A Mizuno-Todd-Ye predictor-corrector infeasible-interior-point method for symmetric optimization with the arc-search strategy

In this paper, we propose a Mizuno-Todd-Ye predictor-corrector infeasible-interior-point method for symmetric optimization using the arc-search strategy. The proposed algorithm searches for optimizers along the ellipses that approximate the central path and ensures that the duality gap and the infeasibility have the same rate of decline. By analyzing, we obtain the iteration complexity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(r\log \varepsilon^{-1})$\end{document}O(rlogε−1) for the Nesterov-Todd direction, where r is the rank of the associated Euclidean Jordan algebra and ε is the required precision. To our knowledge, the obtained complexity bounds coincide with the currently best known theoretical complexity bounds for infeasible symmetric optimization.


Introduction
The purpose of this paper is to propose a Mizuno-Todd-Ye predictor-corrector (MTY-PC) infeasible-interior-point method (infeasible-IPM) for symmetric optimization (SO) by using Euclidean Jordan algebra (EJA). Recently, SO has caused widespread concern, because it provides a unified framework for various convex optimizations including linear optimization (LO), second-order cone optimization (SOCO), and semi-definite optimization (SDO) as special cases. Meanwhile, there are many methods for solving SO. Particularly, the interior-point method (IPM), which was first proposed by Karmarkar [], is an important kind of classification algorithm. There is extensive literature on the analysis of IPMs for SO [-].
Nowadays, it is broadly accepted that the primal-dual IPM is the most efficient IPM and includes the Mehrotra predictor Inspired by their works, we present an O(r log ε - )-iteration complexity MTY-PC algorithm for SO. Moreover, the proposed algorithm will use the infeasible starting, which is found to be easy in practice. This kind of IPM is called infeasible-IPM and is studied in the literature [, , , , -].
Moreover, the proposed algorithm in this paper has another invention, i.e., the arcsearch strategy. Yang [-] first developed the arc-search algorithm that searches for optimizers along an ellipse that is an approximation of the central path and gave some of the advantages of the arc-search algorithm. In order to further study the advantages of the arc-search algorithm, Yang [, ] proposed two infeasible-IPMs for LO and SO, and respectively obtained the O(n / log ε - )-iteration complexity for LO and the O(r / log ε - ) and O(r / log ε - )-iteration complexity, where n is the larger dimension of a standard LO, r is the rank of the associated EJA and ε is the required precision. In order to improve the iteration complexity of infeasible-IPM, we will add the arc-search strategy to the MTY-PC algorithm.
In this paper, we propose an MTY-PC infeasible-IPM for SO. The proposed algorithm uses the arc-search strategy and ensures that the duality gap and the infeasibility have the same rate of decline. By analyzing, we achieve the O(r log ε - ) iteration complexity for the Nesterov-Todd (NT) direction. To our knowledge, this is the best iteration complexity obtained so far for an infeasible SO problem.
The outline of this paper is organized as follows. In Section , we briefly introduce some key results on EJA. In Section , we give some preliminary discussions for an algorithm and propose the algorithm. In Section , we establish the iteration complexity for the proposed algorithm. Finally, we close the paper by some conclusions.

Euclidean Jordan algebra
In order to ensure the integrity of this paper, we give some results for EJA. Most of these can be found in [, ].
EJA is a triple (J , •, ·, · ), where (J , ·, · ) is an n-dimensional inner product space over R and (x, y) → x•y : J ×J → J is a bilinear mapping satisfying the following conditions: We call x • y the Jordan product of x and y and define the inner product as x, y := tr(x • y). If there exists an element e such that x • e = e • x = x for all x ∈ J , then e is called the multiplicative identity element of EJA. For any x ∈ J , the degree of x is denoted by deg(x), which is defined as the smallest integer k such that the set {e, x, x  , . . . , x k } is linearly dependent. The rank of J , simply denoted by r, is the maximum of deg(x) for all x ∈ J . For EJA J , the corresponding cone of squares K := {x  : x ∈ J } is indeed a symmetric cone. A cone is symmetric if and only if it is the cone of squares of some EJA. Moreover, int K denotes the interior of the symmetric cone K.
An idempotent c is a nonzero element of J such that c  = c. An idempotent is primitive if it cannot be written as the sum of two idempotents. Two idempotents c  and c  are orthogonal if c  • c  = . A complete system of orthogonal idempotents is a set {c  , . . . , c k } of idempotents, where c i • c j =  for all i = j, and c  + · · · + c k = e. A complete system of orthogonal primitive idempotents is called a Jordan frame. Theorem . (Spectral decomposition [, Theorem III..]) Let J be EJA with rank r. Then, for every x ∈ J , there exist a Jordan frame {c  , . . . , c r } and real numbers λ  , . . . , λ r such that x, x = λ  i . Since '•' is bilinear for every x ∈ J , there exists a linear operator L x such that, for every y ∈ J , x • y = L x y. In particular, L x e = x and L x x = x  . We say that two elements x, y ∈ J operator commute if L x L y = L y L x . It can be proven that x and s operator commute if and only if they share a common Jordan frame [, The following is a useful proposition of quadratic representation.

Proposition . ([, Proposition ]
) Let x, y, p ∈ int K and definex := Q p x andỹ := Q p - y, then Q x / y, Q y / x and Q˜x/ỹ have the same spectrum.

SO problem and ellipse approximate center
First, we give the standard form of SO and its dual form, as follows: where c ∈ J , b ∈ R m , A is a linear operator that maps J into R m and A * is its adjoint operator such that x, A * y = Ax, y for all x ∈ J , y ∈ R m . Moreover, we denote the sets of optimal solutions of (P) and (D) by P * and D * , and assume that A is surjective and F  = ∅, where F  indicates a primal-dual strict feasibility set that is defined by The Karush-Kuhn-Tucker (KKT) conditions for (P) and (D) are given by where x • s =  is called the complementarity slackness condition.
By relaxing x • s =  with x • s = μe, we obtain where μ = x, s /r >  is called the duality gap.
System () has unique solutions (x(μ), y(μ), s(μ)), the set of which is called the central path, which is denoted by In this paper, we will use the idea of Yang [-], which is that the central path C is replaced by an ellipse , where is defined as follows: where a ∈ R n+m and b ∈ R n+m are the axes of the ellipse perpendicular to each other, and c ∈ R n+m is the center of the ellipse. For the point z = (x, y, s) = (x(θ  ), y(θ  ), s(θ  )) ∈ , we require its first and second derivatives such that Systems () and () do not always have a unique solution due to the fact that x and s do not operator commute in general. To overcome this difficulty, we apply a scaling scheme that follows from [, Lemma ]. For the scaling point p ∈ int K, there are several appropriate choices (see []). In this paper, we select the classical NT-scaling point that is

Foundation of the MTY-PC algorithm
Since the MTY-PC algorithm requires two matrix factorizations and at most three backsolves for each iteration, it is generally divided into two steps, which are the predictor step and the corrector step.
In the predictor step, using p in (), systems () and () are rewritten as By solving systems () and (), we obtain the predictor directions (ẋ,ẏ,ṡ) and (ẍ,ÿ,s) and have the following lemma.
Let (x(θ ), y(θ ),s(θ )) be an arc defined by () passing through a point (x, y,s), and its first and second derivatives at (x, y,s) be (ẋ,ẏ,ṡ) and (ẍ,ÿ,s), which are defined by () and (). Then an ellipsoidal approximation of the central path is given bỹ Using (a), (b), (c), the third equations in () and (), we havẽ where g(θ ) = ( -cos(θ )), ξ =ẋ •s +ṡ •ẍ. Furthermore, using (), we have In what follows, we discuss a method for selecting the predictor step. Firstly, we give the neighborhood that is used in this paper as follows: The neighborhood N F (γ ) has some important properties, which are given in the following proposition. For more details, readers are referred to [].
Now, we give the method of selecting the predictor step, which is to find the largest positiveθ ∈ (, π/] and to satisfy for all θ ∈ (,θ] that In the corrector step, we define (x,ȳ,s) = (Q p -x(θ ), y(θ ), Q ps (θ)) and calculate the corrector direction ( x, y, s) by where r c = ( -sin(θ))μe -x •s. The scaling corrector direction is given by solving the following system: A,r c = ( -sin(θ))μe -x •ŝ. Eventually, the next iteration point is updated by In what follows, we give two useful expressionŝ

Framework of the MTY-PC algorithm
Based on the previous analysis, we state the generic framework of the proposed MTY-PC algorithm in this paper.
Step  If x k ∈ int K, s k ∈ int K and φ k ≤ ε, then stop.
To analyze complexity, we give two remarks for Algorithm .
Proof Using (), (), (a), (b), (c), (), (), by calculating directly, we have In the same way, we have r k+ In what follows, we focus on proving the last inequality and have This completes the proof.
, which implies φ k represents the relative infeasibility at (x k , y k , s k ). Meanwhile, we also have φ k = μ k μ  , which is also the rate of decline of the duality gap μ. Thus, if φ k ≤ ε, then Algorithm  will stop and we obtain an approximate optimal solution of SO.
Remark  For Algorithm , we choose a particular starting point, which is studied by Zhang [, ] and Rangarajan []. In what follows, we give the particular starting point.

Complexity analysis
For simplicity, we will often writex, y,s,x,s,x,ŝ,θ and φ forx k , y k ,s k ,x k ,s k ,x k ,ŝ k ,θ k and φ k , respectively. Moreover, since the NT-scaling point is used in this paper, we can obtain the following special results: In what follows, we give some fundamental lemmas. Firstly, by the proof procedure of Lemma . and Lemma  in [, ], we have the following lemma.

Technical results
In order to achieve the iteration complexity bounds for the proposed Algorithm , we need some technical results.

Multiplying the last equation by
Using the definition of (ǔ,v)  : Using (), () and the fact ẋ -ǔ,ṡ -v = , we have where the last inequality uses the result The proof is completed.
Using Remark  and the proof techniques of Lemma A. in [], we have the following lemma, which gives the upper bound on ζ .
Proof Using Lemmas . and ., we have which completes the proof.
Proof Multiplying the equation of () by L - v and taking norm-squared on both sides, we have where the second equality uses (), the first two inequalities follow from Lemma ., the last two inequalities are due to Lemmas . and .. Using the fact ẍ ,s =  and (), we have Therefore, the proof of the lemma is completed.
The next result follows from Lemmas . and ..

The lower bounds onθ
In this subsection, we will find a lower bounds ofθ to satisfy () and (). They will play a key role in complexity analysis. Letθ  = arg sin( βγ ωr ). If we can prove that () and () hold for all θ ∈ (,θ  ], thenθ  is one of the lower bounds onθ . For this purpose, we first give an important lemma.
From the above analysis, we obtain the result thatθ  is one of the lower bounds onθ .

Corrector step and iteration complexity
It is well known that an important requirement for the MTY-PC algorithm is that the new iteration point must stay in the given neighborhood, which is equivalent to proving (x(θ ),ŝ(θ )) ∈ N F (γ ). In what follows, we will complete this task.