Infeasible interior-point method for symmetric optimization using a positive-asymptotic barrier

Abstract

We propose a new primal-dual infeasible interior-point method for symmetric optimization by using Euclidean Jordan algebras. Different kinds of interior-point methods can be obtained by using search directions based on kernel functions. Some search directions can be also determined by applying an algebraic equivalent transformation on the centering equation of the central path. Using this method we introduce a new search direction, which can not be derived from a usual kernel function. For this reason, we use the new notion of positive-asymptotic kernel function which induces the class of corresponding barriers. In general, the main iterations of the infeasible interior-point methods are composed of one feasibility and several centering steps. We prove that in our algorithm it is enough to take only one centering step in a main iteration in order to obtain a well-defined algorithm. Moreover, we conclude that the algorithm finds solution in polynomial time and has the same complexity as the currently best known infeasible interior-point methods. Finally, we give some numerical results.

Appendix

We present the theory of the Euclidean Jordan algebras and symmetric cones. A more detailed study about this approach can be found in the book of Faraut and Korányi [9]. Let us consider an n-dimensional vector space V over $\mathbb {R}$ and the following bilinear map: $\circ :(x,y) \rightarrow x\circ y\in V$. Then, $(V,\circ )$ is called a Jordan algebra iff for all $x,y \in V$

• i. $x\circ y=y\circ x$;

• ii. $x \circ (x^2 \circ y) = x^2 \circ (x \circ y)$, where $x^2=x \circ x$.

Suppose that there exists an identity element e such that $x \circ e = e \circ x = x, \forall x \in V$. A Jordan algebra is called Euclidean if there exists an associative inner product. Let us introduce the notion of degree of an element x, denoted by deg(x). If r is the smallest integer such that the set $\{e,x, \ldots , x^r\}$ is linearly dependent, for all $x \in V$, then r is called the degree of x. The rank of V is the largest deg(x), for all $x \in V$ and it is denoted as rank(V). We assume that V is a Euclidean Jordan algebra with rank r and we will denote it by V. Let $L(x):V \rightarrow V$ be a linear operator such that for every $y \in V$, $x \circ y = L(x) y$.

We define the quadratic representation of V for each $x \in V$ : $P(x):=2L(x)^2-L(x^2),$ where $L(x)^2:=L(x)L(x)$.

An element $c \in V$ is named indempotent iff $c \ne 0$ and $c^2=c$. An idempotent element c is called primitive iff $c \ne 0$ and $\not \exists \; c_1 \ne 0$ and $c_2 \ne 0$ such that $c = c_1 + c_2$. Two idempotent elements $c_1$ and $c_2$ are orthogonal if $c_1 \circ c_2 = 0.$ A set of primitive idempotents $\{ c_1,\ldots , c_r\}$ is named Jordan frame iff for all primitive idempotents $c_i$ the following hold: $c_i \circ c_j = 0, i\ne j$ and $\sum _{i=1}^{r}c_i=e$. The following theorem plays an important role in the theory of the Euclidean Jordan algebras.

Theorem 3

(Spectral decomposition, Theorem III.1.2, Faraut and Korányi [9]) Let $x \in V$. Then there exists a Jordan frame $\{c_1,\ldots ,c_r\}$ and the real numbers $\lambda _1(x), \ldots , \lambda _r(x)$ such that

$$\begin{aligned} x=\sum _{i=1}^{r}\lambda _i(x)c_i. \end{aligned}$$

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The numbers $\lambda _i(x)$ are the eigenvalues of x.

Theorem 3 enables us to extend the definition of any real-valued univariate continuous function to elements of Euclidean Jordan algebras using eigenvalues. Let us introduce the vector-valued function using the function $\varphi$, which is a real-valued univariate function defined on the interval $(\kappa ^2,+\infty )$ and differentiable on the interval $(\kappa ^2,+\infty )$ such that $\varphi '(t)>0, \forall t>\kappa ^2$. Let $x = \sum _{i=1}^{r}\lambda _i(x)c_i$, where $\{c_1,\ldots ,c_r\}$ is the corresponding Jordan frame. The vector-valued function $\varphi$ is defined in the following way: $\varphi (x):=\varphi (\lambda _1(x))c_1+\ldots +\varphi (\lambda _r(x))c_r$. Similarly, the vector-valued function $\varphi '$ can be given in the following way: $\varphi '(x):=\varphi '(\lambda _1(x))c_1+\ldots +\varphi '(\lambda _r(x))c_r.$

Consider the following notions: the trace, $tr(x):=\sum _{i=1}^{r}\lambda _i(x)$, the square, $x^2:=\sum _{i=1}^{r} \lambda _i(x)^2 c_i$, the square root, $x^{\frac{1}{2}}:=\sum _{i=1}^{r}\sqrt{\lambda _i(x)}c_i$, wherever all $\lambda _i(x) \ge 0$ and the inverse, $x^{-1}:=\sum _{i=1}^{r} \lambda _i(x)^{-1}c_i$, wherever all $\lambda _i \ne 0$. We say that x is invertible, if $x^{-1}$ is defined. Let us consider the trace inner product $\langle x,s \rangle :=tr(x \circ s),$ where $x,s \in V.$ The Frobenius norm, $\Vert \cdot \Vert _F$, is defined by $\Vert x\Vert _F:=\sqrt{\langle x,x \rangle }.$ Then, the following hold: $\Vert x\Vert _F=\sqrt{tr(x^2)}=\sqrt{\sum _{i=1}^{r}\lambda _i^2(x)}.$ Furthermore, let us denote the largest and the smallest eigenvalue of x by $\lambda _{max}(x)$ and $\lambda _{min}(x)$. Then,

$$\begin{aligned} |\lambda _{max}(x)|\le \Vert x\Vert _F \quad \text {and} \quad | \lambda _{min}(x)| \le \Vert x\Vert _F. \end{aligned}$$

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Let us introduce the cone of squares $K =\{x^2:x \in V\}$. The following hold: $x \in K \Leftrightarrow \lambda _i(x) \ge 0, i=1,\ldots ,r$ and $x \in \text{ int }\, K \Leftrightarrow \lambda _i(x) > 0, i=1,\ldots ,r.$

We use the following lemmas in the analysis of the algorithm.

Lemma 12

(Lemma 2.6, Darvay and Takács [6]) Let us consider $x \in V$ and $\Vert x\Vert _F < 1.$ Then,

$$\begin{aligned} e-x \in \text{ int }\, K. \end{aligned}$$

Lemma 13

(Lemma 14, Schmieta and Alizadeh [37]) Let $x,s \in V.$ One has

$$\begin{aligned} \lambda _{min}(x+s)\ge \lambda _{min}(x)+\lambda _{min}(s)\ge \lambda _{min}(x)-\Vert s\Vert _F \end{aligned}$$

and

$$\begin{aligned} \lambda _{max}(x+s)\le \lambda _{max}(x)+\lambda _{max}(s)\le \lambda _{max}(x)+\Vert s\Vert _F \end{aligned}$$

Lemma 14

(Lemma 30, Schmieta and Alizadeh [37]) Let $x,s \in K.$ Then,

$$\begin{aligned} \left\| P(x)^{\frac{1}{2}}s-e\right\| _F \le \Vert x \circ s-e\Vert _F. \end{aligned}$$

Lemma 15

(Theorem 4, Sturm [39]) Let $x,s \in K.$ One has

$$\begin{aligned} \lambda _{min}\left( P(x)^{\frac{1}{2}}s \right) \ge \lambda _{min}(x \circ s). \end{aligned}$$

The following lemma presents the scaling introduced by Nesterov and Todd. This method was first studied for SO by Faybusovich.

Lemma 16

(NT-scaling, Lemma 3.2, Faybusovich [11]) Let $x,s \in \text{ int }\, K.$ Then, there exists a unique $w \in \text{ int }\, K$ such that

$$\begin{aligned} x=P(w)s. \end{aligned}$$

Moreover,

$$\begin{aligned} w=P(x)^{\frac{1}{2}}\left( P(x)^{\frac{1}{2}}s\right) ^{-\frac{1}{2}}\left[ =P(s)^{-\frac{1}{2}}\left( P(s)^{\frac{1}{2}}x \right) ^{\frac{1}{2}}\right] . \end{aligned}$$

The point w is called the scaling point of x and s.

Then, there exists ${{\bar{v}}} \in \text{ int }\, K$ such that

$$\begin{aligned} {{\bar{v}}}=P(w)^{-\frac{1}{2}}x=P(w)^{\frac{1}{2}}s. \end{aligned}$$

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Note that $P(w)^{\frac{1}{2}}$ and it’s inverse $P(w)^{-\frac{1}{2}}$ are automorphisms of $\text{ int }\, K$.