On a primal-dual Newton proximal method for convex quadratic programs

This paper introduces QPDO, a primal-dual method for convex quadratic programs which builds upon and weaves together the proximal point algorithm and a damped semismooth Newton method. The outer proximal regularization yields a numerically stable method, and we interpret the proximal operator as the unconstrained minimization of the primal-dual proximal augmented Lagrangian function. This allows the inner Newton scheme to exploit sparse symmetric linear solvers and multi-rank factorization updates. Moreover, the linear systems are always solvable independently from the problem data and exact linesearch can be performed. The proposed method can handle degenerate problems, provides a mechanism for infeasibility detection, and can exploit warm starting, while requiring only convexity. We present details of our open-source C implementation and report on numerical results against state-of-the-art solvers. QPDO proves to be a simple, robust, and efficient numerical method for convex quadratic programming.

Variable fixing method by weighted average for the continuous quadratic knapsack problem

<p style='text-indent:20px;'>We analyze the method of solving the separable convex continuous quadratic knapsack problem by weighted average from the viewpoint of variable fixing. It is shown that this method, considered as a variant of the variable fixing algorithms, and Kiwiel's variable fixing method generate the same iterates. We further improve the algorithm based on the analysis regarding the semismooth Newton method. Computational results are given and comparisons are made among the state-of-the-art algorithms. Experiments show that our algorithm has significantly good performance; it behaves very much like an <inline-formula><tex-math id="M1">\begin{document}$ O(n) $\end{document}</tex-math></inline-formula> algorithm with a very small constant.</p>

A new semismooth Newton method for solving finite-dimensional quasi-variational inequalities

In this paper, we consider the numerical method for solving finite-dimensional quasi-variational inequalities with both equality and inequality constraints. Firstly, we present a semismooth equation reformulation to the KKT system of a finite-dimensional quasi-variational inequality. Then we propose a semismooth Newton method to solve the equations and establish its global convergence. Finally, we report some numerical results to show the efficiency of the proposed method. Our method can obtain the solution to some problems that cannot be solved by the method proposed in (Facchinei et al. in Comput. Optim. Appl. 62:85–109, 2015). Besides, our method outperforms than the interior point method proposed in (Facchinei et al. in Math. Program. 144:369–412, 2014).

Boundary Element and Augmented Lagrangian Methods for Contact Problem with Coulomb Friction

In this paper, boundary element and augmented Lagrangian methods for Coulomb friction contact problems are presented. Based on the projection technique, both unilateral contact and Coulomb friction conditions are reformulated as fixed point problems. The original problem is deduced to a variational formulation with boundary integral operators. Then, we propose a new augmented Lagrangian method which can be dealt with the semismooth Newton method. Short theoretical results and the algorithm description are given. Numerical simulations show the performance of the method proposed.