This paper is a continuation of the paper Kakihara, Ohara and Tsuchiya by the authors where they demonstrated that the number of iterations of Mizuno-Todd-Ye predictor-corrector primal-dual interior-point methods for SDP and more generally symmetric cone programs is (asymptotically) expressed with an integral over the central trajectory called “curvature integral.” It was shown that the … Read more
We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of … Read more
In this paper we present an infeasible interior-point algorithm for solving linear optimization problems. This algorithm is obtained by modifying the search direction in the algorithm [C. Roos, A full-Newton step ${O}(n)$ infeasible interior-point algorithm for linear optimization, 16(4) 2006, 1110-1136.]. The analysis of our algorithm is much simpler than that of the Roos’s algorithm … Read more
After a brief introduction to Jordan algebras, we present a primal-dual interior-point algorithm for second-order conic optimization that uses full Nesterov-Todd-steps; no line searches are required. The number of iterations of the algorithm is $O(\sqrt{N}\log ({N}/{\varepsilon})$, where $N$ stands for the number of second-order cones in the problem formulation and $\varepsilon$ is the desired accuracy. … Read more
Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed an efficient primal-dual infeasible interior-point 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 … Read more
Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both … Read more
In this paper we present a scaling-invariant interior-point predictor-corrector type algorithm for linear programming (LP) whose iteration-complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the LP constraint matrix. At the predictor stage, the algorithm either takes the step along the standard affine scaling direction or a new … Read more
We prove a new local convergence property of a primal-dual method for solving nonlinear optimization problem. Following a standard interior point approach, the complementarity conditions of the original primal-dual system are perturbed by a parameter which is driven to zero during the iterations. The sequence of iterates is generated by a linearization of the perturbed … Read more
We present a new strategy for choosing primal and dual steplengths in a primal-dual interior-point algorithm for convex quadratic programming. Current implementations often scale steps equally to avoid increases in dual infeasibility between iterations. We propose that this method can be too conservative, while safeguarding an unequally-scaled steplength approach will often require fewer steps toward … Read more
In this paper we develop a practical primal interior decomposition algorithm for two-stage stochastic programming problems. The framework of this algorithm is similar to the framework in Mehrotra and \”{Ozevin} \cite{MO04a,MO04b} and Zhao \cite{GZ01}, however their algorithm is altered in a simple yet fundamental way to achieve practical performance. In particular, this new algorithm weighs … Read more