# A globally convergent primal-dual active-set framework for large-scale convex quadratic optimization

@article{Curtis2015AGC,
title={A globally convergent primal-dual active-set framework for large-scale convex quadratic optimization},
author={Frank E. Curtis and Zheng Han and Daniel P. Robinson},
journal={Computational Optimization and Applications},
year={2015},
volume={60},
pages={311-341}
}
• Published 1 March 2015
• Mathematics, Computer Science
• Computational Optimization and Applications
We present a primal-dual active-set framework for solving large-scale convex quadratic optimization problems (QPs). In contrast to classical active-set methods, our framework allows for multiple simultaneous changes in the active-set estimate, which often leads to rapid identification of the optimal active-set regardless of the initial estimate. The iterates of our framework are the active-set estimates themselves, where for each a primal-dual solution is uniquely defined via a reduced…
49 Citations
• Mathematics
SIAM J. Optim.
• 2016
Three primal-dual active-set (PDAS) methods for solving large-scale instances of an important class of convex quadratic optimization problems (QPs) that allow inexactness in the (reduced) linear system solves at all partitions except optimal ones.
• Mathematics
• 2015
Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints
• Computer Science, Mathematics
SIAM J. Optim.
• 2015
A primal-dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of Kunisch and Rendl and performs well in practice.
• Computer Science, Mathematics
Computational Optimization and Applications
• 2016
The details of a solver for minimizing a strictly convex quadratic objective function subject to general linear constraints are presented and how the linear algebra may be arranged to take computational advantage of sparsity in the second-derivative matrix is shown.
• Computer Science, Mathematics
Comput. Optim. Appl.
• 2017
The details of a solver for minimizing a strictly convex quadratic objective function subject to general linear constraints are presented and how the linear algebra may be arranged to take computational advantage of sparsity in the second-derivative matrix is shown.
• Mathematics, Computer Science
• 2016
A primal feasible active set method is presented for finding the unique solution of a Linear Complementarity Problem (LCP) with a P -matrix, which extends the globally convergent active set method
• Mathematics, Computer Science
• 2017
This paper introduces yet another modified version of this active set method, which aims at maintaining the combinatorial flavour of the original semismooth Newton method and proves global convergence for this modified version and shows it to be competitive on a variety of difficult classes of test problems.
• Computer Science
SIAM J. Optim.
• 2017
A convergence guarantee is proved for the new method for minimizing the sum of a differentiable convex function and an $\ell_1$-norm regularizer and its efficiency is demonstrated on a large set of model prediction problems.
• Computer Science, Mathematics
Optim. Methods Softw.
• 2019
An algorithm is proposed that uses the L-BFGS quasi-Newton approximation of the problem's curvature together with a variant of the weak Wolfe line search to overcome the inherent shortsightedness of the gradient for a non-smooth function.

## References

SHOWING 1-10 OF 47 REFERENCES

• Mathematics
SIAM J. Optim.
• 2006
It is shown that the method converges globally to KKT points under the linear independence constraint qualification (LICQ), and the asymptotic rate of convergence is Q-superlinear under an additional strong second-order sufficient condition (SSOSC) without strict complementarity.
• Mathematics
SIAM J. Optim.
• 2003
A primal-dual active set method for quadratic problems with bound constraints is presented, based on a guess on the active set, that satisfies the first order optimality condition and the complementarity condition.
• Computer Science
Comput. Optim. Appl.
• 1993
This study shows that combining the new algorithm with the nonlinear conjugate gradient method is particularly effective on difficult network problems from the literature.
• Computer Science
SIAM J. Optim.
• 2002
An SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems is discussed and a reduced-Hessian semidefinite QP solver (SQOPT) is discussed.
• Computer Science
SIAM J. Optim.
• 2010
This paper presents a second derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally.
• Mathematics
Math. Program.
• 2016
A new active set method is proposed that performs multiple changes in the active manifold estimate at every iteration, and employs a mechanism for correcting these estimates, when needed.
• Computer Science
Math. Program. Comput.
• 2014
The open-source C++ software package qpOASES is described, which implements a parametric active-set method in a reliable and efficient way and can be used to compute critical points of nonconvex QP problems.
• Computer Science
• 2012
The modified algorithm remains globally convergent and preserves local superlinear convergence provided that a nonmonotone strategy is incorporated and it is proved that the method is globally and locally superlinearly convergent under common assumptions.
• Mathematics
Math. Program.
• 2012
It is shown that the plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) no longer holds when M is a P-matrix of order ≥ 3, since then the algorithm may cycle.