A polynomial primal-dual affine scaling algorithm for symmetric conic optimization

Abstract

The primal-dual Dikin-type affine scaling method was originally proposed for linear optimization and then extended to semidefinite optimization. Here, the method is generalized to symmetric conic optimization using the notion of Euclidean Jordan algebras. The method starts with an interior feasible but not necessarily centered primal-dual solution, and it features both centering and reducing the duality gap simultaneously. The method’s iteration complexity bound is analogous to the semidefinite optimization case. Numerical experiments demonstrate that the method is viable and robust when compared to SeDuMi, MOSEK and SDPT3.

Appendix: Euclidean Jordan algebras

This section gives a brief review of the basic properties of the Euclidean Jordan algebras. For the sake of simplicity, we only provide the necessary concepts which will be required in this paper. For detailed studies of Euclidean Jordan algebras the reader can consult [4] and [25].

Definition 7.1

Let $\mathcal {J}$ be an n-dimensional vector space over the field of real numbers with a bilinear map $(x,s) \rightarrow x \circ s$. Then, $(\mathcal {J},\circ )$ is referred to as the Euclidean Jordan algebra if for all $x,s,z \in \mathcal {J}$

• 1. $x \circ s=s \circ x$,

• 2. $x \circ (x^2 \circ s)=x^2 \circ (x \circ s)$,

• 3. $\langle x,x \rangle > 0$ for all $x \ne 0$,

where $x^2=x \circ x$, and $\langle . , . \rangle$ denotes an inner product defined over $\mathcal {J}$. An identity element is defined for a Euclidean Jordan algebra $\mathcal {J}$, if there exists a unique element e, such that $x \circ e=e \circ x=x$ for all $x \in \mathcal {J}$.

Roughly speaking, a Euclidean Jordan algebra is a commutative algebra over the field of real numbers which is not necessarily associative. Nevertheless, the Euclidean Jordan algebra is power associative; that is $x^{p+q}:=x^p \circ x^q$.

For all $x,s \in \mathcal {J}$, the bilinear map $(x,s) \rightarrow x \circ s$ is characterized by

$$\begin{aligned} L(x)s=x \circ s, \end{aligned}$$

where L(x) denotes a symmetric matrix. In particular, $L(x)e=x$ and $L(x)x=x^2$. The quadratic representation of x is defined as

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

Definition 7.2

The cone of squares of a Euclidean Jordan algebra $\mathcal {J}$ is defined as

$$\begin{aligned} \mathcal {K}(\mathcal {J}):=\{x^2:x \in \mathcal {J}\}, \end{aligned}$$

where $x^2=x \circ x$, and $\mathcal {K}(\mathcal {J})$ is a closed pointed convex cone with nonempty interior.

Example 7.3

Let $\mathcal {J}$ be an n-dimensional vector space over the field of real numbers, where the identity element e and L(x) are defined as

$$\begin{aligned} e_{n \times 1}:=\begin{bmatrix} 1 \\ 0 \\ . \\ . \\ . \\ 0 \\ \end{bmatrix}, \ \ \ L(x):=\begin{bmatrix}&x_0 \ \&x_{1:n-1}^T \\&x_{1:n-1} \ \&x_0 I_{n-1} \end{bmatrix}. \end{aligned}$$

The vector space $\mathcal {J}$ endowed with the bilinear map characterized by L(x) is a Euclidean Jordan algebra. Further, $\mathcal {K}(\mathcal {J}) \equiv \mathcal {L}^n$, where

$$\begin{aligned} \mathcal {L}^n:=\left\{ x \in \mathbb {R}^n: \ x_0 \ge \Vert {x_{1:n-1}}\Vert \right\} , \end{aligned}$$

where $\Vert {.}\Vert$ denotes the $l^2$-norm. The cone $\mathcal {L}^n$ is referred to as Lorentz cone or the second-order cone. In this algebra, the quadratic representation of $x \in \mathcal {K}(\mathcal {J})$ is given by

$$\begin{aligned} P(x):=\begin{bmatrix}&\Vert {x}\Vert ^2 \ \ \ \&2x_0x_{1:n-1}^T \\&2x_0x_{1:n-1} \ \ \ \&\gamma (x) I_{n-1}+2x_{1:n-1}x_{1:n-1}^T \end{bmatrix}, \end{aligned}$$

where $\gamma (x):=x_0^2-\Vert {x_{1:n-1}}\Vert ^2$, and $I_{n-1}$ denotes an identity matrix of size $n-1$.

The following lemma specifies some important properties of the quadratic representation. The interested reader is referred to [4] for more details.

Lemma 7.4

(Proposition II.3.1 in [4], Lemma 8 in [25]) For an invertible $x \in \mathcal {J}$ and integer value t, we have

• 1. $P(x^{-1})=P(x)^{-1}$ and in general $P(x^t)=P(x)^t$,

• 2. $P(x)x^{-1}=x$,

• 3. $P(x)e=x^2$.

We now introduce the concept of eigenvalue and spectral decomposition in Euclidean Jordan algebras. Let r be the smallest integer such that the set $\{e,x,x^2,\ldots ,x^r\}$ is linearly dependent for $x \in \mathcal {J}$. Then, r is denoted as the degree of x, $\deg (x)$. The rank of $\mathcal {J}$ is defined as the largest value of $\deg (x)$ over $x \in \mathcal {J}$.

A nonzero element $q \in \mathcal {J}$ is called idempotent if $q^2=q$. Furthermore, an idempotent is primitive if it is not the sum of two other idempotents. In light of these definition, a Jordan frame is defined as a set of primitive idempotents $\{q_1,\ldots ,q_r\}$, where $q_i \circ q_j=0$ for all $i \ne j$ and $q_1+\ldots +q_r=e$.

Theorem 7.5

(Theorem III.1.2 in [4]) Let $\mathcal {J}$ be a Euclidean Jordan algebra with rank r. Then each $x \in \mathcal {J}$ can be represented as

$$\begin{aligned} x=\lambda _1 q_1+\ldots +\lambda _r q_r, \end{aligned}$$

where $\{q_1, \ldots , q_r\}$ denotes a Jordan frame, and $\lambda _i$ stands for the eigenvalues of x.

Example 7.6

Let $x \in \mathcal {L}^n$. It can be easily shown that

$$\begin{aligned} x^2-2x_0x+(x_0^2-\Vert {x_{1:n-1}}\Vert ^2)e=0. \end{aligned}$$

This implies that $r=2$ for this Euclidean Jordan algebra. Hence, the spectral decomposition for an element $x \in \mathcal {L}^n$ is given by

$$\begin{aligned} x=\lambda _1 q_1+\lambda _2 q_2, \end{aligned}$$

where

$$\begin{aligned} \lambda _1=x_0-\Vert {x_{1:n-1}}\Vert , \ \text {and} \ \ \lambda _2=x_0+\Vert {x_{1:n-1}}\Vert ,\\ q_1=\tfrac{1}{2} \begin{pmatrix} 1\\ -\frac{x_{1:n-1}}{\Vert {x_{1:n-1}}\Vert } \end{pmatrix}, \ \text {and} \ \ q_2=\tfrac{1}{2} \begin{pmatrix} 1\\ \frac{x_{1:n-1}}{\Vert {x_{1:n-1}}\Vert } \end{pmatrix}. \end{aligned}$$

We can now extend the definition of any real valued function f(.) to elements of the Euclidean Jordan algebras by

$$\begin{aligned} f(x):=f(\lambda _1)q_1+\cdots +f(\lambda _r)q_r. \end{aligned}$$

In particular, we have

$$\begin{aligned} x^{\tfrac{1}{2}}&:=\lambda _1^{\tfrac{1}{2}} q_1+\cdots +\lambda _r^{\tfrac{1}{2}} q_r,\\ x^{-1}&:=\lambda _1^{-1}q_1+\cdots +\lambda _r^{-1}q_r. \end{aligned}$$

Note that $x^{-1} \circ x=e$. Further, x is called invertible if all the eigenvalues of x are nonzeros.

In light of the definitions given so far, trace, determinant and norms of x are formally defined as follows.

Definition 7.7

Let $x \in \mathcal {J}$ and $\lambda _1,\cdots ,\lambda _r$ be the eigenvalues of x. Then,

• 1. ${{\mathrm{tr}}}(x):=\lambda _1+\lambda _2+\cdots +\lambda _r$,

• 2. $\det (x):=\lambda _1 \lambda _2 \cdots \lambda _r$,

• 3. $\langle x,y \rangle :={{\mathrm{tr}}}(x \circ y)$,

• 4. $\Vert {x}\Vert _F:=\sqrt{\lambda _1^2+\lambda _2^2+\cdots +\lambda _r^2}$,

• 5. $\Vert {x}\Vert _2:=\max _{i} |\lambda _i|$.

We now state an important theorem from [2] which is adopted from [4].

Theorem 7.8

(Theorem 8.3.6 in [2, 4]) Let $\mathcal{J}$ be an Euclidean Jordan algebra. Then $\mathcal{J}$ falls into one of the following categories:

• 1. An n-dimensional vector space over the field of real numbers, where

  $$\begin{aligned} x \circ s := \begin{bmatrix} x^Ts\\x_0s_{1:n-1}+s_0x_{1:n-1}\end{bmatrix}. \end{aligned}$$

• 2. The space of $n \times n$ real symmetric matrices, where

  $$\begin{aligned} X \circ S := \frac{XS+SX}{2}. \end{aligned}$$

  (18)

  for symmetric matrices X and S.

• 3. The space of $n \times n$ complex Hermitian matrices, where the Jordan product is defined as in (18) for Hermitian matrices X and S.

• 4. The space of $n \times n$ Hermitian matrices with quaternion entries, where the Jordan product is defined as in (18) for quaternion Hermitian matrices X and S.

• 5. The space of $3 \times 3$ Hermitian matrices with octonion entries, known as Albert algebra, where the Jordan product is defined as in (18).

In the rest of this section, we review some technical lemmas (without their proofs) which are necessary for the complexity analysis of the Dikin-type algorithm. As stated at the begining part of this paper, it is assumed that $\mathcal {K}$ is a symmetric cone with nonempty interior $\mathcal {K}_+$.

Lemma 7.9

(Lemma 2.13 in [10]) Assume that $x,s \in \mathcal {J}$ and ${{\mathrm{tr}}}(x \circ s)=0$. Then, we have

$$\begin{aligned} -\tfrac{1}{4} \Vert {x+s}\Vert _F^2 e \preceq _{\mathcal {K}} x \circ s \preceq _{\mathcal {K}} \tfrac{1}{4} \Vert {x+s}\Vert ^2_F e. \end{aligned}$$

Lemma 7.10

(Lemma 2.15 in [10]) Assume that $0 \prec _{\mathcal {K}} x \circ s$, where $x,s \in \mathcal {J}$. Then, $\det (x) \ne 0$.

Lemma 7.11

(Lemma 2.17 in [10]) Let $x \in \mathcal {J}$ and $0 \prec _{\mathcal {K}} s$. Then, we have

$$\begin{aligned} \lambda _{\min }(x) {{\mathrm{tr}}}(s) \le {{\mathrm{tr}}}(x \circ s) \le \lambda _{\max }(x) {{\mathrm{tr}}}(s). \end{aligned}$$

The following lemma points out a nice property of the quadratic representation.

Lemma 7.12

(Theorem III.2.1 and Proposition III.2.2 in [4]) Let $x \in \mathcal {J}$. Then, L(x) is positive definite (semidefinite) if and only if $0 \prec _{\mathcal {K}} x$ ($0 \preceq _{\mathcal {K}} x$). Further, $P(x)\mathcal {K}_+=\mathcal {K}_+$ if x is invertible.

In fact, Lemma 7.12 states that for each interior solution $0 \prec _{\mathcal {K}} x$ and $0 \prec _{\mathcal {K}} s$, P(x)s is an invertible linear map from $\mathcal {K}_+$ to $\mathcal {K}_+$. All this hints that the NT search directions obtained from (13) are well-defined.

Lemma 7.13

(Proposition 21 in [25]) Let $0 \prec _{\mathcal {K}} x$ and $0 \prec _{\mathcal {K}} s$, and w be the scaling point of x and s as defined in (5). Then,

• 1. ${{\mathrm{tr}}}(P(w)^{-\tfrac{1}{2}}x \circ P(w)^{\tfrac{1}{2}}s)={{\mathrm{tr}}}(x \circ s)$,

• 2. $P(x^{\tfrac{1}{2}})s \sim P(s^{\tfrac{1}{2}})x$,

• 3. $P(\tilde{x}^{\tfrac{1}{2}})\tilde{s} \sim P(x^{\tfrac{1}{2}})s$,

where $\tilde{x}:=P(w)^{-\tfrac{1}{2}}x$ and $\tilde{s}:=P(w)^{\tfrac{1}{2}}s$.

Lemma 7.14

(Lemma 30 in [25]) Let $0 \prec _{\mathcal {K}} x$ and $0 \prec _{\mathcal {K}} s$. Then, we have

• 1. $\lambda _{\min }(P(x)^{\tfrac{1}{2}}s) \ge \lambda _{\min } (x \circ s)$,

• 2. $\lambda _{\max }(P(x)^{\tfrac{1}{2}}s) \le \lambda _{\max } (x \circ s)$.