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An Infeasible InteriorPoint Algorithm with Full NesterovTodd Step for Semidefinite Programming
"... This paper proposes an infeasible interiorpoint algorithm with full NesterovTodd step for semidefinite programming, which is an extension of the work of Roos (SIAM J. Optim., 16(4):1110– 1136, 2006). The polynomial bound coincides with that of infeasible interiorpoint methods for linear programmi ..."
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This paper proposes an infeasible interiorpoint algorithm with full NesterovTodd step for semidefinite programming, which is an extension of the work of Roos (SIAM J. Optim., 16(4):1110– 1136, 2006). The polynomial bound coincides with that of infeasible interiorpoint methods for linear programming, namely, O(n log n/ε).
A New FullNewton step O (n) Infeasible InteriorPoint Algorithm for Semidefinite Optimization ∗
"... Interiorpoint methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed an efficient primaldual infeasible interiorpoint algorithm with full Newton steps for linear optimization problems. ..."
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Interiorpoint methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed an efficient primaldual infeasible interiorpoint algorithm with full Newton steps for linear optimization problems. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of fullNewton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates in at most O ( n log n ε steps either by finding an εsolution or by determining that the primaldual problem pair has no optimal solution with vanishing duality gap satisfying a condition in terms of ζ.
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"... A fullNewton step infeasible interiorpoint algorithm for linear programming based on a kernel function ..."
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A fullNewton step infeasible interiorpoint algorithm for linear programming based on a kernel function
An Infeasible InteriorPoint Algorithm with fullNewton Step for Linear Optimization
"... In this paper we present an infeasible interiorpoint algorithm for solving linear optimization problems. This algorithm is obtained by modifying the search direction in the algorithm [8]. The analysis of our algorithm is much simpler than that of the algorithm [8] at some places. The iteration boun ..."
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In this paper we present an infeasible interiorpoint algorithm for solving linear optimization problems. This algorithm is obtained by modifying the search direction in the algorithm [8]. The analysis of our algorithm is much simpler than that of the algorithm [8] at some places. The iteration bound of the algorithm is as good as the best known iteration bound O ( n log 1 ε for IIPMs.
Full NesterovTodd Step PrimalDual InteriorPoint Methods for SecondOrder Cone
"... After a brief introduction to Jordan algebras, we present a primaldual interiorpoint algorithm for secondorder conic optimization that uses full NesterovToddsteps; no line searches are required. The number of iterations of the algorithm is O ( √ N log(N/ε), where N stands for the number of sec ..."
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After a brief introduction to Jordan algebras, we present a primaldual interiorpoint algorithm for secondorder conic optimization that uses full NesterovToddsteps; no line searches are required. The number of iterations of the algorithm is O ( √ N log(N/ε), where N stands for the number of secondorder cones in the problem formulation and ε is the desired accuracy. The bound coincides with the currently best iteration bound for secondorder conic optimization. We also generalize an infeasible interiorpoint method for linear optimization [26] to secondorder conic optimization. As usual for infeasible interiorpoint methods the starting point depends on a positive number ζ. The algorithm either finds an εsolution in at most O (N log(N/ε)) steps or determines that the primaldual problem pair has no optimal solution with vanishing duality gap satisfying a condition in terms of ζ. 1