AN INFEASIBLE FULL NT-STEP INTERIOR POINT METHOD FOR CIRCULAR OPTIMIZATION

. In this paper, we design a primal-dual infeasible interior-point method for circular optimization that uses only full Nesterov-Todd steps. Each main iteration of the algorithm consisted of one so-called feasibility step. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.


1.
Introduction. Circular optimization (CO) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of a finite number of circular cones. Mathematically, a typical circular cone in R nj has the form Q nj θj := (x j 0 ;x j ) : where θ ∈ (0, π 2 ) is a given angle. Let Q n θ ⊆ R n be the Cartesian product of several circular cones, i.e., Q n1,...,n N θ1,...,θ N : where n = N j=1 n j . We consider CO problem in the standard form min N j=1 c j , x j θj :
Defining Without loss of generality, we assume that A has full row rank, i.e., rank(A) = m.
Due to the fact that Q n θ is self-dual [1, Lemma 2], the dual problem of (P) is given by It is well known that CO includes linear optimization (LO) as a special case. Even though CO is less general than symmetric cone optimization (SCO), which includes LO, second-order cone optimization (SOCO) and semidefinite optimization (SDO). The study of primal-dual interior-point methods (IPMs) for SCO was started by Nesterov and Todd [20]. Faybusovich [5] invoked Euclidean Jordan algebra to analyze a variety of search directions for SCO. Schmieta and Alizadeh [25] used the Euclidean Jordan algebraic framework to extend the analysis of the Monteiro-Zhang family [19] to all symmetric cones. Recently, CO and circular cone complementarity problems (CCCP) as the special cases of SCO and symmetric cone complementarity problems (SCCP) have received considerable attention from researchers because of its wide range of applications. Bai et al. [3] developed a primal-dual kernel function-based IPM for CO. Zhou et al. [29] and Chi et al. [4] studied the variational analysis of CO and circular cone eigenvalue complementarity problems (CCECP), respectively. Miao et al. [18] considered the constructions of complementarity functions and merit functions for CCCP.
The first polynomial primal-dual full Newton-step IPM for LO was proposed by Roos et al. [24]. In their work, the small-update path-following methods based on Newton direction is shown to have O( √ n log n ) iteration-complexity. In [26,28], Wang et al. and in [27], Wang and Lesaja extend full Newton step path-following algorithm for LO to SCO, convex quadratic symmetric cone optimization (CQSCO) and the Cartesian P * (κ)-symmetric cone complementarity problems (SCLCP) using the Nesterov-Todd (NT) direction. Kheirfam and Mahdavi-Amiri [16] and Kheirfam [6] proposed some variants of Roos et al.'s algorithm [24] for SCLCP and the Cartesian P * (κ)-SCLCP. Recently, Kheirfam [15] analyzed the primal-dual path-following algorithm for CO introduced in [1] by using the NT-directions and established the corresponding complexity bound.
Each main iteration of the aforementioned IIPMs is composed of one so-called feasibility step and a several centering steps to get an -optimal solution of the underlying problem. Recently, Roos [23] and Kheirfam [12,13,14] proposed IIPMs for LO, HLCP, the Cartesian P * (κ)-SCLCP and SCO so that their algorithms do not need centering steps and take only one feasibility step in order to get a new iterate close enough to the central path.
Motivated by Roos [23] and Kheirfam [12,13,14], we present a full-NT step IIPM for CO. Each main iteration of the proposed algorithm is consisted of only one feasibility step. Moreover, we analyze the algorithm and derive the iteration-complexity bound which matches the currently best-known iteration bound for IIPMs.
The remainder of this paper is organized as follows: In section 2, we provide a brief introduction to the theory of Euclidean Jordan algebra on the circular cones. In section 3, we give the perturbed problems and our new algorithm. The convergence analysis of the algorithm is shown in section 4. We obtain the complexity bound of the algorithm in subsection 4.3. Finally, the paper will end with some concluding remarks follow in section 5.
In the sequel, we generalize the above definitions to the case where N > 1, when the circular cone underlying Q n θ is the Cartesian product of N circular cones Q nj θj . For any x ∈ R n , the algebra (R n , θ, •) is defined as a direct product of the Jordan algebras (R nj , θ j , •) as Obviously, if e j is the identity element in the Jordan algebra for the jth circular cone, then e = (e 1 ; . . . ; e N ) is the identity element in (R n , θ, •). Moreover, tr(e) = 2N . The arrow-shaped matrix L θ (x) and the quadratic representation P θ (x) of (R n , θ, •) with respect to θ can be respectively adjusted to tr(x j ).
3. An infeasible algorithm. In this section, we present a full NT-step IIPM for CO. To this end, in next subsection we introduce the perturbed problems.
3.1. The perturbed problems. In accordance with the available results on IIPMs, let (x * , y * , s * ) be an optimal solution of (P) and (D) such that x * + s * Q n θ ξe, and consider the starting point ( where ξ is a (positive) number. For any ν θ , with ν θ ∈ (0, 1], we consider the perturbed problem pair (P ν θ ) and (D ν θ ) as follows: where r 0 p = b − (Ax 0 ) θ and r 0 d = c − A T y 0 − s 0 denote the initial values of the primal and dual residual vectors, respectively. The Karush-Kuhn-Tucker (KKT) optimality conditions for the perturbed problem pair (P ν θ ) and (D ν θ ) are given as and reduce the parameters µ θ and ν θ to zero, until an -solution of the problem pair (P) and (D) is obtained. Assuming P θ (·) as the quadratic representation of R n with respect to θ, considering w = P θ (x as the NT-scaling point of x and s with respect to θ, and using ( Lemma 28] with u = w − 1 2 , we may rewrite system (2) equivalently as follows: Note that, if ν θ = 1, then x = x 0 yields a strictly feasible solution of (P ν θ ), and (y, s) = (y 0 , s 0 ) is a strictly feasible solution of (D ν θ ). This means that both perturbed problems (P ν θ ) and (D ν θ ) satisfy the interior-point condition (IPC) for ν θ = 1. Therefore, system (2) has a unique solution for each µ θ > 0 (see [1]) and the set of all such solutions form a guide line, so-called central path, to the -solution of the problem pair (P) and (D). In what follows, the parameters µ θ and ν θ always satisfy the relation µ θ = µ 0 θ ν θ . Lemma 3.1. Let the original problems (P) and (D) be feasible and 0 < ν θ ≤ 1. Then, the perturbed problems (P ν θ ) and (D ν θ ) satisfy the IPC.
3.2. The algorithm. We measure proximity to the central path of the perturbed problem pair (P ν θ ) and (D ν θ ) by the quantity As an immediate consequence, we have the following lemma.

Upper bound forω(v).
Lemma 4.5. The solution (v 1 , z, v 2 ) of the linear system Av 1 = 0, Proof. From the first two equations of (10), it follows that v 1 , v 2 θ = 0. Taking norm-squared with respect to θ of the third equation, we get θ,F . This proves the lemma. Lemma 4.6. Let (d x , ∆y, d s ) be a solution of (7). Then, we have where w is the NT-scaling point of x and s with respect to θ and ρ(δ) is defined as in Lemma 3.2.
In the same way, it follows that Substitution of the last two inequalities in (13) and finally in (12) gives where the second inequality follows from the triangular inequality and (4) and the last inequality is due to Lemma 3.2. This implies the inequality in lemma.
Using Lemma 4.7, (15) and Lemma 4.6, we obtain the following upper bound for ω(v):ω 4.2. Values for β and τ . The main goal of this section is to devote some positive values for the parameters β and τ such that the proposed algorithm in Fig. 1 is well-defined. That is, after a full NT-step, the iterate (x + , y + , s + ) is strictly feasible and δ(x + , s + ; µ + θ ) ≤ τ . Due to Lemma 4.4, this will hold if ω(v) < 1 − β, Using (16), the inequality (17) One easily verifies that the left-hand side expression in (19) is increasing with respect to δ. Hence, assuming δ ≤ τ , it suffices to have Choosing τ = 1 16 , one may easily check that the inequality (20) is satisfied if β = 1 20N . This means that the iterates (x + , y + , s + ) are strictly feasible. Noting the left-hand side of (20) provides an upper bound forω(v), we then have Therefore, the algorithm is well-defined in the sense that δ(v) ≤ τ is maintained in all iterations.
Thus, we may state without further proof the main result of the paper as follows: Theorem 4.8. Let (P) and (D) be feasible and ξ > 0 such that x * + s * Q n θ ξe for some optimal solutions x * of (P) and (y * , s * ) of (D). Then, after at most 20N log max tr((x 0 • s 0 ) θ ), r 0 p θ,F , r 0 d θ,F .
iterations the algorithm finds an -solution of (P) and (D).

Conclusion.
In this paper, we presented a full NT-step IIPM for CO. In the proposed algorithm, after one feasibility step, the new generated iterate is strictly feasible and close enough to the central path. The polynomial iteration complexity of the algorithm is established.