Abstract
An arc search interior-point algorithm for monotone symmetric cone linear complementarity problem is presented. The algorithm estimates the central path by an ellipse and follows an ellipsoidal approximation of the central path to reach an "-approximate solution of the problem in a wide neighborhood of the central path. The convergence analysis of the algorithm is derived. Furthermore, we prove that the algorithm has the complexity bound O ( p rL) using Nesterov-Todd search direction and O (rL) by the xs and sx search directions. The obtained iteration complexities coincide with the best-known ones obtained by any proposed interior- point algorithm for this class of mathematical problems.
Highlights
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, ◦)
To this end, motivated by Yang [19, 20], we first estimate the central path of monotone symmetric cone linear complementarity problem (SCLCP) by an ellipse and using the wide neighborhood given by Ai and Zhang [1], we propose an arc search feasible interior-point algorithm for monotone SCLCPs
This paper proposed an arc search feasible interior-point algorithm for monotone SCLCPs
Summary
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, ◦). The first arc search interior-point algorithms were suggested by Yang [19,20] for convex quadratic optimization (CQO) problems and LO problems These algorithms utilize the first and second-order derivatives to construct an ellipse for approximating the central path. Pirhaji et al [24], using the arc search strategy and the wide neighborhood given by Ai and Zhang [1], suggested an infeasible interior-point algorithm for monotone LCPs. Based on a commutative class of search directions and the Ai-Zhang’s wide neighborhood, Yang et al [14] generalized their proposed arc search algorithm for LO problems [25] to SO problems.
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