Symmetric cone optimization (SCO) problem is a convex optimization problem, minimization of a linear function over the intersection of an affine subspace and a symmetric cone. The SCO problem is considered to be an important class of convex optimization problems, because it includes linear optimization (LO), second-order cone optimization (SOCO), and semidefinite optimization (SDO) as special cases. In recent years, these problems have attracted the attention of many scholars in optimization.

Among various approaches for solving SCO problem, interior-point methods (IPMs) are the most popular algorithms and have been widely used to obtain the best iteration complexity results. The basic idea for solving SCO problems using IPMs was first proposed by Nesterov and Nemirovskii [1]. Faybusovich [2] extended a primal-dual IPM for SDO to SCO using the Euclidean Jordan algebra. Consequently, several efficient IPMs for SCO have been developed; see, e.g., [3]-[10].

Predictor-corrector methods, the most studied variants of interior-point methods, are being used as Mehrotra’s predictor-corrector algorithm [11] in a number of optimization packages. In these methods, by solving linear systems with the same coefficient matrix, three directions are produced: predictor, corrector and centering. The moving direction is determined by a combination of these directions. Zhang and Zhang [12] first established convergence theory and complexity bounds for two Mehrotra-type second-order algorithms. Later, Zhang and Zhang [13] extended the idea of [12] to SCO and showed the complexity bound of O ( r 3 / 2 log ϵ − 1 ) for the Nesterov-Todd search direction. Many primary IPMs are based on the small neighborhood or the negative infinity large neighborhood. Among all existing path-following interior-point algorithms, the theoretical iteration complexity of wide neighborhood algorithms is the worst, but in practice, these algorithms perform better than small neighborhood algorithms. Recently, Ai and Zhang [14] considered an IPM for linear complementarity problem (LCP), which works in a wide neighborhood and has the same complexity as the best known small neighborhood IPMs. Applying an interesting idea, they decomposed the classical Newton search direction into the nonnegative and nonpositive parts corresponding to the nonnegative and the nonpositive parts of the right-hand side. They showed that if these two directions are equipped with different and appropriate step sizes, then the method enjoys the low iteration bound of O ( n log ϵ − 1 ) . Liu et al. [15] modified the Ai-Zhang’s primal-dual path-following interior-point method of [14] to gain a class of Mehrotra-type predictor-corrector algorithm for monotone LCPs. Here, we are to modify the Ai-Zhang’s primal-dual interior point method to present two Mehrotra-type predictor-corrector algorithms. First, we present a Mehrotra-type predictor-corrector algorithm generalizing the approach of [15] for LCP to SCO. Then, we propose the second Mehrotra-type predictor-corrector algorithm for SCO, by adding two corrector steps to the Ai-Zhang path-following algorithm [14], to improve the performance. Finally, we show that the use of corrector steps does not impair the worst-case complexity of the algorithm.

Here, we recall briefly the basic concept and useful results of Euclidean Jordan algebra. We refer the reader to [16] for details.

Definition 1.Let J be an n -dimensional vector space over the field of real numbers with a multiplication“ ∘ ”where the map ∘ : ( x , y ) ↦ x ∘ y is bilinear from J × J to J .The triple ( J , ∘ , 〈 ⋅ , ⋅ 〉 ) is a Euclidean Jordan algebra if forall x , y , z ∈ J ,we have:

1) x ∘ y = y ∘ x ;

2) x ∘ ( x 2 ∘ y ) = x 2 ∘ ( x ∘ y ) , where x 2 : = x ∘ x ;

3) 〈 x ∘ y , z 〉 = 〈 y , x ∘ z 〉 .

We assume that there exists an identity element e ∈ J such that e ∘ x = x ∘ e = x , for all x ∈ J . The degree of x ∈ J , denoted by deg ( x ) , is the smallest integer k such that { e , x , ⋯ , x k } is linearly dependent. The rank of J , denoted by rank ( J ) , is defined as the maximum deg ( x ) over all x ∈ J .

An element c is idempotent if c ≠ 0 and c 2 = c . An idempotent c is primitive if it is nonzero and can not be expressed by sum of two other nonzero idempotents. Two idempotents c i and c j are said to be pairwise orthogonal if c i ∘ c j = 0 . A set of orthogonal primitive idempotents { c 1 , c 2 , ⋯ , c k } is called a Jordan frame if c i ∘ c j = 0 , i ≠ j , and c 1 + ⋯ + c k = e . For x ∈ J , let k be the smallest positive integer such that { e , x , ⋯ , x k } is linearly dependent; k is called the degree of x and is denoted by deg ( x ) . The rank of J , denoted by rank ( J ) , is defined by r : = max { deg ( x ) : x ∈ J } .

A cone is symmetric if and only if it is the cone of squares of a Euclidean Jordan algebra, which is denoted by K : = { x 2 : x ∈ J } . For a given x ∈ J , the Lyapunov transformation L : J → J is defined by L ( x ) y : = x ∘ y , for every y ∈ J . Furthermore, we define the quadratic representation of x in J by Q ( x ) : = 2 L ( x ) 2 − L ( x 2 ) , where L ( x ) 2 = L ( x ) L ( x ) .

The spectral decomposition theorem (Theorem III.1.2 of [16]) of a Euclidean Jordan algebra J states that for any x ∈ J , there exists a Jordan frame { c 1 , c 2 , ⋯ , c r } and real numbers { λ 1 , ⋯ , λ r } (the eigenvalues of x ) such that x = λ 1 c 1 + ⋯ + λ r c r . Some generated functions by the eigenvalues of x are: the Frobenius norm on J is defined by ‖ x ‖ : = 〈 x , x 〉 = ( ∑ i = 1 r λ i 2 ) 1 / 2 , and T r ( x ) = ∑ i = 1 r λ i . Moreover, since T r ( x ∘ y ) is a symmetric positive definite quadratic form, the inner product can be defined as 〈 x , y 〉 : = T r ( x ∘ y ) . The inverse is x − 1 : = ∑ i = 1 r λ i − 1 c i , whenever all λ i ≠ 0 . Accordingly, x + = ∑ i = 1 r λ i + c i , where λ i + = max { λ i , 0 } , for i = 1 , 2 , ⋯ , r . We define x − = x − x + . We call x ∈ J positive semidefinite (positive definite), denoted by x ≻ _ 0 ( x ≻ 0 ) , if all eigenvalues of J are nonnegative (positive). Moreover, x ≻ _ 0 ( x ≻ 0 ) if and only if x ∈ K ( x ∈ int K ) .

Let J be a Euclidean Jordan algebra of dimension n with rank r , and K be the cone of squares corresponding to J . Consider the following primal and dual problems in the standard forms:

(P) min 〈 c , x 〉 : A x = b ,   x ∈ K ,

and

(D) max 〈 b , y 〉 : A * y + s = c ,   s ∈ K ,   y ∈ ℝ m ,

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 〉 = 〈 A x , y 〉 , for all x ∈ J .

The primal-dual feasible set is defined as

and we assume that the interior of the primal-dual feasible set,

is nonempty; that is, the primal-dual pair of the SCO problems satisfies the interior-point condition (IPC). Under the IPC, finding an optimal solution of (P) and (D) is equivalent to solving the following system [17]:

The basic idea of a path-following interior-point algorithm is to replace the third equation in (1), the so-called complementarity condition, by the parameterized equation x ∘ s = τ μ e , for μ = 〈 x , s 〉 / r , and τ ∈ ( 0 , 1 ) . Thus, we consider the following system:

Due to the IPC, a unique solution ( x ( μ ) , y ( μ ) , s ( μ ) ) of the perturbed system (2) exists for each μ > 0 [17]. This solution is called the μ -center for the SCO problem. The set of μ -centers provides a path, which is called the central path of the SCO problem. The μ -center approximates an optimal solution of the given problem pair as μ goes to zero. Applying the Newton method for the perturbed system of (2), we get the following equations:

It has been shown that the reason for the system (3) not having a unique solution is the fact that x and s do not operator commute, in general; that is, L ( x ) L ( s ) ≠ L ( s ) L ( x ) .

To overcome this defect, we use the scaling scheme based on the following fact (Lemma 28 of [6]). Let u be invertible. Then, x ∘ s = μ e if and only if Q ( u ) x ∘ Q ( u − 1 ) s = μ e .

In our work here, we choose

which gives the Nesterov-Todd (NT) direction. For the NT direction [18], we have Q ( u ) x = Q ( u − 1 ) s . The system (3) can be rewritten as follows:

Most interior point algorithms work in the classical negative infinity large neighborhood, defined by

where w = Q ( x 1 / 2 ) s and γ ∈ ( 0 , 1 ) . Our proposed algorithms start from a feasible point in a wide neighborhood of the central path due to [14][19], as follows:

This neighborhood is even larger than the N ∞ − ( 1 − γ ) neighborhood, since it can be verified that

Here, we describe our algorithms in detail. In accordance with the Ai-Zhang’s original idea [14], we decompose the Newton direction into two separate parts corresponding to the positive and negative parts of the right-hand side of the third equality of the system (4); i.e., τ μ e − x ˜ ∘ s ˜ . Therefore, in our algorithms, we compute the predictor directions ( Δ x ˜ − , Δ y − , Δ s ˜ − ) and ( Δ x ˜ + , Δ y + , Δ s ˜ + ) respectively by solving the following two systems:

and