Search

Following the central-path, as a guide line to optimal solution of mathematical problems, is one of the main difficulty of interior-point methods in practice. These iterative methods, follow the central path step by step to get close enough to the optimal solution of underlying problem. Based on estimating the central path by an ellipse, we propose an infeasible interior-point method for linear complementarity problem. In each iteration, the algorithm follows the ellipsoidal approximation of the central-path to find an ε-approximate solution of the problem. We prove that under certain conditions the proposed algorithm is well-defined and the generated points by the algorithm converge to an ε-approximate solution of the linear complementarity problem.

In this paper, a corrector-predictor interior-point algorithm is proposed for symmetric optimization. The algorithm approximates the central path by an ellipse, follows the ellipsoidal approximation of the central-path step by step and generates a sequence of iterates in a wide neighborhood of the central-path. Using the machinery of Euclidean Jordan algebra and the commutative class of search directions, the convergence analysis of the algorithm is shown and it is proved that the algorithm has the complexity bound O(rL)for the well-known Nesterov-Todd search direction and O(rL)for the xsand sxsearch directions.

In this paper, we propose a primal–dual interior-point method for semidefinite optimization problems. The algorithm is based on a new class of search directions and the Ai-Zhang’s wide neighborhood for monotone linear complementarity problems. The theoretical complexity of the new algorithm is calculated. It is investigated that the proposed algorithm has polynomial iteration complexity $$O(\sqrt{n}L)$$ O(nL) and coincides with the best known iteration bound for semidefinite optimization problems.

We investigate the effect of making equivalent transformation for the central path from $$x\diamond s=\mu e$$ x⋄s=μe to $$x\diamond s=\mu v$$ x⋄s=μv , and we obtain a new class of search directions in both version of feasible and infeasible interior point algorithms for linear optimization over Cartesian symmetric cones. We propose a new modified infeasible interior-point algorithm based on elimination of the centering steps. Moreover, the proposed algorithm finds an $$\varepsilon $$ ε -optimal solution of the underlying problem in polynomial time complexity and its iteration bound coincides with the best known iteration bound for this class of problems.