On a Class of Superlinearly Convergent Polynomial Time Interior Point Methods for Sufficient LCP

Abstract

Abstract. A new class of infeasible interior point methods for solving sufficient linear complementarity problems requiring one matrix factorization and m backsolves at each iteration is proposed and analyzed. The algorithms from this class use a large (N − ∞ ) neighborhood of an infeasible central path associated with the complementarity problem and an initial positive, but not necessarily feasible, starting point. The Q-order of convergence of the complementarity gap, the residual, and the iteration sequence is m + 1 for problems that admit a strict complementarity solution and (m + 1)/2 for general sufficient linear complementarity problems. The methods do not depend on the handicap κ of the sufficient LCP. If the starting point is feasible (or "almost" feasible) the proposed algo-Key words. linear complementarity, interior point, affine scaling, large neighborhood, superlinear convergence AMS subject classifications. 90C51, 65K05, 49M15, 90C05, 90C201. Introduction. Interior point methods have provided the first polynomialtime algorithms for solving linear programming (LP) and other classes of convex optimization problems. Numerical experiments performed over the past two decades show that the most efficient interior point methods for LP are primal-dual path following interior point methods. The fact that these methods, as opposed to the ellipsoid method, perform much better in practice than indicated by the (worst case) computational complexity bounds, is explained in part by their superlinear convergence. The first interior point method having both polynomial complexity and superlinear convergence was the predictor-corrector method of Mizuno, Todd and Ye (MTY). This method was proposed for LP in [16], where it was shown to have O( √ nL) iteration complexity. Shortly after that, Ye et al. [45], and independently Mehrotra [14], proved that the duality gap of the iterates produced by MTY converges quadratically to zero. MTY was generalized to LCP in [9], and the resulting algorithm was proved to have O( √ nL) iteration complexity under general conditions, and superlinear convergence under the assumption that the LCP has a (perhaps not unique) strictly complementary solution (i.e., the LCP is nondegenerate) and the iteration sequence converges. From [3] it follows that the latter assumption always holds. Subsequently Ye and Anstreicher [44] proved that MTY converges quadratically assuming only that the LCP is nondegenerate. The nondegeneracy assumption is not restrictive since according to [17] a large class of interior point methods, which contains MTY, can have only linear convergence if this assumption is violated. In [9,14,16,45,44] one assumes that the starting point for the MTY algorithm is strictly feasible. A generalization of the MTY algorithm for infeasible starting points was proposed in [21,20] for LP, and in [22] for monotone LCP. The proposed algorithm requires two matrix factorizations and at most three backsolves per iteration. Its computational complexity depends

A full Nesterov–Todd step infeasible-interior-point algorithm for CartesianP*(κ) horizontal linear complementarity problems over symmetric cones

“…The following result extends Lemma 3.2 in [28] to context of Jordan algebras. 3 If SCHLCP is P * (κ), then for any w ∈ int K and any a ∈ J the linear system (29) has a unique solution (u, v) ∈ J × J for which the following estimate holds: and a = a (1)T , a (2)T , .…”

“…Lemma 3.3 in [28]) If SCHLCP is P * (κ), then for any w ∈ int K and any a, b ∈ J , the linear system (30) has a unique solution t = (u, v) ∈ J × J and the following inequality is satisfied:…”

Corrector-predictor methods for sufficient linear complementarity problems

“…Most of them have been known in one form or another in the interior point literature. We are using the formulations from [14].…”

“…As evidenced by (14), (16), and (18), Algorithm 1 depends on the handicap κ of the HLCP, so that Algorithm 1 can be applied only to sufficient HLCPs for which the handicap, or at least an upper bound for the handicap, is known. In what follows, we will show that by interchanging the roles of the predictor and the corrector, we obtain an interior point method for sufficient HLCPs that does not depend on the handicap κ.…”

Infeasible Interior-Point Methods for Linear Optimization Based on Large Neighborhood

“…We show that the complexity of our algorithm based on this kernel function is slightly worse than the complexity of the algorithm introduced and discussed by Salahi et al [7], who used another variant of this function. Note that the best-known iteration bound for the IIPMs that use a large neighborhood of the central path is due to Potra and Stoer [8] for a class of superlinearly convergent IIPMs for sufficient linear complementarity problems (LCP).…”