We present an interior-point trust-funnel algorithm for solving large-scale nonlinear optimization problems. The method is based on an approach proposed by Gould and Toint (Math Prog 122(1):155–196, 2010) that focused on solving equality constrained problems. Our method is similar in that it achieves global convergence guarantees by combining a trust-region methodology with a funnel mechanism, … Read more
This work focuses on the iterative solution of sequences of KKT linear systems arising in interior point methods applied to large convex quadratic programming problems. This task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method. Constraint preconditioners are very effective … Read more
Solving semidefinite programs (SDP) in a short time is the key to managing various mathematical optimization problems in practical time. The matrix-completion primal-dual interior-point method (MC-PDIPM) extracts a structural sparsity of input SDP by factorizing the variable matrices, and it shrinks the computation time. In this paper, we propose a new factorization based on the … Read more
The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation. A reduction in the number of calls … Read more
We present an infeasible primal-dual interior point method for semidefinite optimization problems, making use of constraint reduction. We show that the algorithm is globally convergent and has polynomial complexity, the first such complexity result for primal-dual constraint reduction algorithms for any class of problems. Our algorithm is a modification of one with no constraint reduction … Read more
Interior-point methods in semidefinite programming (SDP) require the solution of a sequence of linear systems which are used to derive the search directions. Safeguards are typically required in order to handle rank-deficient Jacobians and free variables. We generalize the primal-dual regularization of \cite{friedlander-orban-2012} to SDP and show that it is possible to recover an optimal … Read more
We consider the minimization of a convex function on a compact polyhedron defined by linear equality constraints and nonnegative variables. We define the Levenberg-Marquardt (L-M) and central trajectories starting at the analytic center and using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of … Read more
In this paper we will discuss two variants of an inexact feasible interior point algorithm for convex quadratic programming. We will consider two different neighbourhoods: a (small) one induced by the use of the Euclidean norm which yields a short-step algorithm and a symmetric one induced by the use of the infinity norm which yields … Read more
We study the numerical solution of the problem $\min_{X \ge 0} \|BX-c\|2$, where $X$ is a symmetric square matrix, and $B$ a linear operator, such that $B^*B$ is invertible. With $\rho$ the desired fractional duality gap, we prove $O(\sqrt{m}\log\rho^{-1})$ iteration complexity for a simple primal-dual interior point method directly based on those for linear programs … Read more
Three interior point methods are proposed for sufficient horizontal linear complementarity problems (HLCP): a large update path following algorithm, a first order corrector-predictor method, and a second order corrector-predictor method. All algorithms produce sequences of iterates in the wide neighborhood of the central path introduced by Ai and Zhang. The algorithms do not depend on … Read more