New complexity analysis of interior-point methods for the Cartesian P ∗ ( κ ) -SCLCP

Abstract

In this paper, we give a unified analysis for both large-and small-update interior-point methods for the Cartesian P * (κ)-linear complementarity problem over symmetric cones based on a finite barrier. The proposed finite barrier is used both for determining the search directions and for measuring the distance between the given iterate and the μ-center for the algorithm. The symmetry of the resulting search directions is forced by using the Nesterov-Todd scaling scheme. By means of Euclidean Jordan algebras, together with the feature of the finite kernel function, we derive the iteration bounds that match the currently best known iteration bounds for large-and small-update methods. Furthermore, our algorithm and its polynomial iteration complexity analysis provide a unified treatment for a class of primal-dual interior-point methods and their complexity analysis. MSC: 90C33; 90C51

Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

“…In this context, Bai et al [7] introduced in 2002 an IPM based on an exponential barrier term which has a finite value at the boundary of the feasible region. The growth term of this finite KF was later parametrized by El Ghami et al [23] and the approach was extended to solve different types of optimization problems including SDP [26], convex quadratic programming (CQP) [15], P * (κ) horizontal linear complementarity problem [8] and Cartesian P * (κ)−linear complementarity problem over symmetric cones (SCLCP) [54].…”

An arc search interior-point algorithm for monotone linear complementarity problems over symmetric cones

“…Let (V, •) be an n-dimensional Euclidean Jordan algebra (EJA) with rank r equipped with the standard inner product x, s := tr(x • s) and assume that K is the symmetric cone related to EJA (V, •). In this paper, similar to [8,9,22], we define the monotone symmetric cone linear complementarity problem (SCLCP) in the standard form as follows: The problem of finding a pair (x, s) ∈ K × K such that s = M(x) + q, x • s = 0, where q ∈ V and M : V −→ V is a positive semidefinite linear operator. That is, M(x), x ≥ 0 for x ∈ V. Since V is EJA, we can consider the matrix representation M(x) = M x in which M ∈ R n×n is positive semidefinite with respect to the inner product •, • in (V, •).…”

“…Gu et al [23] extended the proposed algorithm by Roos [16] for linear optimization (LO) to SO problems. Wang et al [22] generalized the Gu et al's feasible algorithm for SO [6] to the Cartesian P * (K)-SCLCP and obtained the complexity bound O ( √ rL) for their algorithm.…”