _i^{-1}c_i$ if all $\lambda _i\ne 0$;

• Trace $\mathbf {Tr}(x)=\sum _{i=1}^{r}\lambda _i$;

• Determinant $det(x)=\prod _{i=1}^{r}\lambda _i$;

• Frobenius norm $\Vert x\Vert :=\sqrt{\langle x,\,x\rangle }=\left( \sum _{i=1}^{r}\lambda _i^2\right) ^{1/2}$;

• Metric projection: $x^+=\sum _{i=1}^{r}\lambda _i^+c_i$, where $\lambda _i^+=\max \left\{ \lambda _i,\,0\right\}$, for $i=1,\,2,\,\ldots ,\,r$. Moreover, $x^-=x-x^+.$

We present some technical Lemmas, which are used frequently during the analysis.

Lemma 5.1

(Lemma 13 in [23]) Let $x\in \mathcal {J},$ then we obtain the smallest and the largest eigenvalue as

$$\begin{aligned} \lambda _{\min }(x)=\min _{u}\frac{\langle u,\,x\circ u\rangle }{\langle u,\,u\rangle }, ~~~\lambda _{\max }(x)=\max _{u}\frac{\langle u,\,x\circ u\rangle }{\langle u,\,u\rangle }. \end{aligned}$$

Lemma 5.2

(Lemma 2.13 in [9]) Let $x,\,s\in \mathcal {J}$ with $\langle x,\,s\rangle =0$. Then one has

$$\begin{aligned} \left\| x\circ s \right\| \le 2^{-3/2}\left\| x+s \right\| ^2. \end{aligned}$$

Lemma 5.3

(Lemma 2.15 in [9]) Let $x\circ s\in \text{ int }~\mathcal {K}$, then $det(x)\ne 0$.

For any $x,\,y \in \mathcal {J}$, Lyapunov transformation $L(x): \mathcal {J}\rightarrow \mathcal {J}$ is defined as

$$\begin{aligned} L(x)y:=x\circ y. \end{aligned}$$

The vectors x and y are said to be operator commute if L(x) and L(y) commute, i.e., $L(x)L(y)=L(y)L(x)$. In other words, x and y operator commute if $x\circ (y\circ z)=y\circ (x\circ z)$, for all $z\in \mathcal {J}$. This fact can be generalized in the following theorem which will be used in the Sect. 3.2.

Theorem 5.2

(Theorem 27 in [23]) Let x and y be two elements of a Euclidean Jordan algebra $\mathcal {J}$. Then x and y operator commute if and only if there is a Jordan frame $\left\{ c_1,\,c_2,\,\ldots ,\,c_r\right\}$ such that $x=\sum _{i=1}^{r}\lambda _ic_i$ and $y=\sum _{i=1}^{r}\xi _ic_i$.

Additionally, we define the quadratic representation of x

$$\begin{aligned} Q(x):=2L(x)^2-L(x^2), \end{aligned}$$

where $L(x)^2=L(x)L(x)$. The quadratic representation is an essential concept in the theory of Jordan algebras and will play a key role in our subsequent analysis. The following are two useful properties about the quadratic representation of an element belong to $\text{ int }~\mathcal {K}$.

Lemma 5.4

(Lemma 28 in [23]) Let $x,\,s\in \text{ int }~\mathcal {K}$ and p be invertible. Then $x \circ s=\mu e$ if and only if $Q(p)x\circ Q(p^{-1})s=\mu e$.

Lemma 5.5

(Proposition 2.9 in [17]) Let $x,\,s\in \text{ int }~\mathcal {K}$. If x and s are operator commute then $Q(x^{1/2})s=x\circ s$.